3.378 \(\int \frac{(d+e x)^{9/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=300 \[ \frac{(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{3 e \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 c^2}-\frac{3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac{(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \]

[Out]

(-3*e*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4*b^4*c^2) - ((d + e*x)^(7/2)*(b*d + (2*
c*d - b*e)*x))/(2*b^2*(b*x + c*x^2)^2) + ((d + e*x)^(3/2)*(b*c*d^2*(12*c*d - 13*b*e) + (2*c*d - b*e)*(12*c^2*d
^2 - 12*b*c*d*e - b^2*e^2)*x))/(4*b^4*c*(b*x + c*x^2)) - (3*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*Arc
Tanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.519851, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {738, 818, 824, 826, 1166, 208} \[ \frac{(d+e x)^{3/2} \left (x (2 c d-b e) \left (-b^2 e^2-12 b c d e+12 c^2 d^2\right )+b c d^2 (12 c d-13 b e)\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{3 e \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 c^2}-\frac{3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac{(d+e x)^{7/2} (x (2 c d-b e)+b d)}{2 b^2 \left (b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]

[Out]

(-3*e*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*Sqrt[d + e*x])/(4*b^4*c^2) - ((d + e*x)^(7/2)*(b*d + (2*
c*d - b*e)*x))/(2*b^2*(b*x + c*x^2)^2) + ((d + e*x)^(3/2)*(b*c*d^2*(12*c*d - 13*b*e) + (2*c*d - b*e)*(12*c^2*d
^2 - 12*b*c*d*e - b^2*e^2)*x))/(4*b^4*c*(b*x + c*x^2)) - (3*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*Arc
Tanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

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 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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] && Ne
Q[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 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)^{9/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}-\frac{\int \frac{(d+e x)^{5/2} \left (\frac{1}{2} d (12 c d-13 b e)-\frac{1}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2}\\ &=-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\int \frac{\sqrt{d+e x} \left (-\frac{3}{4} c d^2 \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )+\frac{3}{4} e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{2 b^4 c}\\ &=-\frac{3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt{d+e x}}{4 b^4 c^2}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{3}{4} c^2 d^3 \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )-\frac{3}{4} e \left (8 c^4 d^4-16 b c^3 d^3 e+7 b^2 c^2 d^2 e^2+b^3 c d e^3+b^4 e^4\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^2}\\ &=-\frac{3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt{d+e x}}{4 b^4 c^2}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} c^2 d^3 e \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )+\frac{3}{4} d e \left (8 c^4 d^4-16 b c^3 d^3 e+7 b^2 c^2 d^2 e^2+b^3 c d e^3+b^4 e^4\right )-\frac{3}{4} e \left (8 c^4 d^4-16 b c^3 d^3 e+7 b^2 c^2 d^2 e^2+b^3 c d e^3+b^4 e^4\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{b^4 c^2}\\ &=-\frac{3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt{d+e x}}{4 b^4 c^2}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\left (3 (c d-b e)^3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 c^2}+\frac{\left (3 c d^3 \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5}\\ &=-\frac{3 e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt{d+e x}}{4 b^4 c^2}-\frac{(d+e x)^{7/2} (b d+(2 c d-b e) x)}{2 b^2 \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 (12 c d-13 b e)+(2 c d-b e) \left (12 c^2 d^2-12 b c d e-b^2 e^2\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{3 d^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.641668, size = 275, normalized size = 0.92 \[ \frac{\frac{b \sqrt{d+e x} \left (b^3 c^2 d \left (-17 d^2 e x-2 d^3+33 d e^2 x^2+3 e^3 x^3\right )+b^2 c^3 d^2 x \left (8 d^2-73 d e x+21 e^2 x^2\right )-5 b^4 c e^3 x^2 (d+e x)-3 b^5 e^4 x^2+12 b c^4 d^3 x^2 (3 d-4 e x)+24 c^5 d^4 x^3\right )}{c^2 x^2 (b+c x)^2}-3 d^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{3 (c d-b e)^{5/2} \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{5/2}}}{4 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(9/2)/(b*x + c*x^2)^3,x]

