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

Optimal. Leaf size=322 \[ \frac{\sqrt{d+e x} (3 b e+8 c d)}{4 b^2 d^2 x (b+c x)^2}-\frac{3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{5/2}}+\frac{3 c^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}+\frac{3 c \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 d^2 (b+c x) (c d-b e)^2}+\frac{c \sqrt{d+e x} \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (c d-b e)}-\frac{\sqrt{d+e x}}{2 b d x^2 (b+c x)^2} \]

[Out]

(c*(12*c^2*d^2 - 7*b*c*d*e - 3*b^2*e^2)*Sqrt[d + e*x])/(4*b^3*d^2*(c*d - b*e)*(b
+ c*x)^2) - Sqrt[d + e*x]/(2*b*d*x^2*(b + c*x)^2) + ((8*c*d + 3*b*e)*Sqrt[d + e
*x])/(4*b^2*d^2*x*(b + c*x)^2) + (3*c*(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*d^2*(c*d - b*e)^2*(b + c*x)) - (3*(16*c^2*d^2 + 4*b*
c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(5/2)) + (3*c^(5/2)*(1
6*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
b*e]])/(4*b^5*(c*d - b*e)^(5/2))

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Rubi [A]  time = 1.36622, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{d+e x} (3 b e+8 c d)}{4 b^2 d^2 x (b+c x)^2}-\frac{3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 d^{5/2}}+\frac{3 c^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}+\frac{3 c \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )}{4 b^4 d^2 (b+c x) (c d-b e)^2}+\frac{c \sqrt{d+e x} \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (c d-b e)}-\frac{\sqrt{d+e x}}{2 b d x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*(12*c^2*d^2 - 7*b*c*d*e - 3*b^2*e^2)*Sqrt[d + e*x])/(4*b^3*d^2*(c*d - b*e)*(b
+ c*x)^2) - Sqrt[d + e*x]/(2*b*d*x^2*(b + c*x)^2) + ((8*c*d + 3*b*e)*Sqrt[d + e
*x])/(4*b^2*d^2*x*(b + c*x)^2) + (3*c*(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*d^2*(c*d - b*e)^2*(b + c*x)) - (3*(16*c^2*d^2 + 4*b*
c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(5/2)) + (3*c^(5/2)*(1
6*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
b*e]])/(4*b^5*(c*d - b*e)^(5/2))

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

[Out]

