3.2295 \(\int \frac{(d+e x)^{7/2}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=691 \[ \frac{e \sqrt{d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{\left (c^2 e^2 \left (-3 b d \left (d \sqrt{b^2-4 a c}+12 a e\right )+2 a e \left (13 d \sqrt{b^2-4 a c}-10 a e\right )+3 b^2 d^2\right )-2 c^3 d^2 e \left (-d \sqrt{b^2-4 a c}-18 a e+8 b d\right )+b c e^3 \left (-5 b d \sqrt{b^2-4 a c}-13 a e \sqrt{b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (c^2 e^2 \left (3 b d \left (d \sqrt{b^2-4 a c}-12 a e\right )-2 a e \left (13 d \sqrt{b^2-4 a c}+10 a e\right )+3 b^2 d^2\right )-2 c^3 d^2 e \left (d \sqrt{b^2-4 a c}-18 a e+8 b d\right )+b c e^3 \left (5 b d \sqrt{b^2-4 a c}+13 a e \sqrt{b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e (d+e x)^{3/2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

[Out]

(e*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e))*Sqrt[d + e*x])/(c^2*(b^2 - 4*a*c)) + (e*(2*c*d - b*e)*(d + e*
x)^(3/2))/(c*(b^2 - 4*a*c)) - ((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^
2)) + ((8*c^4*d^4 - 3*b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 2*c^3*d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 18*a*e) + b
*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 - 4*a*c]*d + 19*a*b*e - 13*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e^2*(3*b^2*d^2 + 2*a*e*
(13*Sqrt[b^2 - 4*a*c]*d - 10*a*e) - 3*b*d*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e
*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]) - ((8*c^4*d^4 - 3*b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 2*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d -
18*a*e) + b*c*e^3*(5*b^2*d + 5*b*Sqrt[b^2 - 4*a*c]*d + 19*a*b*e + 13*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e^2*(3*b^2*d
^2 + 3*b*d*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) - 2*a*e*(13*Sqrt[b^2 - 4*a*c]*d + 10*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]
*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e])

________________________________________________________________________________________

Rubi [A]  time = 15.9623, antiderivative size = 691, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {738, 824, 826, 1166, 208} \[ \frac{e \sqrt{d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{\left (c^2 e^2 \left (-3 b d \left (d \sqrt{b^2-4 a c}+12 a e\right )+2 a e \left (13 d \sqrt{b^2-4 a c}-10 a e\right )+3 b^2 d^2\right )-2 c^3 d^2 e \left (-d \sqrt{b^2-4 a c}-18 a e+8 b d\right )+b c e^3 \left (-5 b d \sqrt{b^2-4 a c}-13 a e \sqrt{b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (c^2 e^2 \left (3 b d \left (d \sqrt{b^2-4 a c}-12 a e\right )-2 a e \left (13 d \sqrt{b^2-4 a c}+10 a e\right )+3 b^2 d^2\right )-2 c^3 d^2 e \left (d \sqrt{b^2-4 a c}-18 a e+8 b d\right )+b c e^3 \left (5 b d \sqrt{b^2-4 a c}+13 a e \sqrt{b^2-4 a c}+19 a b e+5 b^2 d\right )-3 b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+8 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e (d+e x)^{3/2} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^2,x]

[Out]

(e*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e))*Sqrt[d + e*x])/(c^2*(b^2 - 4*a*c)) + (e*(2*c*d - b*e)*(d + e*
x)^(3/2))/(c*(b^2 - 4*a*c)) - ((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^
2)) + ((8*c^4*d^4 - 3*b^3*(b - Sqrt[b^2 - 4*a*c])*e^4 - 2*c^3*d^2*e*(8*b*d - Sqrt[b^2 - 4*a*c]*d - 18*a*e) + b
*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 - 4*a*c]*d + 19*a*b*e - 13*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e^2*(3*b^2*d^2 + 2*a*e*
(13*Sqrt[b^2 - 4*a*c]*d - 10*a*e) - 3*b*d*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e
*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]) - ((8*c^4*d^4 - 3*b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 2*c^3*d^2*e*(8*b*d + Sqrt[b^2 - 4*a*c]*d -
18*a*e) + b*c*e^3*(5*b^2*d + 5*b*Sqrt[b^2 - 4*a*c]*d + 19*a*b*e + 13*a*Sqrt[b^2 - 4*a*c]*e) + c^2*e^2*(3*b^2*d
^2 + 3*b*d*(Sqrt[b^2 - 4*a*c]*d - 12*a*e) - 2*a*e*(13*Sqrt[b^2 - 4*a*c]*d + 10*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]
*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(5/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e])

