3.384 \(\int \frac{1}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=398 \[ \frac{5 b e+8 c d}{4 b^2 d^2 x (b+c x)^2 \sqrt{d+e x}}-\frac{3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{7/2}}+\frac{3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}+\frac{c (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 (b+c x) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{c \left (-5 b^2 e^2-5 b c d e+12 c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)}-\frac{1}{2 b d x^2 (b+c x)^2 \sqrt{d+e x}} \]

[Out]

(3*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2))/(4*b^4*d
^3*(c*d - b*e)^3*Sqrt[d + e*x]) + (c*(12*c^2*d^2 - 5*b*c*d*e - 5*b^2*e^2))/(4*b^
3*d^2*(c*d - b*e)*(b + c*x)^2*Sqrt[d + e*x]) - 1/(2*b*d*x^2*(b + c*x)^2*Sqrt[d +
e*x]) + (8*c*d + 5*b*e)/(4*b^2*d^2*x*(b + c*x)^2*Sqrt[d + e*x]) + (c*(2*c*d - b
*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))/(4*b^4*d^2*(c*d - b*e)^2*(b + c*x)*Sq
rt[d + e*x]) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sq
rt[d]])/(4*b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(7/2))

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Rubi [A]  time = 1.91608, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{5 b e+8 c d}{4 b^2 d^2 x (b+c x)^2 \sqrt{d+e x}}-\frac{3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{7/2}}+\frac{3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}+\frac{c (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 (b+c x) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{c \left (-5 b^2 e^2-5 b c d e+12 c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)}-\frac{1}{2 b d x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]

[Out]

(3*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2))/(4*b^4*d
^3*(c*d - b*e)^3*Sqrt[d + e*x]) + (c*(12*c^2*d^2 - 5*b*c*d*e - 5*b^2*e^2))/(4*b^
3*d^2*(c*d - b*e)*(b + c*x)^2*Sqrt[d + e*x]) - 1/(2*b*d*x^2*(b + c*x)^2*Sqrt[d +
e*x]) + (8*c*d + 5*b*e)/(4*b^2*d^2*x*(b + c*x)^2*Sqrt[d + e*x]) + (c*(2*c*d - b
*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))/(4*b^4*d^2*(c*d - b*e)^2*(b + c*x)*Sq
rt[d + e*x]) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sq
rt[d]])/(4*b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(7/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.12517, size = 251, normalized size = 0.63 \[ \frac{1}{4} \left (-\frac{3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^5 d^{7/2}}+\frac{3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^5 (c d-b e)^{7/2}}+\sqrt{d+e x} \left (\frac{c^4 (19 b e-12 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac{7 b e+12 c d}{b^4 d^3 x}+\frac{2 c^4}{b^3 (b+c x)^2 (c d-b e)^2}-\frac{2}{b^3 d^2 x^2}-\frac{8 e^5}{d^3 (d+e x) (c d-b e)^3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)^3),x]

[Out]

(Sqrt[d + e*x]*(-2/(b^3*d^2*x^2) + (12*c*d + 7*b*e)/(b^4*d^3*x) + (2*c^4)/(b^3*(
c*d - b*e)^2*(b + c*x)^2) + (c^4*(-12*c*d + 19*b*e))/(b^4*(-(c*d) + b*e)^3*(b +
c*x)) - (8*e^5)/(d^3*(c*d - b*e)^3*(d + e*x))) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5
*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2
- 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b
^5*(c*d - b*e)^(7/2)))/4