Timed out

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Mathematica [A]  time = 1.62322, size = 214, normalized size = 0.66 \[ \frac{-\frac{3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{d^{5/2}}+\frac{3 c^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{5/2}}+b \sqrt{d+e x} \left (\frac{3 c^3 (4 c d-5 b e)}{(b+c x) (c d-b e)^2}+\frac{2 b c^3}{(b+c x)^2 (c d-b e)}+\frac{3 (b e+4 c d)}{d^2 x}-\frac{2 b}{d x^2}\right )}{4 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.036, size = 526, normalized size = 1.6 \[ -{\frac{15\,{e}^{2}{c}^{4}}{4\,{b}^{3} \left ( cex+be \right ) ^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e{c}^{5} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( cex+be \right ) ^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }}-{\frac{17\,{e}^{2}{c}^{3}}{4\,{b}^{3} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) }\sqrt{ex+d}}+3\,{\frac{e{c}^{4}\sqrt{ex+d}d}{{b}^{4} \left ( cex+be \right ) ^{2} \left ( be-cd \right ) }}-{\frac{63\,{e}^{2}{c}^{3}}{4\,{b}^{3} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+27\,{\frac{e{c}^{4}d}{{b}^{4} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{c}^{5}{d}^{2}}{{b}^{5} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) \sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{3}{4\,{b}^{3}{x}^{2}{d}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}c}{e{b}^{4}{x}^{2}d}}-{\frac{5}{4\,{b}^{3}{x}^{2}d}\sqrt{ex+d}}-3\,{\frac{c\sqrt{ex+d}}{e{b}^{4}{x}^{2}}}-{\frac{3\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){d}^{-{\frac{5}{2}}}}-3\,{\frac{ce}{{b}^{4}{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{c}^{2}}{{b}^{5}\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15/4*e^2*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)+3*e*c^
5/b^4/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*d-17/4*e^2*c^3/b^3
/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)+3*e*c^4/b^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+
d)^(1/2)*d-63/4*e^2*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arct
an(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))+27*e*c^4/b^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)
/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*d-12*c^5/b^5/(b
^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*
c)^(1/2))*d^2+3/4/b^3/x^2/d^2*(e*x+d)^(3/2)+3/e/b^4/x^2/d*(e*x+d)^(3/2)*c-5/4/b^
3/x^2/d*(e*x+d)^(1/2)-3/e/b^4/x^2*(e*x+d)^(1/2)*c-3/4*e^2/b^3/d^(5/2)*arctanh((e
*x+d)^(1/2)/d^(1/2))-3*e/b^4/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*c-12/b^5/d^(
1/2)*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*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.45718, 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*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/8*(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)*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)) - 2*(2*b^4*c^2
*d^3 - 4*b^5*c*d^2*e + 2*b^6*d*e^2 - 3*(8*b*c^5*d^3 - 12*b^2*c^4*d^2*e + 2*b^3*c
^3*d*e^2 + b^4*c^2*e^3)*x^3 - (36*b^2*c^4*d^3 - 55*b^3*c^3*d^2*e + 10*b^4*c^2*d*
e^2 + 6*b^5*c*e^3)*x^2 - (8*b^3*c^3*d^3 - 13*b^4*c^2*d^2*e + 2*b^5*c*d*e^2 + 3*b
^6*e^3)*x)*sqrt(e*x + d)*sqrt(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)*log(((e*x + 2*d
)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/(((b^5*c^4*d^4 - 2*b^6*c^3*d^3*e + b^7*c^2*d^
2*e^2)*x^4 + 2*(b^6*c^3*d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e^2)*x^3 + (b^7*c^2*d^
4 - 2*b^8*c*d^3*e + b^9*d^2*e^2)*x^2)*sqrt(d)), 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)*sq
rt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x +
d)*c)) - 2*(2*b^4*c^2*d^3 - 4*b^5*c*d^2*e + 2*b^6*d*e^2 - 3*(8*b*c^5*d^3 - 12*b^
2*c^4*d^2*e + 2*b^3*c^3*d*e^2 + b^4*c^2*e^3)*x^3 - (36*b^2*c^4*d^3 - 55*b^3*c^3*
d^2*e + 10*b^4*c^2*d*e^2 + 6*b^5*c*e^3)*x^2 - (8*b^3*c^3*d^3 - 13*b^4*c^2*d^2*e
+ 2*b^5*c*d*e^2 + 3*b^6*e^3)*x)*sqrt(e*x + d)*sqrt(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)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/(((b^5*c^4*d^4 - 2*b^6*c
^3*d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(b^6*c^3*d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e