[Out]

((b*Sqrt[d + e*x]*(-3*b^5*e^4*x^2 + 24*c^5*d^4*x^3 + 12*b*c^4*d^3*x^2*(3*d - 4*e*x) - 5*b^4*c*e^3*x^2*(d + e*x
) + b^2*c^3*d^2*x*(8*d^2 - 73*d*e*x + 21*e^2*x^2) + b^3*c^2*d*(-2*d^3 - 17*d^2*e*x + 33*d*e^2*x^2 + 3*e^3*x^3)
))/(c^2*x^2*(b + c*x)^2) - 3*d^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + (
3*(c*d - b*e)^(5/2)*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(5/
2))/(4*b^5)

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Maple [B]  time = 0.27, size = 703, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)/(c*x^2+b*x)^3,x)

[Out]

-3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^5*c^3+3/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)
*c)^(1/2))*d+21/4*e^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d^2-111/4*e^2/b^3*c/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d^3+33*e/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan((e
*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*d^4-12/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*
d^5*c^3-31/4*e^2/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(3/2)*d^3+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d^4*c^3+15/2*e^4/
b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2-15*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*d^3+45/4*e^2/b^3/(c*e*x+b*e)^2*c^2*
(e*x+d)^(1/2)*d^4-5/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(3/2)-3/4*e^6*b/(c*e*x+b*e)^2/c^2*(e*x+d)^(1/2)+3/4*e^5/c^2/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))+3/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d+21/4*e
^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(3/2)*d^2-17/4*d^3/b^3/x^2*(e*x+d)^(3/2)+3/e*d^4/b^4/x^2*(e*x+d)^(3/2)*c+15/4*d
^4/b^3/x^2*(e*x+d)^(1/2)-3/e*d^5/b^4/x^2*(e*x+d)^(1/2)*c-63/4*e^2*d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))+2
7*e*d^(7/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12*d^(9/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 12.6822, size = 4871, normalized size = 16.24 \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)^(9/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d
^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^
3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(
e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 3*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16
*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*
e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4
*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4
*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*
x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(6*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3
+ b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^
3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4)*x^2)*sqrt(-(c*d - b*e)/c
)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2
*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e
+ 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b^4*c^2*d^4 - (24*b*c^5
*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (36*b^2*c^4*d^4 - 73*b^3*c
^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b^4*c^2*d^3*e)*x)*sqrt(e*
x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/8*(6*((16*c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2
)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21
*b^4*c^2*d^2*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((16*c^6*d^4 - 28*b*c^5*d^3*e + 9*b^2*c^4
*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*d^2*e^2 + 2*b^4
*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*d*e^3 + b^6*e^4
)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(2
*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3 - (3
6*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 - 17*b
^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), 1/4*(3*((16*c^6*d^4 - 28*b*c^5*d^
3*e + 9*b^2*c^4*d^2*e^2 + 2*b^3*c^3*d*e^3 + b^4*c^2*e^4)*x^4 + 2*(16*b*c^5*d^4 - 28*b^2*c^4*d^3*e + 9*b^3*c^3*
d^2*e^2 + 2*b^4*c^2*d*e^3 + b^5*c*e^4)*x^3 + (16*b^2*c^4*d^4 - 28*b^3*c^3*d^3*e + 9*b^4*c^2*d^2*e^2 + 2*b^5*c*
d*e^3 + b^6*e^4)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 3*((16*
c^6*d^4 - 36*b*c^5*d^3*e + 21*b^2*c^4*d^2*e^2)*x^4 + 2*(16*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2)*
x^3 + (16*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 21*b^4*c^2*d^2*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d)
- (2*b^4*c^2*d^4 - (24*b*c^5*d^4 - 48*b^2*c^4*d^3*e + 21*b^3*c^3*d^2*e^2 + 3*b^4*c^2*d*e^3 - 5*b^5*c*e^4)*x^3
- (36*b^2*c^4*d^4 - 73*b^3*c^3*d^3*e + 33*b^4*c^2*d^2*e^2 - 5*b^5*c*d*e^3 - 3*b^6*e^4)*x^2 - (8*b^3*c^3*d^4 -
17*b^4*c^2*d^3*e)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)]

