3.24.2 \(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^3} \, dx\) [2302]

Optimal. Leaf size=751 \[ -\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (96 c^4 d^4-b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d-9 a b e+8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt {b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt {b^2-4 a c} d+38 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (96 c^4 d^4-b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d-9 a b e-8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt {b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt {b^2-4 a c} d-5 a e\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

-1/2*(e*x+d)^(5/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2+1/4*(12*b*c*d*(3*a*e^2+c*d^2)-4*a*c
*e*(5*a*e^2+7*c*d^2)-b^2*(a*e^3+11*c*d^2*e)+(-b*e+2*c*d)*(12*c^2*d^2+b^2*e^2-4*c*e*(-2*a*e+3*b*d))*x)*(e*x+d)^
(1/2)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-1/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2))
)^(1/2))*(96*c^4*d^4-b^3*e^4*(b-(-4*a*c+b^2)^(1/2))-8*c^3*d^2*e*(24*b*d-19*a*e-3*d*(-4*a*c+b^2)^(1/2))-2*b*c*e
^3*(5*b^2*d-9*a*b*e-5*b*d*(-4*a*c+b^2)^(1/2)+8*a*e*(-4*a*c+b^2)^(1/2))+2*c^2*e^2*(53*b^2*d^2+4*a*e*(5*a*e+4*d*
(-4*a*c+b^2)^(1/2))-2*b*d*(38*a*e+9*d*(-4*a*c+b^2)^(1/2))))/c^(3/2)/(-4*a*c+b^2)^(5/2)*2^(1/2)/(2*c*d-e*(b-(-4
*a*c+b^2)^(1/2)))^(1/2)+1/8*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(96*
c^4*d^4-b^3*e^4*(b+(-4*a*c+b^2)^(1/2))-8*c^3*d^2*e*(24*b*d-19*a*e+3*d*(-4*a*c+b^2)^(1/2))-2*b*c*e^3*(5*b^2*d-9
*a*b*e+5*b*d*(-4*a*c+b^2)^(1/2)-8*a*e*(-4*a*c+b^2)^(1/2))+2*c^2*e^2*(53*b^2*d^2-4*a*e*(-5*a*e+4*d*(-4*a*c+b^2)
^(1/2))+2*b*d*(-38*a*e+9*d*(-4*a*c+b^2)^(1/2))))/c^(3/2)/(-4*a*c+b^2)^(5/2)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(
1/2)))^(1/2)

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Rubi [A]
time = 10.18, antiderivative size = 751, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 832, 840, 1180, 214} \begin {gather*} \frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\left (-8 c^3 d^2 e \left (-3 d \sqrt {b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (-2 b d \left (9 d \sqrt {b^2-4 a c}+38 a e\right )+4 a e \left (4 d \sqrt {b^2-4 a c}+5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (-5 b d \sqrt {b^2-4 a c}+8 a e \sqrt {b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (b-\sqrt {b^2-4 a c}\right )+96 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 )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\left (-8 c^3 d^2 e \left (3 d \sqrt {b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (2 b d \left (9 d \sqrt {b^2-4 a c}-38 a e\right )-4 a e \left (4 d \sqrt {b^2-4 a c}-5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (5 b d \sqrt {b^2-4 a c}-8 a e \sqrt {b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (\sqrt {b^2-4 a c}+b\right )+96 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 )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*((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)^2) + (Sqrt[d + e*x]*(1
2*b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(7*c*d^2 + 5*a*e^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^2
+ b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((96*c^4*d^4 - b^3*(b - Sqrt
[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2
- 4*a*c]*d - 9*a*b*e + 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*
e) - 2*b*d*(9*Sqrt[b^2 - 4*a*c]*d + 38*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b
^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((96*c^4*d
^4 - b^3*(b + Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^
2*d + 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e - 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 2*b*d*(9*Sqrt[b^2
- 4*a*c]*d - 38*a*e) - 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d - 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c]
)*e])

Rule 214

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

Rule 752

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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], 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] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 840