^2)*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*e + b^9*d^2*e^2)*x^2)*sqrt(d)), 1/8*(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)*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)) - 2*(2*b^4*c^2*d^3 - 4*b^
5*c*d^2*e + 2*b^6*d*e^2 - 3*(8*b*c^5*d^3 - 12*b^2*c^4*d^2*e + 2*b^3*c^3*d*e^2 +
b^4*c^2*e^3)*x^3 - (36*b^2*c^4*d^3 - 55*b^3*c^3*d^2*e + 10*b^4*c^2*d*e^2 + 6*b^5
*c*e^3)*x^2 - (8*b^3*c^3*d^3 - 13*b^4*c^2*d^2*e + 2*b^5*c*d*e^2 + 3*b^6*e^3)*x)*
sqrt(e*x + d)*sqrt(-d) + 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)*arctan(d/(sqrt(e*x + d)*s
qrt(-d))))/(((b^5*c^4*d^4 - 2*b^6*c^3*d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(b^6*c^3*
d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e^2)*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*e + b^9*
d^2*e^2)*x^2)*sqrt(-d)), 1/4*(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)*sqrt(-c/(c*d - b
*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) - (2*b^4*c^2*d^
3 - 4*b^5*c*d^2*e + 2*b^6*d*e^2 - 3*(8*b*c^5*d^3 - 12*b^2*c^4*d^2*e + 2*b^3*c^3*
d*e^2 + b^4*c^2*e^3)*x^3 - (36*b^2*c^4*d^3 - 55*b^3*c^3*d^2*e + 10*b^4*c^2*d*e^2
+ 6*b^5*c*e^3)*x^2 - (8*b^3*c^3*d^3 - 13*b^4*c^2*d^2*e + 2*b^5*c*d*e^2 + 3*b^6*
e^3)*x)*sqrt(e*x + d)*sqrt(-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)*arctan(d/(sqrt(e*
x + d)*sqrt(-d))))/(((b^5*c^4*d^4 - 2*b^6*c^3*d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(
b^6*c^3*d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e^2)*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*
e + b^9*d^2*e^2)*x^2)*sqrt(-d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222106, size = 836, normalized size = 2.6 \[ -\frac{3 \,{\left (16 \, c^{5} d^{2} - 36 \, b c^{4} d e + 21 \, b^{2} c^{3} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \,{\left (b^{5} c^{2} d^{2} - 2 \, b^{6} c d e + b^{7} e^{2}\right )} \sqrt{-c^{2} d + b c e}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{5} d^{3} e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{5} d^{4} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{5} d^{5} e - 24 \, \sqrt{x e + d} c^{5} d^{6} e - 36 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{4} d^{2} e^{2} + 144 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{4} d^{3} e^{2} - 180 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{4} d^{4} e^{2} + 72 \, \sqrt{x e + d} b c^{4} d^{5} e^{2} + 6 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{2} c^{3} d e^{3} - 73 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c^{3} d^{2} e^{3} + 136 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c^{3} d^{3} e^{3} - 69 \, \sqrt{x e + d} b^{2} c^{3} d^{4} e^{3} + 3 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{3} c^{2} e^{4} +{\left (x e + d\right )}^{\frac{5}{2}} b^{3} c^{2} d e^{4} - 24 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} c^{2} d^{2} e^{4} + 18 \, \sqrt{x e + d} b^{3} c^{2} d^{3} e^{4} + 6 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} c e^{5} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} c d e^{5} + 8 \, \sqrt{x e + d} b^{4} c d^{2} e^{5} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{5} e^{6} - 5 \, \sqrt{x e + d} b^{5} d e^{6}}{4 \,{\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac{3 \,{\left (16 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*(16*c^5*d^2 - 36*b*c^4*d*e + 21*b^2*c^3*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c
^2*d + b*c*e))/((b^5*c^2*d^2 - 2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) + 1/
4*(24*(x*e + d)^(7/2)*c^5*d^3*e - 72*(x*e + d)^(5/2)*c^5*d^4*e + 72*(x*e + d)^(3
/2)*c^5*d^5*e - 24*sqrt(x*e + d)*c^5*d^6*e - 36*(x*e + d)^(7/2)*b*c^4*d^2*e^2 +
144*(x*e + d)^(5/2)*b*c^4*d^3*e^2 - 180*(x*e + d)^(3/2)*b*c^4*d^4*e^2 + 72*sqrt(
x*e + d)*b*c^4*d^5*e^2 + 6*(x*e + d)^(7/2)*b^2*c^3*d*e^3 - 73*(x*e + d)^(5/2)*b^
2*c^3*d^2*e^3 + 136*(x*e + d)^(3/2)*b^2*c^3*d^3*e^3 - 69*sqrt(x*e + d)*b^2*c^3*d
^4*e^3 + 3*(x*e + d)^(7/2)*b^3*c^2*e^4 + (x*e + d)^(5/2)*b^3*c^2*d*e^4 - 24*(x*e
+ d)^(3/2)*b^3*c^2*d^2*e^4 + 18*sqrt(x*e + d)*b^3*c^2*d^3*e^4 + 6*(x*e + d)^(5/
2)*b^4*c*e^5 - 10*(x*e + d)^(3/2)*b^4*c*d*e^5 + 8*sqrt(x*e + d)*b^4*c*d^2*e^5 +
3*(x*e + d)^(3/2)*b^5*e^6 - 5*sqrt(x*e + d)*b^5*d*e^6)/((b^4*c^2*d^4 - 2*b^5*c*d
^3*e + b^6*d^2*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b
*d*e)^2) + 3/4*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))
/(b^5*sqrt(-d)*d^2)