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} \left (4 c d^2-7 b d e+10 a e^2\right )-\frac{3}{2} e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac{e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} \left (4 c^2 d^3-3 a b e^3-c d e (7 b d-16 a e)\right )-\frac{1}{2} e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x\right )}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac{e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt{d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac{e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{\frac{1}{2} \left (4 c^3 d^4+3 a b^2 e^4-c^2 d^2 e (7 b d-18 a e)-5 a c e^3 (b d+2 a e)\right )+\frac{1}{2} e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) x}{\sqrt{d+e x} \left (a+b x+c x^2\right )} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac{e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt{d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac{e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{1}{2} d e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+\frac{1}{2} e \left (4 c^3 d^4+3 a b^2 e^4-c^2 d^2 e (7 b d-18 a e)-5 a c e^3 (b d+2 a e)\right )+\frac{1}{2} e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{c^2 \left (b^2-4 a c\right )}\\ &=\frac{e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt{d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac{e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (8 c^4 d^4-3 b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d+\sqrt{b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d+5 b \sqrt{b^2-4 a c} d+19 a b e+13 a \sqrt{b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+3 b d \left (\sqrt{b^2-4 a c} d-12 a e\right )-2 a e \left (13 \sqrt{b^2-4 a c} d+10 a e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac{\left (8 c^4 d^4-3 b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d-\sqrt{b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d-5 b \sqrt{b^2-4 a c} d+19 a b e-13 a \sqrt{b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+2 a e \left (13 \sqrt{b^2-4 a c} d-10 a e\right )-3 b d \left (\sqrt{b^2-4 a c} d+12 a e\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \sqrt{b^2-4 a c} e+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}\\ &=\frac{e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt{d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac{e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (8 c^4 d^4-3 b^3 \left (b-\sqrt{b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d-\sqrt{b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d-5 b \sqrt{b^2-4 a c} d+19 a b e-13 a \sqrt{b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+2 a e \left (13 \sqrt{b^2-4 a c} d-10 a e\right )-3 b d \left (\sqrt{b^2-4 a c} d+12 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{\left (8 c^4 d^4-3 b^3 \left (b+\sqrt{b^2-4 a c}\right ) e^4-2 c^3 d^2 e \left (8 b d+\sqrt{b^2-4 a c} d-18 a e\right )+b c e^3 \left (5 b^2 d+5 b \sqrt{b^2-4 a c} d+19 a b e+13 a \sqrt{b^2-4 a c} e\right )+c^2 e^2 \left (3 b^2 d^2+3 b d \left (\sqrt{b^2-4 a c} d-12 a e\right )-2 a e \left (13 \sqrt{b^2-4 a c} d+10 a e\right )\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\right )}{\sqrt{2} c^{5/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}

Mathematica [A]  time = 6.4917, size = 1116, normalized size = 1.62 \[ -\frac{\left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right ) (d+e x)^{9/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac{\frac{2 \left (\frac{2 \left (\frac{2 \left (\frac{4 \left (\frac{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left (\frac{105}{32} c^2 e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )-\frac{2 c \left (\frac{105}{32} c^2 e \left (c d^2-b e d+a e^2\right ) \left (4 c^3 d^4-7 b c^2 e d^3+18 a c^2 e^2 d^2-5 a b c e^3 d+3 a b^2 e^4-10 a^2 c e^4\right )-\frac{105}{32} c^2 d e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )\right )-\frac{105}{32} c^2 e (2 c d-b e) (b e-2 c d) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )}{\sqrt{b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}\right )}{\sqrt{2} \sqrt{c} \left (-2 c d+b e+\sqrt{b^2-4 a c} e\right )}+\frac{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e} \left (\frac{105}{32} e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right ) c^2+\frac{2 c \left (\frac{105}{32} c^2 e \left (c d^2-b e d+a e^2\right ) \left (4 c^3 d^4-7 b c^2 e d^3+18 a c^2 e^2 d^2-5 a b c e^3 d+3 a b^2 e^4-10 a^2 c e^4\right )-\frac{105}{32} c^2 d e (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )\right )-\frac{105}{32} c^2 e (2 c d-b e) (b e-2 c d) \left (c d^2-b e d+a e^2\right ) \left (c^2 d^2-b c e d-3 b^2 e^2+13 a c e^2\right )}{\sqrt{b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right )}{\sqrt{2} \sqrt{c} \left (-2 c d+b e-\sqrt{b^2-4 a c} e\right )}\right )}{c}-\frac{105}{8} c e \left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d+3 b^2 e^2-10 a c e^2\right ) \sqrt{d+e x}\right )}{3 c}-\frac{35}{4} c e (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{3/2}\right )}{5 c}-7 c e \left (c d^2-e (b d-a e)\right ) (d+e x)^{5/2}\right )}{7 c}-e (2 c d-b e) (d+e x)^{7/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^2,x]