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 1180

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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (12 c d^2-11 b d e+10 a e^2\right )+\frac {1}{2} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{4} \left (-48 c^3 d^4-a b^2 e^4+4 c^2 d^2 e (21 b d-19 a e)-5 c e^2 \left (7 b^2 d^2-12 a b d e+4 a^2 e^2\right )\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{4} d e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )+\frac {1}{4} e \left (-48 c^3 d^4-a b^2 e^4+4 c^2 d^2 e (21 b d-19 a e)-5 c e^2 \left (7 b^2 d^2-12 a b d e+4 a^2 e^2\right )\right )-\frac {1}{4} e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 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 \left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (96 c^4 d^4-b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d-9 a b e-8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt {b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt {b^2-4 a c} d-5 a e\right )\right )\right ) \text {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 )}{8 c \left (b^2-4 a c\right )^{5/2}}+\frac {\left (96 c^4 d^4-b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d-9 a b e+8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt {b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt {b^2-4 a c} d+38 a e\right )\right )\right ) \text {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 )}{8 c \left (b^2-4 a c\right )^{5/2}}\\ &=-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (12 b c d \left (c d^2+3 a e^2\right )-4 a c e \left (7 c d^2+5 a e^2\right )-b^2 \left (11 c d^2 e+a e^3\right )+(2 c d-b e) \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (96 c^4 d^4-b^3 \left (b-\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d-3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d-5 b \sqrt {b^2-4 a c} d-9 a b e+8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+4 a e \left (4 \sqrt {b^2-4 a c} d+5 a e\right )-2 b d \left (9 \sqrt {b^2-4 a c} d+38 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 )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (96 c^4 d^4-b^3 \left (b+\sqrt {b^2-4 a c}\right ) e^4-8 c^3 d^2 e \left (24 b d+3 \sqrt {b^2-4 a c} d-19 a e\right )-2 b c e^3 \left (5 b^2 d+5 b \sqrt {b^2-4 a c} d-9 a b e-8 a \sqrt {b^2-4 a c} e\right )+2 c^2 e^2 \left (53 b^2 d^2+2 b d \left (9 \sqrt {b^2-4 a c} d-38 a e\right )-4 a e \left (4 \sqrt {b^2-4 a c} d-5 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 )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [A]
time = 15.97, size = 720, normalized size = 0.96 \begin {gather*} \frac {-4 \sqrt {c} \sqrt {b^2-4 a c} \sqrt {d+e x} \left (b^4 e^3 x^2+b^2 \left (a^2 e^3+c^2 d x \left (-8 d^2+55 d e x-10 e^2 x^2\right )+a c e \left (7 d^2-58 d e x+5 e^2 x^2\right )\right )+4 c \left (5 a^3 e^3-6 c^3 d^3 x^3-a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )\right )-4 b c \left (a^2 e^2 (9 d-7 e x)+9 c^2 d^2 x^2 (d-e x)+a c \left (5 d^3-14 d^2 e x+11 d e^2 x^2-4 e^3 x^3\right )\right )+b^3 \left (2 a e^3 x+c \left (2 d^3+13 d^2 e x-16 d e^2 x^2-e^3 x^3\right )\right )\right )-\sqrt {4 c d+2 \left (-b+\sqrt {b^2-4 a c}\right ) e} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) (a+x (b+c x))^2 \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 {4 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+8 c^2 d e \left (-18 b d+3 \sqrt {b^2-4 a c} d+13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) (a+x (b+c x))^2 \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 )}{16 c^{3/2} \left (b^2-4 a c\right )^{5/2} (a+x (b+c x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

(-4*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*(b^4*e^3*x^2 + b^2*(a^2*e^3 + c^2*d*x*(-8*d^2 + 55*d*e*x - 10*e^2*
x^2) + a*c*e*(7*d^2 - 58*d*e*x + 5*e^2*x^2)) + 4*c*(5*a^3*e^3 - 6*c^3*d^3*x^3 - a*c^2*d*x*(10*d^2 + d*e*x + 8*
e^2*x^2) + a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)) - 4*b*c*(a^2*e^2*(9*d - 7*e*x) + 9*c^2*d^2*x^2*(d - e*x) +
a*c*(5*d^3 - 14*d^2*e*x + 11*d*e^2*x^2 - 4*e^3*x^3)) + b^3*(2*a*e^3*x + c*(2*d^3 + 13*d^2*e*x - 16*d*e^2*x^2 -
e^3*x^3))) - Sqrt[4*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(96*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2
*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*
a*Sqrt[b^2 - 4*a*c]*e))*(a + x*(b + c*x))^2*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^
2 - 4*a*c]*e]] + Sqrt[4*c*d - 2*(b + Sqrt[b^2 - 4*a*c])*e]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c
^2*d*e*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt
[b^2 - 4*a*c]*d + 13*a*e)))*(a + x*(b + c*x))^2*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt
[b^2 - 4*a*c])*e]])/(16*c^(3/2)*(b^2 - 4*a*c)^(5/2)*(a + x*(b + c*x))^2)