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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)**(9/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.36662, size = 852, normalized size = 2.84 \begin{align*} \frac{3 \,{\left (16 \, c^{2} d^{5} - 36 \, b c d^{4} e + 21 \, b^{2} d^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d}} - \frac{3 \,{\left (16 \, c^{5} d^{5} - 44 \, b c^{4} d^{4} e + 37 \, b^{2} c^{3} d^{3} e^{2} - 7 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4} - b^{5} e^{5}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \, \sqrt{-c^{2} d + b c e} b^{5} c^{2}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{5} d^{4} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{5} d^{5} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{5} d^{6} e - 24 \, \sqrt{x e + d} c^{5} d^{7} e - 48 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{4} d^{3} e^{2} + 180 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{4} d^{4} e^{2} - 216 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{4} d^{5} e^{2} + 84 \, \sqrt{x e + d} b c^{4} d^{6} e^{2} + 21 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{3} d^{2} e^{3} - 136 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{3} d^{3} e^{3} + 217 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{3} d^{4} e^{3} - 102 \, \sqrt{x e + d} b^{2} c^{3} d^{5} e^{3} + 3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} c^{2} d e^{4} + 24 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c^{2} d^{2} e^{4} - 74 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c^{2} d^{3} e^{4} + 45 \, \sqrt{x e + d} b^{3} c^{2} d^{4} e^{4} - 5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} c e^{5} + 10 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} c d e^{5} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} c d^{2} e^{5} - 3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{5} e^{6} + 6 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} d e^{6} - 3 \, \sqrt{x e + d} b^{5} d^{2} e^{6}}{4 \,{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2} b^{4} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

3/4*(16*c^2*d^5 - 36*b*c*d^4*e + 21*b^2*d^3*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) - 3/4*(16*c^5*d
^5 - 44*b*c^4*d^4*e + 37*b^2*c^3*d^3*e^2 - 7*b^3*c^2*d^2*e^3 - b^4*c*d*e^4 - b^5*e^5)*arctan(sqrt(x*e + d)*c/s
qrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c^2) + 1/4*(24*(x*e + d)^(7/2)*c^5*d^4*e - 72*(x*e + d)^(5/2)*c
^5*d^5*e + 72*(x*e + d)^(3/2)*c^5*d^6*e - 24*sqrt(x*e + d)*c^5*d^7*e - 48*(x*e + d)^(7/2)*b*c^4*d^3*e^2 + 180*
(x*e + d)^(5/2)*b*c^4*d^4*e^2 - 216*(x*e + d)^(3/2)*b*c^4*d^5*e^2 + 84*sqrt(x*e + d)*b*c^4*d^6*e^2 + 21*(x*e +
d)^(7/2)*b^2*c^3*d^2*e^3 - 136*(x*e + d)^(5/2)*b^2*c^3*d^3*e^3 + 217*(x*e + d)^(3/2)*b^2*c^3*d^4*e^3 - 102*sq
rt(x*e + d)*b^2*c^3*d^5*e^3 + 3*(x*e + d)^(7/2)*b^3*c^2*d*e^4 + 24*(x*e + d)^(5/2)*b^3*c^2*d^2*e^4 - 74*(x*e +
d)^(3/2)*b^3*c^2*d^3*e^4 + 45*sqrt(x*e + d)*b^3*c^2*d^4*e^4 - 5*(x*e + d)^(7/2)*b^4*c*e^5 + 10*(x*e + d)^(5/2
)*b^4*c*d*e^5 - 5*(x*e + d)^(3/2)*b^4*c*d^2*e^5 - 3*(x*e + d)^(5/2)*b^5*e^6 + 6*(x*e + d)^(3/2)*b^5*d*e^6 - 3*
sqrt(x*e + d)*b^5*d^2*e^6)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c^2)