[Out]

-(((d + e*x)^(9/2)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a +
b*x + c*x^2))) - (-(e*(2*c*d - b*e)*(d + e*x)^(7/2)) + (2*(-7*c*e*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(5/2) + (2
*((-35*c*e*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(3/2))/4 + (2*((-105*c*e*(c*d^2 - b*d*e + a*e^2)*(2
*c^2*d^2 - 2*b*c*d*e + 3*b^2*e^2 - 10*a*c*e^2)*Sqrt[d + e*x])/8 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]
*((105*c^2*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 - ((-105*c
^2*e*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + 2
*c*((-105*c^2*d*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + (10
5*c^2*e*(c*d^2 - b*d*e + a*e^2)*(4*c^3*d^4 - 7*b*c^2*d^3*e + 18*a*c^2*d^2*e^2 - 5*a*b*c*d*e^3 + 3*a*b^2*e^4 -
10*a^2*c*e^4))/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2
- 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*
e]*((105*c^2*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + ((-105
*c^2*e*(2*c*d - b*e)*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 +
2*c*((-105*c^2*d*e*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(c^2*d^2 - b*c*d*e - 3*b^2*e^2 + 13*a*c*e^2))/32 + (
105*c^2*e*(c*d^2 - b*d*e + a*e^2)*(4*c^3*d^4 - 7*b*c^2*d^3*e + 18*a*c^2*d^2*e^2 - 5*a*b*c*d*e^3 + 3*a*b^2*e^4
- 10*a^2*c*e^4))/32))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b
^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(3*c)))/(5*c)))/(7*c))/((b^2 - 4
*a*c)*(c*d^2 - b*d*e + a*e^2))

________________________________________________________________________________________

Maple [B]  time = 0.294, size = 3838, normalized size = 5.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x)

[Out]

-4*e*c^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e
*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^4-13/2*e^4/c/(4*a*c-b^2)*2^(1/2)/((b*e
-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))
*c)^(1/2))*a*b-3/2*e^3/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^
(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d^2-3/2*e^3/(4*a*c-
b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^
(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d^2+3/2*e^5/c^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)
*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2))*b^4-19/2*e^5/c/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c
-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^2-
5/2*e^4/c/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((
e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*d+8*e^2*c/(4*a*c-b^2)/(-e^2*(4*a*c-
b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+
(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^3-18*e^3*c/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(
-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1
/2))*a*d^2-18*e^3*c/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2
)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*d^2+18*e^4/(4*a*c-b^2)/(-e^
2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b
*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b*d+18*e^4/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+
2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2)
)*c)^(1/2))*a*b*d-19/2*e^5/c/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2)
)*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*b^2-5/2*e^4/c/(4*a
*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)
*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3*d+8*e^2*c/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)
*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4
*a*c-b^2))^(1/2))*c)^(1/2))*b*d^3+5/2*e^3/c/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2
)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2*d-5/2*e^3/c/(4*a*c-b^2)
*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2))*b^2*d+3/2*e^5/c^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*
a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^4
-4*e*c^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(
(e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^4+13/2*e^4/c/(4*a*c-b^2)*2^(1/2)/((
-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^
(1/2))*c)^(1/2))*a*b-13*e^3/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d
)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a*d+2*e^3/c^2*(e*x+d)^(1/2)+13*e^3/(4*a*c-b
^2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4
*a*c-b^2))^(1/2))*c)^(1/2))*a*d+3/2*e^4/c^2/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)
*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^3-3/2*e^4/c^2/(4*a*c-b^2)*2^
(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2))*b^3+3*e^3/c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*b^2*d-3/2*e^2/(4*a*c-
b^2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(
4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2+10*e^5/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*
c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a^2-3*e
^3/c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*b^2*d^2-e*c/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4
*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^
3+e*c/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-
2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^3+3*e^4/c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*a*b-
e^5/c^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*a*b^2+e^4/c^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2
)*(e*x+d)^(1/2)*b^3*d+10*e^5/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2
))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a^2+3/2*e^2/(4*a*
c-b^2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-
e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d^2-3*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*b*d^2+4*e^2/
(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*b*d^3-e^4/c^2/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d
)^(3/2)*b^3+2*e*c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*d^3+2*e^5/c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4
*a*c-b^2)*(e*x+d)^(1/2)*a^2-2*e*c/(c*e^2*x^2+b*e^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(1/2)*d^4-6*e^3/(c*e^2*x^2+b*e
^2*x+a*e^2)/(4*a*c-b^2)*(e*x+d)^(3/2)*a*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^2, x)