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Maple [A]
time = 0.78, size = 1277, normalized size = 1.70 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*((-1/8*(16*a*b*c*e^3-32*a*c^2*d*e^2-b^3*e^3-10*b^2*c*d*e^2+36*b*c^2*d^2*e-24*c^3*d^3)/e^4/(16*a^2*c^2-8*
a*b^2*c+b^4)*(e*x+d)^(7/2)-1/8*(36*a^2*c^2*e^4+5*a*b^2*c*e^4-92*a*b*c^2*d*e^3+92*a*c^3*d^2*e^2+b^4*e^4-13*b^3*
c*d*e^3+85*b^2*c^2*d^2*e^2-144*b*c^3*d^3*e+72*c^4*d^4)/c/e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)-1/4/c*(1
4*a^2*b*c*e^5-28*a^2*c^2*d*e^4+a*b^3*e^5-34*a*b^2*c*d*e^4+96*a*b*c^2*d^2*e^3-64*a*c^3*d^3*e^2-b^4*d*e^4+21*b^3
*c*d^2*e^3-74*b^2*c^2*d^3*e^2+90*b*c^3*d^4*e-36*c^4*d^5)/e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)-1/8*(a*e
^2-b*d*e+c*d^2)/c*(20*a^2*c*e^4+a*b^2*e^4-44*a*b*c*d*e^3+44*a*c^2*d^2*e^2-b^3*d*e^3+25*b^2*c*d^2*e^2-48*b*c^2*
d^3*e+24*c^3*d^4)/e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2))/((e*x+d)^2*c+b*e*(e*x+d)-2*c*d*(e*x+d)+e^2*a-b
*d*e+c*d^2)^2+1/2/e^4/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/8*(-40*e^4*a^2*c^2-18*a*b^2*c*e^4+152*a*b*c^2*d*e^3-152*d^
2*e^2*c^3*a+b^4*e^4+10*b^3*c*d*e^3-106*b^2*c^2*d^2*e^2+192*b*c^3*d^3*e-96*c^4*d^4-16*(-e^2*(4*a*c-b^2))^(1/2)*
a*b*c*e^3+32*(-e^2*(4*a*c-b^2))^(1/2)*a*c^2*d*e^2+(-e^2*(4*a*c-b^2))^(1/2)*b^3*e^3+10*(-e^2*(4*a*c-b^2))^(1/2)
*b^2*c*d*e^2-36*(-e^2*(4*a*c-b^2))^(1/2)*b*c^2*d^2*e+24*(-e^2*(4*a*c-b^2))^(1/2)*c^3*d^3)/c/(-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(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2
*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/8*(40*e^4*a^2*c^2+18*a*b^2*c*e^4-152*a*b*c^2*d*e^3+152*d^2*e^2*c^3*a-b^4*e^4-
10*b^3*c*d*e^3+106*b^2*c^2*d^2*e^2-192*b*c^3*d^3*e+96*c^4*d^4-16*(-e^2*(4*a*c-b^2))^(1/2)*a*b*c*e^3+32*(-e^2*(
4*a*c-b^2))^(1/2)*a*c^2*d*e^2+(-e^2*(4*a*c-b^2))^(1/2)*b^3*e^3+10*(-e^2*(4*a*c-b^2))^(1/2)*b^2*c*d*e^2-36*(-e^
2*(4*a*c-b^2))^(1/2)*b*c^2*d^2*e+24*(-e^2*(4*a*c-b^2))^(1/2)*c^3*d^3)/c/(-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(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/
2))*c)^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((x*e + d)^(7/2)/(c*x^2 + b*x + a)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8619 vs. \(2 (694) = 1388\).
time = 9.50, size = 8619, normalized size = 11.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