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Maple [A]  time = 0.04, size = 530, normalized size = 1.3 \[ 2\,{\frac{{e}^{5}}{{d}^{3} \left ( be-cd \right ) ^{3}\sqrt{ex+d}}}+{\frac{19\,{e}^{2}{c}^{5}}{4\,{b}^{3} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-3\,{\frac{e{c}^{6} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}}+{\frac{21\,{e}^{3}{c}^{4}}{4\,{b}^{2} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-{\frac{33\,{e}^{2}{c}^{5}d}{4\,{b}^{3} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+3\,{\frac{e{c}^{6}\sqrt{ex+d}{d}^{2}}{{b}^{4} \left ( be-cd \right ) ^{3} \left ( cex+be \right ) ^{2}}}+{\frac{99\,{e}^{2}{c}^{4}}{4\,{b}^{3} \left ( be-cd \right ) ^{3}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-33\,{\frac{e{c}^{5}d}{{b}^{4} \left ( be-cd \right ) ^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+12\,{\frac{{c}^{6}{d}^{2}}{{b}^{5} \left ( be-cd \right ) ^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{7}{4\,{d}^{3}{b}^{3}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{d}^{2}{b}^{4}{x}^{2}}}-{\frac{9}{4\,{d}^{2}{b}^{3}{x}^{2}}\sqrt{ex+d}}-3\,{\frac{c\sqrt{ex+d}}{de{b}^{4}{x}^{2}}}-{\frac{15\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{7}{2}}}}-9\,{\frac{ce}{{d}^{5/2}{b}^{4}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{c}^{2}}{{d}^{3/2}{b}^{5}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x)^3,x)

[Out]