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Fricas [B]  time = 64.651, size = 22507, normalized size = 32.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

-1/2*(sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((32*c^7*d^7 -
112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b
^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 +
150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 + (b^6*c^5
- 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 1
1*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e
^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^
2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81
*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a
^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*log(sqrt(1/2)*(70*(b^4*c^
6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^6*e^4 - 210*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^5*e^5 + 21*(13*b^6*c^4 -
106*a*b^4*c^5 + 224*a^2*b^2*c^6 - 32*a^3*c^7)*d^4*e^6 - 28*(7*b^7*c^3 - 59*a*b^5*c^4 + 136*a^2*b^3*c^5 - 48*a^
3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^2 - 146*a*b^6*c^3 + 1289*a^2*b^4*c^4 - 4072*a^3*b^2*c^5 + 4240*a^4*c^6)*d^2*e^8
+ 3*(27*b^9*c - 474*a*b^7*c^2 + 3026*a^2*b^5*c^3 - 8368*a^3*b^3*c^4 + 8480*a^4*b*c^5)*d*e^9 - (27*b^10 - 459*a
*b^8*c + 2961*a^2*b^6*c^2 - 8818*a^3*b^4*c^3 + 11360*a^4*b^2*c^4 - 4000*a^5*c^5)*e^10 - (8*(b^6*c^8 - 12*a*b^4
*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^3 - 12*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^2*e
- 2*(b^8*c^6 - 28*a*b^6*c^7 + 240*a^2*b^4*c^8 - 832*a^3*b^2*c^9 + 1024*a^4*c^10)*d*e^2 + (3*b^9*c^5 - 52*a*b^
7*c^6 + 336*a^2*b^5*c^7 - 960*a^3*b^3*c^8 + 1024*a^4*b*c^9)*e^3)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 +
980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 -
1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^
4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*
b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 1
2*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)
*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 +
5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^
7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8
)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6
)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*
a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*
c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550
*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a
*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) - 2*(560*c^8*d^10*e^3 - 2800*b*c^7*d^9*e^4 + 7*(647*b^2*c^6 + 1012*a*
c^7)*d^8*e^5 - 28*(47*b^3*c^5 + 1012*a*b*c^6)*d^7*e^6 - (3329*b^4*c^4 - 35844*a*b^2*c^5 - 27488*a^2*c^6)*d^6*e
^7 + (2833*b^5*c^3 - 8356*a*b^3*c^4 - 82464*a^2*b*c^5)*d^5*e^8 + (9*b^6*c^2 - 14273*a*b^4*c^3 + 77982*a^2*b^2*
c^4 + 33464*a^3*c^5)*d^4*e^9 - (675*b^7*c - 9414*a*b^5*c^2 + 18524*a^2*b^3*c^3 + 66928*a^3*b*c^4)*d^3*e^10 + (
189*b^8 - 999*a*b^6*c - 8127*a^2*b^4*c^2 + 40196*a^3*b^2*c^3 + 10000*a^4*c^4)*d^2*e^11 - (378*a*b^7 - 3645*a^2
*b^5*c + 6732*a^3*b^3*c^2 + 10000*a^4*b*c^3)*d*e^12 + (189*a^2*b^6 - 1971*a^3*b^4*c + 5625*a^4*b^2*c^2 - 2500*
a^5*c^3)*e^13)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*