-1/8*(sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^
2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16
*a^3*b*c^3)*x)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4
+ 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 -
880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c +
280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a
^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 +
42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^
6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 64
0*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*
c^6 - 64*a^3*c^7)*d^4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(167*b^8
*c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^
2 + 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 569
6*a^3*b^4*c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 - (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640
*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a
^3*b^5*c^8 + 1280*a^4*b^3*c^9 - 1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 -
6400*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (b^13*c^3 - 72*a*b^11*c^4 +
1200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*
c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^
4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^
2*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*
(7*b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a
*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 3
5*a*b^5*c + 280*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c
^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)
*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 +
160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b
^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432
*(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*
c^4 + 24016*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10 - 2*(518*b^6*c -
10131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 +
100000*a^3*b*c^3)*d*e^12 + (35*a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(x*e + d)
) - sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a
^3*b*c^3)*x)*sqrt((4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 +
20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 88
0*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 2
80*a^2*b^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4
*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 4
2*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*
c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*
a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(-1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c
^6 - 64*a^3*c^7)*d^4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(167*b^8*
c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2
+ 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696
*a^3*b^4*c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3067 vs. \(2 (694) = 1388\).
time = 3.44, size = 3067, normalized size = 4.08 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(192*(b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*d^5*e - 480*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*
b^3*c^8 - 64*a^3*b*c^9)*d^4*e^2 + 4*(101*b^8*c^5 - 1136*a*b^6*c^6 + 3936*a^2*b^4*c^7 - 2816*a^3*b^2*c^8 - 4864
*a^4*c^9)*d^3*e^3 - 6*(21*b^9*c^4 - 176*a*b^7*c^5 + 96*a^2*b^5*c^6 + 2304*a^3*b^3*c^7 - 4864*a^4*b*c^8)*d^2*e^
4 - (24*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 2*(5*b^2*c + 16*a*c^2)*d*e^3 + (b^3 - 16*a*b*c)*e^4)*(b^4*c*e - 8*a*b^2
*c^2*e + 16*a^2*c^3*e)^2 + 4*(2*b^10*c^3 + 23*a*b^8*c^4 - 448*a^2*b^6*c^5 + 1888*a^3*b^4*c^6 - 2048*a^4*b^2*c^
7 - 1280*a^5*c^8)*d*e^5 - 2*(24*(b^2*c^5 - 4*a*c^6)*sqrt(b^2 - 4*a*c)*d^4*e - 48*(b^3*c^4 - 4*a*b*c^5)*sqrt(b^
2 - 4*a*c)*d^3*e^2 + (25*b^4*c^3 - 56*a*b^2*c^4 - 176*a^2*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (b^5*c^2 + 40*a*b^3
*c^3 - 176*a^2*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (a*b^4*c^2 + 16*a^2*b^2*c^3 - 80*a^3*c^4)*sqrt(b^2 - 4*a*c)*e^
5)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e) + (b^11*c^2 - 30*a*b^9*c^3 + 224*a^2*b^7*c^4 - 448*a^3*b^5*c^5