2*e^5/d^3/(b*e-c*d)^3/(e*x+d)^(1/2)+19/4*e^2*c^5/b^3/(b*e-c*d)^3/(c*e*x+b*e)^2*(
e*x+d)^(3/2)-3*e*c^6/b^4/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*d+21/4*e^3*c^4/
b^2/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)-33/4*e^2*c^5/b^3/(b*e-c*d)^3/(c*e*x+
b*e)^2*(e*x+d)^(1/2)*d+3*e*c^6/b^4/(b*e-c*d)^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2+9
9/4*e^2*c^4/b^3/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d
)*c)^(1/2))-33*e*c^5/b^4/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/
((b*e-c*d)*c)^(1/2))*d+12*c^6/b^5/(b*e-c*d)^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+
d)^(1/2)/((b*e-c*d)*c)^(1/2))*d^2+7/4/d^3/b^3/x^2*(e*x+d)^(3/2)+3/e/d^2/b^4/x^2*
(e*x+d)^(3/2)*c-9/4/d^2/b^3/x^2*(e*x+d)^(1/2)-3/e/d/b^4/x^2*(e*x+d)^(1/2)*c-15/4
*e^2/d^(7/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))-9*e/d^(5/2)/b^4*arctanh((e*x+d)^
(1/2)/d^(1/2))*c-12/d^(3/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.30436, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/8*(3*((16*c^7*d^5 - 44*b*c^6*d^4*e + 33*b^2*c^5*d^3*e^2)*x^4 + 2*(16*b*c^6*d
^5 - 44*b^2*c^5*d^4*e + 33*b^3*c^4*d^3*e^2)*x^3 + (16*b^2*c^5*d^5 - 44*b^3*c^4*d
^4*e + 33*b^4*c^3*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(d)*sqrt(c/(c*d - b*e))*log((c
*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b))
- 3*((16*c^7*d^5 - 36*b*c^6*d^4*e + 17*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^2*e^3 + 3*
b^4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 + 2*(16*b*c^6*d^5 - 36*b^2*c^5*d^4*e + 17*b^3
*c^4*d^3*e^2 + 5*b^4*c^3*d^2*e^3 + 3*b^5*c^2*d*e^4 - 5*b^6*c*e^5)*x^3 + (16*b^2*
c^5*d^5 - 36*b^3*c^4*d^4*e + 17*b^4*c^3*d^3*e^2 + 5*b^5*c^2*d^2*e^3 + 3*b^6*c*d*
e^4 - 5*b^7*e^5)*x^2)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d
)/x) + 2*(2*b^4*c^3*d^5 - 6*b^5*c^2*d^4*e + 6*b^6*c*d^3*e^2 - 2*b^7*d^2*e^3 - 3*
(8*b*c^6*d^4*e - 16*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^2*e^3 + 3*b^4*c^3*d*e^4 - 5*b^
5*c^2*e^5)*x^4 - (24*b*c^6*d^5 - 12*b^2*c^5*d^4*e - 58*b^3*c^4*d^3*e^2 + 33*b^4*
c^3*d^2*e^3 + 13*b^5*c^2*d*e^4 - 30*b^6*c*e^5)*x^3 - (36*b^2*c^5*d^5 - 65*b^3*c^
4*d^4*e + 7*b^4*c^3*d^3*e^2 + 23*b^5*c^2*d^2*e^3 - b^6*c*d*e^4 - 15*b^7*e^5)*x^2
- (8*b^3*c^4*d^5 - 19*b^4*c^3*d^4*e + 9*b^5*c^2*d^3*e^2 + 7*b^6*c*d^2*e^3 - 5*b
^7*d*e^4)*x)*sqrt(d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e^2 - b^8
*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2 - b^9*c
*d^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*d^3*e^3)
*x^2)*sqrt(e*x + d)*sqrt(d)), 1/8*(6*((16*c^7*d^5 - 44*b*c^6*d^4*e + 33*b^2*c^5*
d^3*e^2)*x^4 + 2*(16*b*c^6*d^5 - 44*b^2*c^5*d^4*e + 33*b^3*c^4*d^3*e^2)*x^3 + (1
6*b^2*c^5*d^5 - 44*b^3*c^4*d^4*e + 33*b^4*c^3*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(d
)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c
)) + 3*((16*c^7*d^5 - 36*b*c^6*d^4*e + 17*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^2*e^3 +
3*b^4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 + 2*(16*b*c^6*d^5 - 36*b^2*c^5*d^4*e + 17*b
^3*c^4*d^3*e^2 + 5*b^4*c^3*d^2*e^3 + 3*b^5*c^2*d*e^4 - 5*b^6*c*e^5)*x^3 + (16*b^
2*c^5*d^5 - 36*b^3*c^4*d^4*e + 17*b^4*c^3*d^3*e^2 + 5*b^5*c^2*d^2*e^3 + 3*b^6*c*
d*e^4 - 5*b^7*e^5)*x^2)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)
*d)/x) - 2*(2*b^4*c^3*d^5 - 6*b^5*c^2*d^4*e + 6*b^6*c*d^3*e^2 - 2*b^7*d^2*e^3 -
3*(8*b*c^6*d^4*e - 16*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^2*e^3 + 3*b^4*c^3*d*e^4 - 5*
b^5*c^2*e^5)*x^4 - (24*b*c^6*d^5 - 12*b^2*c^5*d^4*e - 58*b^3*c^4*d^3*e^2 + 33*b^
4*c^3*d^2*e^3 + 13*b^5*c^2*d*e^4 - 30*b^6*c*e^5)*x^3 - (36*b^2*c^5*d^5 - 65*b^3*