c^3)*x)*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4
*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*
(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*
a^3*b*c^3)*e^7 + (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^
7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2
*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 11
16*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 5
50*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c
^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))
*log(-sqrt(1/2)*(70*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^6*e^4 - 210*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*
d^5*e^5 + 21*(13*b^6*c^4 - 106*a*b^4*c^5 + 224*a^2*b^2*c^6 - 32*a^3*c^7)*d^4*e^6 - 28*(7*b^7*c^3 - 59*a*b^5*c^
4 + 136*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^2 - 146*a*b^6*c^3 + 1289*a^2*b^4*c^4 - 4072*a^3*b^2*c
^5 + 4240*a^4*c^6)*d^2*e^8 + 3*(27*b^9*c - 474*a*b^7*c^2 + 3026*a^2*b^5*c^3 - 8368*a^3*b^3*c^4 + 8480*a^4*b*c^
5)*d*e^9 - (27*b^10 - 459*a*b^8*c + 2961*a^2*b^6*c^2 - 8818*a^3*b^4*c^3 + 11360*a^4*b^2*c^4 - 4000*a^5*c^5)*e^
10 - (8*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^3 - 12*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3
*c^9 - 64*a^3*b*c^10)*d^2*e - 2*(b^8*c^6 - 28*a*b^6*c^7 + 240*a^2*b^4*c^8 - 832*a^3*b^2*c^9 + 1024*a^4*c^10)*d
*e^2 + (3*b^9*c^5 - 52*a*b^7*c^6 + 336*a^2*b^5*c^7 - 960*a^3*b^3*c^8 + 1024*a^4*b*c^9)*e^3)*sqrt((1225*c^8*d^8
*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*
b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11
+ 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 +
1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a
^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e
+ 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^
5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 -
120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 + (b^6*c^5 - 12*a*b^4*c^6 +
48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8
- 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*
c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*
c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*
c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*
a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) - 2*(560*c^8*d^10*e^3 - 2800*b*c^7*d^9*e^4
+ 7*(647*b^2*c^6 + 1012*a*c^7)*d^8*e^5 - 28*(47*b^3*c^5 + 1012*a*b*c^6)*d^7*e^6 - (3329*b^4*c^4 - 35844*a*b^2
*c^5 - 27488*a^2*c^6)*d^6*e^7 + (2833*b^5*c^3 - 8356*a*b^3*c^4 - 82464*a^2*b*c^5)*d^5*e^8 + (9*b^6*c^2 - 14273
*a*b^4*c^3 + 77982*a^2*b^2*c^4 + 33464*a^3*c^5)*d^4*e^9 - (675*b^7*c - 9414*a*b^5*c^2 + 18524*a^2*b^3*c^3 + 66
928*a^3*b*c^4)*d^3*e^10 + (189*b^8 - 999*a*b^6*c - 8127*a^2*b^4*c^2 + 40196*a^3*b^2*c^3 + 10000*a^4*c^4)*d^2*e
^11 - (378*a*b^7 - 3645*a^2*b^5*c + 6732*a^3*b^3*c^2 + 10000*a^4*b*c^3)*d*e^12 + (189*a^2*b^6 - 1971*a^3*b^4*c
+ 5625*a^4*b^2*c^2 - 2500*a^5*c^3)*e^13)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c
^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35
*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 -
60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5
*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*
c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 1
