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

Optimal. Leaf size=252 \[ \frac{3 c \sqrt{d+e x} (2 c d-b e)}{b^4 (b+c x)}+\frac{c \sqrt{d+e x} (12 c d-7 b e)}{4 b^3 (b+c x)^2}+\frac{\sqrt{d+e x} (8 c d-5 b e)}{4 b^2 x (b+c x)^2}+\frac{3 \sqrt{c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c d-b e}}-\frac{3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 b x^2 (b+c x)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.20163, antiderivative size = 252, 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 \[ \frac{3 c \sqrt{d+e x} (2 c d-b e)}{b^4 (b+c x)}+\frac{c \sqrt{d+e x} (12 c d-7 b e)}{4 b^3 (b+c x)^2}+\frac{\sqrt{d+e x} (8 c d-5 b e)}{4 b^2 x (b+c x)^2}+\frac{3 \sqrt{c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c d-b e}}-\frac{3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5 \sqrt{d}}-\frac{d \sqrt{d+e x}}{2 b x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.348725, size = 197, normalized size = 0.78 \[ \frac{-\frac{3 \left (b^2 e^2-12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}+\frac{3 \sqrt{c} \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c d-b e}}+\frac{b \sqrt{d+e x} \left (b^3 (-(2 d+5 e x))+b^2 c x (8 d-19 e x)-12 b c^2 x^2 (e x-3 d)+24 c^3 d x^3\right )}{x^2 (b+c x)^2}}{4 b^5} \]

Antiderivative was successfully verified.

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

[Out]

((b*Sqrt[d + e*x]*(24*c^3*d*x^3 + b^2*c*x*(8*d - 19*e*x) - 12*b*c^2*x^2*(-3*d +
e*x) - b^3*(2*d + 5*e*x)))/(x^2*(b + c*x)^2) - (3*(16*c^2*d^2 - 12*b*c*d*e + b^2
*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d] + (3*Sqrt[c]*(16*c^2*d^2 - 20*b*c*
d*e + 5*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/Sqrt[c*d - b*
e])/(4*b^5)