c^4*d^4*e + 7*b^4*c^3*d^3*e^2 + 23*b^5*c^2*d^2*e^3 - b^6*c*d*e^4 - 15*b^7*e^5)*x
^2 - (8*b^3*c^4*d^5 - 19*b^4*c^3*d^4*e + 9*b^5*c^2*d^3*e^2 + 7*b^6*c*d^2*e^3 - 5
*b^7*d*e^4)*x)*sqrt(d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e^2 - b
^8*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2 - b^9
*c*d^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*d^3*e^
3)*x^2)*sqrt(e*x + d)*sqrt(d)), -1/8*(3*((16*c^7*d^5 - 44*b*c^6*d^4*e + 33*b^2*c
^5*d^3*e^2)*x^4 + 2*(16*b*c^6*d^5 - 44*b^2*c^5*d^4*e + 33*b^3*c^4*d^3*e^2)*x^3 +
(16*b^2*c^5*d^5 - 44*b^3*c^4*d^4*e + 33*b^4*c^3*d^3*e^2)*x^2)*sqrt(e*x + d)*sqr
t(-d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e - 2*(c*d - b*e)*sqrt(e*x + d)
*sqrt(c/(c*d - b*e)))/(c*x + b)) - 6*((16*c^7*d^5 - 36*b*c^6*d^4*e + 17*b^2*c^5*
d^3*e^2 + 5*b^3*c^4*d^2*e^3 + 3*b^4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 + 2*(16*b*c^6
*d^5 - 36*b^2*c^5*d^4*e + 17*b^3*c^4*d^3*e^2 + 5*b^4*c^3*d^2*e^3 + 3*b^5*c^2*d*e
^4 - 5*b^6*c*e^5)*x^3 + (16*b^2*c^5*d^5 - 36*b^3*c^4*d^4*e + 17*b^4*c^3*d^3*e^2
+ 5*b^5*c^2*d^2*e^3 + 3*b^6*c*d*e^4 - 5*b^7*e^5)*x^2)*sqrt(e*x + d)*arctan(d/(sq
rt(e*x + d)*sqrt(-d))) + 2*(2*b^4*c^3*d^5 - 6*b^5*c^2*d^4*e + 6*b^6*c*d^3*e^2 -
2*b^7*d^2*e^3 - 3*(8*b*c^6*d^4*e - 16*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^2*e^3 + 3*b^
4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 - (24*b*c^6*d^5 - 12*b^2*c^5*d^4*e - 58*b^3*c^4
*d^3*e^2 + 33*b^4*c^3*d^2*e^3 + 13*b^5*c^2*d*e^4 - 30*b^6*c*e^5)*x^3 - (36*b^2*c
^5*d^5 - 65*b^3*c^4*d^4*e + 7*b^4*c^3*d^3*e^2 + 23*b^5*c^2*d^2*e^3 - b^6*c*d*e^4
- 15*b^7*e^5)*x^2 - (8*b^3*c^4*d^5 - 19*b^4*c^3*d^4*e + 9*b^5*c^2*d^3*e^2 + 7*b
^6*c*d^2*e^3 - 5*b^7*d*e^4)*x)*sqrt(-d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^
7*c^3*d^4*e^2 - b^8*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*
c^2*d^4*e^2 - b^9*c*d^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*
e^2 - b^10*d^3*e^3)*x^2)*sqrt(e*x + d)*sqrt(-d)), 1/4*(3*((16*c^7*d^5 - 44*b*c^6
*d^4*e + 33*b^2*c^5*d^3*e^2)*x^4 + 2*(16*b*c^6*d^5 - 44*b^2*c^5*d^4*e + 33*b^3*c
^4*d^3*e^2)*x^3 + (16*b^2*c^5*d^5 - 44*b^3*c^4*d^4*e + 33*b^4*c^3*d^3*e^2)*x^2)*
sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b
*e))/(sqrt(e*x + d)*c)) + 3*((16*c^7*d^5 - 36*b*c^6*d^4*e + 17*b^2*c^5*d^3*e^2 +
5*b^3*c^4*d^2*e^3 + 3*b^4*c^3*d*e^4 - 5*b^5*c^2*e^5)*x^4 + 2*(16*b*c^6*d^5 - 36
*b^2*c^5*d^4*e + 17*b^3*c^4*d^3*e^2 + 5*b^4*c^3*d^2*e^3 + 3*b^5*c^2*d*e^4 - 5*b^
6*c*e^5)*x^3 + (16*b^2*c^5*d^5 - 36*b^3*c^4*d^4*e + 17*b^4*c^3*d^3*e^2 + 5*b^5*c
^2*d^2*e^3 + 3*b^6*c*d*e^4 - 5*b^7*e^5)*x^2)*sqrt(e*x + d)*arctan(d/(sqrt(e*x +
d)*sqrt(-d))) - (2*b^4*c^3*d^5 - 6*b^5*c^2*d^4*e + 6*b^6*c*d^3*e^2 - 2*b^7*d^2*e
^3 - 3*(8*b*c^6*d^4*e - 16*b^2*c^5*d^3*e^2 + 5*b^3*c^4*d^2*e^3 + 3*b^4*c^3*d*e^4
- 5*b^5*c^2*e^5)*x^4 - (24*b*c^6*d^5 - 12*b^2*c^5*d^4*e - 58*b^3*c^4*d^3*e^2 +
33*b^4*c^3*d^2*e^3 + 13*b^5*c^2*d*e^4 - 30*b^6*c*e^5)*x^3 - (36*b^2*c^5*d^5 - 65
*b^3*c^4*d^4*e + 7*b^4*c^3*d^3*e^2 + 23*b^5*c^2*d^2*e^3 - b^6*c*d*e^4 - 15*b^7*e
^5)*x^2 - (8*b^3*c^4*d^5 - 19*b^4*c^3*d^4*e + 9*b^5*c^2*d^3*e^2 + 7*b^6*c*d^2*e^
3 - 5*b^7*d*e^4)*x)*sqrt(-d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e
^2 - b^8*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2
- b^9*c*d^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*
d^3*e^3)*x^2)*sqrt(e*x + d)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (b + c x\right )^{3} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Integral(1/(x**3*(b + c*x)**3*(d + e*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.22879, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^3*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Done