4*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^
3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5
*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3
+ 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48
*a^2*b^2*c^7 - 64*a^3*c^8))*log(sqrt(1/2)*(70*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*d^6*e^4 - 210*(b^5*c^5 - 8*
a*b^3*c^6 + 16*a^2*b*c^7)*d^5*e^5 + 21*(13*b^6*c^4 - 106*a*b^4*c^5 + 224*a^2*b^2*c^6 - 32*a^3*c^7)*d^4*e^6 - 2
8*(7*b^7*c^3 - 59*a*b^5*c^4 + 136*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^2 - 146*a*b^6*c^3 + 1289*a^
2*b^4*c^4 - 4072*a^3*b^2*c^5 + 4240*a^4*c^6)*d^2*e^8 + 3*(27*b^9*c - 474*a*b^7*c^2 + 3026*a^2*b^5*c^3 - 8368*a
^3*b^3*c^4 + 8480*a^4*b*c^5)*d*e^9 - (27*b^10 - 459*a*b^8*c + 2961*a^2*b^6*c^2 - 8818*a^3*b^4*c^3 + 11360*a^4*
b^2*c^4 - 4000*a^5*c^5)*e^10 + (8*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^3 - 12*(b^7*c^7 -
12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^2*e - 2*(b^8*c^6 - 28*a*b^6*c^7 + 240*a^2*b^4*c^8 - 832*a^3*
b^2*c^9 + 1024*a^4*c^10)*d*e^2 + (3*b^9*c^5 - 52*a*b^7*c^6 + 336*a^2*b^5*c^7 - 960*a^3*b^3*c^8 + 1024*a^4*b*c^
9)*e^3)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a
*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 -
1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(2
7*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2
- 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt((32*
c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^
3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b
^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 -
(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^
2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c
^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 55
32*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e
^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^
11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)) - 2*(560*c^8*d^1
0*e^3 - 2800*b*c^7*d^9*e^4 + 7*(647*b^2*c^6 + 1012*a*c^7)*d^8*e^5 - 28*(47*b^3*c^5 + 1012*a*b*c^6)*d^7*e^6 - (
3329*b^4*c^4 - 35844*a*b^2*c^5 - 27488*a^2*c^6)*d^6*e^7 + (2833*b^5*c^3 - 8356*a*b^3*c^4 - 82464*a^2*b*c^5)*d^
5*e^8 + (9*b^6*c^2 - 14273*a*b^4*c^3 + 77982*a^2*b^2*c^4 + 33464*a^3*c^5)*d^4*e^9 - (675*b^7*c - 9414*a*b^5*c^
2 + 18524*a^2*b^3*c^3 + 66928*a^3*b*c^4)*d^3*e^10 + (189*b^8 - 999*a*b^6*c - 8127*a^2*b^4*c^2 + 40196*a^3*b^2*
c^3 + 10000*a^4*c^4)*d^2*e^11 - (378*a*b^7 - 3645*a^2*b^5*c + 6732*a^3*b^3*c^2 + 10000*a^4*b*c^3)*d*e^12 + (18
9*a^2*b^6 - 1971*a^3*b^4*c + 5625*a^4*b^2*c^2 - 2500*a^5*c^3)*e^13)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*c^2 - 4*
a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^
5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 1
4*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*
d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*c^2 - 420*a^3*b*c^3)*e^7 - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7
- 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5
+ 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^
3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12
- 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b
^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b