- 768*a^4*b^3*c^6 + 2560*a^5*b*c^7)*e^6)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^2*d - 16*a*b^2*c^3*d
+ 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e + sqrt((2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c^4*d
- b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e)^2 - 4*(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e
+ 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)*(b^4*c^2 - 8*a*b^2*c^
3 + 16*a^2*c^4)))/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)))/(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*
(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*sqrt(b^2 - 4*a*c)*d - (b^8*c - 16*a*b^6*c^2 + 96*a^2*b^
4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5 + (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*sqrt(b^2 - 4*a*
c))*e)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e)*abs(c)) - 1/4*(192*(b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9
- 64*a^3*c^10)*d^5*e - 480*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b*c^9)*d^4*e^2 + 4*(101*b^8*c^5
- 1136*a*b^6*c^6 + 3936*a^2*b^4*c^7 - 2816*a^3*b^2*c^8 - 4864*a^4*c^9)*d^3*e^3 - 6*(21*b^9*c^4 - 176*a*b^7*c^5
+ 96*a^2*b^5*c^6 + 2304*a^3*b^3*c^7 - 4864*a^4*b*c^8)*d^2*e^4 - (24*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 2*(5*b^2*c
+ 16*a*c^2)*d*e^3 + (b^3 - 16*a*b*c)*e^4)*(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e)^2 + 4*(2*b^10*c^3 + 23*a*b
^8*c^4 - 448*a^2*b^6*c^5 + 1888*a^3*b^4*c^6 - 2048*a^4*b^2*c^7 - 1280*a^5*c^8)*d*e^5 + 2*(24*(b^2*c^5 - 4*a*c^
6)*sqrt(b^2 - 4*a*c)*d^4*e - 48*(b^3*c^4 - 4*a*b*c^5)*sqrt(b^2 - 4*a*c)*d^3*e^2 + (25*b^4*c^3 - 56*a*b^2*c^4 -
176*a^2*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (b^5*c^2 + 40*a*b^3*c^3 - 176*a^2*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (
a*b^4*c^2 + 16*a^2*b^2*c^3 - 80*a^3*c^4)*sqrt(b^2 - 4*a*c)*e^5)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e) +
(b^11*c^2 - 30*a*b^9*c^3 + 224*a^2*b^7*c^4 - 448*a^3*b^5*c^5 - 768*a^4*b^3*c^6 + 2560*a^5*b*c^7)*e^6)*arctan(2
*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2
*b*c^3*e - sqrt((2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e)^2 - 4
*(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^
2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^2)*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)))/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2
*c^4)))/(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3
*c^5)*sqrt(b^2 - 4*a*c)*d + (b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5 - (b^7*c -
12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*sqrt(b^2 - 4*a*c))*e)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e
)*abs(c)) + 1/4*(24*(x*e + d)^(7/2)*c^4*d^3*e - 72*(x*e + d)^(5/2)*c^4*d^4*e + 72*(x*e + d)^(3/2)*c^4*d^5*e -
24*sqrt(x*e + d)*c^4*d^6*e - 36*(x*e + d)^(7/2)*b*c^3*d^2*e^2 + 144*(x*e + d)^(5/2)*b*c^3*d^3*e^2 - 180*(x*e +
d)^(3/2)*b*c^3*d^4*e^2 + 72*sqrt(x*e + d)*b*c^3*d^5*e^2 + 10*(x*e + d)^(7/2)*b^2*c^2*d*e^3 + 32*(x*e + d)^(7/
2)*a*c^3*d*e^3 - 85*(x*e + d)^(5/2)*b^2*c^2*d^2*e^3 - 92*(x*e + d)^(5/2)*a*c^3*d^2*e^3 + 148*(x*e + d)^(3/2)*b
^2*c^2*d^3*e^3 + 128*(x*e + d)^(3/2)*a*c^3*d^3*e^3 - 73*sqrt(x*e + d)*b^2*c^2*d^4*e^3 - 68*sqrt(x*e + d)*a*c^3
*d^4*e^3 + (x*e + d)^(7/2)*b^3*c*e^4 - 16*(x*e + d)^(7/2)*a*b*c^2*e^4 + 13*(x*e + d)^(5/2)*b^3*c*d*e^4 + 92*(x
*e + d)^(5/2)*a*b*c^2*d*e^4 - 42*(x*e + d)^(3/2)*b^3*c*d^2*e^4 - 192*(x*e + d)^(3/2)*a*b*c^2*d^2*e^4 + 26*sqrt
(x*e + d)*b^3*c*d^3*e^4 + 136*sqrt(x*e + d)*a*b*c^2*d^3*e^4 - (x*e + d)^(5/2)*b^4*e^5 - 5*(x*e + d)^(5/2)*a*b^
2*c*e^5 - 36*(x*e + d)^(5/2)*a^2*c^2*e^5 + 2*(x*e + d)^(3/2)*b^4*d*e^5 + 68*(x*e + d)^(3/2)*a*b^2*c*d*e^5 + 56
*(x*e + d)^(3/2)*a^2*c^2*d*e^5 - sqrt(x*e + d)*b^4*d^2*e^5 - 70*sqrt(x*e + d)*a*b^2*c*d^2*e^5 - 64*sqrt(x*e +
d)*a^2*c^2*d^2*e^5 - 2*(x*e + d)^(3/2)*a*b^3*e^6 - 28*(x*e + d)^(3/2)*a^2*b*c*e^6 + 2*sqrt(x*e + d)*a*b^3*d*e^
6 + 64*sqrt(x*e + d)*a^2*b*c*d*e^6 - sqrt(x*e + d)*a^2*b^2*e^7 - 20*sqrt(x*e + d)*a^3*c*e^7)/((b^4*c - 8*a*b^2
*c^2 + 16*a^2*c^3)*((x*e + d)^2*c - 2*(x*e + d)...