_______________________________________________________________________________________

Maple [A]  time = 0.032, size = 414, normalized size = 1.6 \[ -{\frac{7\,{e}^{2}{c}^{2}}{4\,{b}^{3} \left ( cex+be \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{e{c}^{3} \left ( ex+d \right ) ^{3/2}d}{{b}^{4} \left ( cex+be \right ) ^{2}}}-{\frac{9\,{e}^{3}c}{4\,{b}^{2} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}+{\frac{21\,{e}^{2}{c}^{2}d}{4\,{b}^{3} \left ( cex+be \right ) ^{2}}\sqrt{ex+d}}-3\,{\frac{e{c}^{3}\sqrt{ex+d}{d}^{2}}{{b}^{4} \left ( cex+be \right ) ^{2}}}-{\frac{15\,{e}^{2}c}{4\,{b}^{3}}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+15\,{\frac{e{c}^{2}d}{{b}^{4}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-12\,{\frac{{d}^{2}{c}^{3}}{{b}^{5}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{5}{4\,{b}^{3}{x}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+3\,{\frac{ \left ( ex+d \right ) ^{3/2}cd}{e{b}^{4}{x}^{2}}}-3\,{\frac{c\sqrt{ex+d}{d}^{2}}{e{b}^{4}{x}^{2}}}+{\frac{3\,d}{4\,{b}^{3}{x}^{2}}\sqrt{ex+d}}-{\frac{3\,{e}^{2}}{4\,{b}^{3}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{d}}}}+9\,{\frac{e\sqrt{d}c}{{b}^{4}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-12\,{\frac{{d}^{3/2}{c}^{2}}{{b}^{5}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.336439, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(3*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 20*
b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e + 5*b^4*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*d - 12*(2*b*c^3*d - b^2*c^2*e)*x
^3 - (36*b^2*c^2*d - 19*b^3*c*e)*x^2 - (8*b^3*c*d - 5*b^4*e)*x)*sqrt(e*x + d)*sq
rt(d) + 3*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*
b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*lo
g(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/((b^5*c^2*x^4 + 2*b^6*c*x^3 + b^
7*x^2)*sqrt(d)), 1/8*(6*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16
*b*c^3*d^2 - 20*b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e
+ 5*b^4*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*(2*b^4*d - 12*(2*b*c^3*d - b^2*c^2*e)*x^3 - (36*b
^2*c^2*d - 19*b^3*c*e)*x^2 - (8*b^3*c*d - 5*b^4*e)*x)*sqrt(e*x + d)*sqrt(d) + 3*
((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*b^2*c^2*d*
e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*log(((e*x +
2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/((b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)*sqr
t(d)), 1/8*(3*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2
- 20*b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e + 5*b^4*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)*sq
rt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(2*b^4*d - 12*(2*b*c^3*d - b^2*c
^2*e)*x^3 - (36*b^2*c^2*d - 19*b^3*c*e)*x^2 - (8*b^3*c*d - 5*b^4*e)*x)*sqrt(e*x
+ d)*sqrt(-d) + 6*((16*c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d
^2 - 12*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)
*x^2)*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/((b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)
*sqrt(-d)), 1/4*(3*((16*c^4*d^2 - 20*b*c^3*d*e + 5*b^2*c^2*e^2)*x^4 + 2*(16*b*c^
3*d^2 - 20*b^2*c^2*d*e + 5*b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 20*b^3*c*d*e + 5*b
^4*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*d - 12*(2*b*c^3*d - b^2*c^2*e)*x^3 - (36*b^2*c^2
*d - 19*b^3*c*e)*x^2 - (8*b^3*c*d - 5*b^4*e)*x)*sqrt(e*x + d)*sqrt(-d) + 3*((16*
c^4*d^2 - 12*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 12*b^2*c^2*d*e + b
^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 12*b^3*c*d*e + b^4*e^2)*x^2)*arctan(d/(sqrt(e*
x + d)*sqrt(-d))))/((b^5*c^2*x^4 + 2*b^6*c*x^3 + b^7*x^2)*sqrt(-d))]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229615, size = 529, normalized size = 2.1 \[ -\frac{3 \,{\left (16 \, c^{3} d^{2} - 20 \, b c^{2} d e + 5 \, b^{2} c e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{4 \, \sqrt{-c^{2} d + b c e} b^{5}} + \frac{3 \,{\left (16 \, c^{2} d^{2} - 12 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{4 \, b^{5} \sqrt{-d}} + \frac{24 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d e - 72 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{2} e + 72 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e - 24 \, \sqrt{x e + d} c^{3} d^{4} e - 12 \,{\left (x e + d\right )}^{\frac{7}{2}} b c^{2} e^{2} + 72 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} d e^{2} - 108 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d^{2} e^{2} + 48 \, \sqrt{x e + d} b c^{2} d^{3} e^{2} - 19 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} c e^{3} + 46 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c d e^{3} - 27 \, \sqrt{x e + d} b^{2} c d^{2} e^{3} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{4} + 3 \, \sqrt{x e + d} b^{3} d e^{4}}{4 \,{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/4*(16*c^3*d^2 - 20*b*c^2*d*e + 5*b^2*c*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*
d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5) + 3/4*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2)
*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*(x*e + d)^(7/2)*c^3*d*e
- 72*(x*e + d)^(5/2)*c^3*d^2*e + 72*(x*e + d)^(3/2)*c^3*d^3*e - 24*sqrt(x*e + d
)*c^3*d^4*e - 12*(x*e + d)^(7/2)*b*c^2*e^2 + 72*(x*e + d)^(5/2)*b*c^2*d*e^2 - 10
8*(x*e + d)^(3/2)*b*c^2*d^2*e^2 + 48*sqrt(x*e + d)*b*c^2*d^3*e^2 - 19*(x*e + d)^
(5/2)*b^2*c*e^3 + 46*(x*e + d)^(3/2)*b^2*c*d*e^3 - 27*sqrt(x*e + d)*b^2*c*d^2*e^
3 - 5*(x*e + d)^(3/2)*b^3*e^4 + 3*sqrt(x*e + d)*b^3*d*e^4)/(((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)