^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8))*log(-sqrt(1/2)*(70*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*
d^6*e^4 - 210*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^5*e^5 + 21*(13*b^6*c^4 - 106*a*b^4*c^5 + 224*a^2*b^2*c^
6 - 32*a^3*c^7)*d^4*e^6 - 28*(7*b^7*c^3 - 59*a*b^5*c^4 + 136*a^2*b^3*c^5 - 48*a^3*b*c^6)*d^3*e^7 - 6*(3*b^8*c^
2 - 146*a*b^6*c^3 + 1289*a^2*b^4*c^4 - 4072*a^3*b^2*c^5 + 4240*a^4*c^6)*d^2*e^8 + 3*(27*b^9*c - 474*a*b^7*c^2
+ 3026*a^2*b^5*c^3 - 8368*a^3*b^3*c^4 + 8480*a^4*b*c^5)*d*e^9 - (27*b^10 - 459*a*b^8*c + 2961*a^2*b^6*c^2 - 88
18*a^3*b^4*c^3 + 11360*a^4*b^2*c^4 - 4000*a^5*c^5)*e^10 + (8*(b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^
3*c^11)*d^3 - 12*(b^7*c^7 - 12*a*b^5*c^8 + 48*a^2*b^3*c^9 - 64*a^3*b*c^10)*d^2*e - 2*(b^8*c^6 - 28*a*b^6*c^7 +
240*a^2*b^4*c^8 - 832*a^3*b^2*c^9 + 1024*a^4*c^10)*d*e^2 + (3*b^9*c^5 - 52*a*b^7*c^6 + 336*a^2*b^5*c^7 - 960*
a^3*b^3*c^8 + 1024*a^4*b*c^9)*e^3)*sqrt((1225*c^8*d^8*e^6 - 4900*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^
6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 - 2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(6
6*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 110
0*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*
a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12
- 64*a^3*c^13)))*sqrt((32*c^7*d^7 - 112*b*c^6*d^6*e + 14*(7*b^2*c^5 + 20*a*c^6)*d^5*e^2 + 35*(b^3*c^4 - 20*a*
b*c^5)*d^4*e^3 - 70*(b^4*c^3 - 6*a*b^2*c^4 - 8*a^2*c^5)*d^3*e^4 + 14*(b^5*c^2 + 5*a*b^3*c^3 - 60*a^2*b*c^4)*d^
2*e^5 + 7*(3*b^6*c - 40*a*b^4*c^2 + 150*a^2*b^2*c^3 - 120*a^3*c^4)*d*e^6 - (9*b^7 - 105*a*b^5*c + 385*a^2*b^3*
c^2 - 420*a^3*b*c^3)*e^7 - (b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*sqrt((1225*c^8*d^8*e^6 - 490
0*b*c^7*d^7*e^7 + 980*(6*b^2*c^6 + 11*a*c^7)*d^6*e^8 - 490*(b^3*c^5 + 66*a*b*c^6)*d^5*e^9 - 14*(241*b^4*c^4 -
2103*a*b^2*c^5 - 1569*a^2*c^6)*d^4*e^10 + 28*(66*b^5*c^3 - 178*a*b^3*c^4 - 1569*a^2*b*c^5)*d^3*e^11 + 7*(27*b^
6*c^2 - 1116*a*b^4*c^3 + 5532*a^2*b^2*c^4 - 1100*a^3*c^5)*d^2*e^12 - 14*(27*b^7*c - 351*a*b^5*c^2 + 1197*a^2*b
^3*c^3 - 550*a^3*b*c^4)*d*e^13 + (81*b^8 - 918*a*b^6*c + 3051*a^2*b^4*c^2 - 2550*a^3*b^2*c^3 + 625*a^4*c^4)*e^
14)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64
*a^3*c^8)) - 2*(560*c^8*d^10*e^3 - 2800*b*c^7*d^9*e^4 + 7*(647*b^2*c^6 + 1012*a*c^7)*d^8*e^5 - 28*(47*b^3*c^5
+ 1012*a*b*c^6)*d^7*e^6 - (3329*b^4*c^4 - 35844*a*b^2*c^5 - 27488*a^2*c^6)*d^6*e^7 + (2833*b^5*c^3 - 8356*a*b^
3*c^4 - 82464*a^2*b*c^5)*d^5*e^8 + (9*b^6*c^2 - 14273*a*b^4*c^3 + 77982*a^2*b^2*c^4 + 33464*a^3*c^5)*d^4*e^9 -
(675*b^7*c - 9414*a*b^5*c^2 + 18524*a^2*b^3*c^3 + 66928*a^3*b*c^4)*d^3*e^10 + (189*b^8 - 999*a*b^6*c - 8127*a
^2*b^4*c^2 + 40196*a^3*b^2*c^3 + 10000*a^4*c^4)*d^2*e^11 - (378*a*b^7 - 3645*a^2*b^5*c + 6732*a^3*b^3*c^2 + 10
000*a^4*b*c^3)*d*e^12 + (189*a^2*b^6 - 1971*a^3*b^4*c + 5625*a^4*b^2*c^2 - 2500*a^5*c^3)*e^13)*sqrt(e*x + d))
+ 2*(b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - 2*(b^2*c - 4*a*c^2)*e^3*x^2 - (3*a*b^2 - 10*a^2*c)*e^3 + (2*c
^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (3*b^3 - 11*a*b*c)*e^3)*x)*sqrt(e*x + d))/(a*b^2*c^2 - 4*
a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

Timed out