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Mupad [B]
time = 25.58, size = 2500, normalized size = 3.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x)

[Out]

log((e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)^2*(35*b^6*e^6 + 27648*c^6*d^6 + 6400*a^3*c^3*e^6 + 76032*a*c^5*
d^4*e^2 + 9456*a^2*b^2*c^2*e^6 + 57024*a^2*c^4*d^2*e^4 + 84672*b^2*c^4*d^4*e^2 - 31104*b^3*c^3*d^3*e^3 + 972*b
^4*c^2*d^2*e^4 - 1176*a*b^4*c*e^6 - 82944*b*c^5*d^5*e + 756*b^5*c*d*e^5 - 152064*a*b*c^4*d^3*e^3 - 9504*a*b^3*
c^2*d*e^5 - 57024*a^2*b*c^3*d*e^5 + 85536*a*b^2*c^3*d^2*e^4))/(64*c*(4*a*c - b^2)^6) - (2^(1/2)*((2^(1/2)*((c*
e^3*(24*c^3*d^4 + a*b^2*e^4 + 20*a^2*c*e^4 - b^3*d*e^3 + 44*a*c^2*d^2*e^2 + 25*b^2*c*d^2*e^2 - 48*b*c^2*d^3*e
- 44*a*b*c*d*e^3))/(4*a*c - b^2) - (2^(1/2)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 +
4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a
^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^
7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*
a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*
c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 -
21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c
*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 +
144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7
*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a
^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2
+ 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^
2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e - 630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*
c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 21560*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^
3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 10321920*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c
^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b
*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 + 2580480*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*
c - b^2)^10))^(1/2))/2)*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(
1/2) + 92160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128*b^11*c^6*d^6*e - 737280*a^2
*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^
3*e^7 + 34880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 1863680*a^7*b^3*c^7*e^7 + 137
62560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^
14*c^3*d^3*e^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a*b^15*c*e^7 - 25*a*c*e^7
*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21*b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^
5*e^2 + 1209600*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4*d^2*e^5 + 11182080*a^3
*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 2576000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18
063360*a^4*b^4*c^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 1908480*a^4*b^7*c^6*
d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5*b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896
*a^5*b^5*c^7*d^2*e^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 322560*a*b^9*c^7*d^6*e -
630*a*b^14*c^2*d*e^6 - 16515072*a^5*b*c^11*d^6*e + 403200*a*b^10*c^6*d^5*e^2 - 201600*a*b^11*c^5*d^4*e^3 + 215
60*a*b^12*c^4*d^3*e^4 + 7980*a*b^13*c^3*d^2*e^5 + 2580480*a^2*b^7*c^8*d^6*e + 840*a^2*b^12*c^3*d*e^6 - 1032192
0*a^3*b^5*c^9*d^6*e + 84000*a^3*b^10*c^4*d*e^6 + 20643840*a^4*b^3*c^10*d^6*e - 846720*a^4*b^8*c^5*d*e^6 + 3472
896*a^5*b^6*c^6*d*e^6 - 34406400*a^6*b*c^10*d^4*e^3 - 6236160*a^6*b^4*c^7*d*e^6 - 18923520*a^7*b*c^9*d^2*e^5 +
2580480*a^7*b^2*c^8*d*e^6)/(c^3*(4*a*c - b^2)^10))^(1/2))/16 + ((d + e*x)^(1/2)*(b^8*e^10 + 800*a^4*c^4*e^10
+ 4608*c^8*d^8*e^2 + 13440*a*c^7*d^6*e^4 - 18432*b*c^7*d^7*e^3 + 314*a^2*b^4*c^2*e^10 + 208*a^3*b^2*c^3*e^10 +
12320*a^2*c^6*d^4*e^6 + 4032*a^3*c^5*d^2*e^8 + 28896*b^2*c^6*d^6*e^4 - 22176*b^3*c^5*d^5*e^5 + 8330*b^4*c^4*d
^4*e^6 - 1204*b^5*c^3*d^3*e^7 - 42*b^6*c^2*d^2*e^8 - 36*a*b^6*c*e^10 + 20*b^7*c*d*e^9 + 15456*a^2*b^2*c^4*d^2*
e^8 - 40320*a*b*c^6*d^5*e^5 - 196*a*b^5*c^2*d*e^9 - 4032*a^3*b*c^4*d*e^9 + 44240*a*b^2*c^5*d^4*e^6 - 21280*a*b
^3*c^4*d^3*e^7 + 4116*a*b^4*c^3*d^2*e^8 - 24640*a^2*b*c^5*d^3*e^7 - 3136*a^2*b^3*c^3*d*e^9))/(8*c*(4*a*c - b^2
)^4))*(-(b^17*e^7 + 4718592*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 92160*a*b^8
*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*...

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