3.371 \(\int \frac{(d+e x)^{5/2}}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=159 \[ -\frac{(c d-b e)^{3/2} (b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}+\frac{d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{d+e x} (c d-b e) (2 c d-b e)}{b^2 c (b+c x)}-\frac{d (d+e x)^{3/2}}{b x (b+c x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.622378, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(c d-b e)^{3/2} (b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{3/2}}+\frac{d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{d+e x} (c d-b e) (2 c d-b e)}{b^2 c (b+c x)}-\frac{d (d+e x)^{3/2}}{b x (b+c x)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(b*x + c*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 59.733, size = 138, normalized size = 0.87 \[ - \frac{d \left (d + e x\right )^{\frac{3}{2}}}{b x \left (b + c x\right )} - \frac{\sqrt{d + e x} \left (b e - 2 c d\right ) \left (b e - c d\right )}{b^{2} c \left (b + c x\right )} - \frac{d^{\frac{3}{2}} \left (5 b e - 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{\frac{3}{2}} \left (b e + 4 c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(c*x**2+b*x)**2,x)

[Out]

-d*(d + e*x)**(3/2)/(b*x*(b + c*x)) - sqrt(d + e*x)*(b*e - 2*c*d)*(b*e - c*d)/(b
**2*c*(b + c*x)) - d**(3/2)*(5*b*e - 4*c*d)*atanh(sqrt(d + e*x)/sqrt(d))/b**3 +
(b*e - c*d)**(3/2)*(b*e + 4*c*d)*atan(sqrt(c)*sqrt(d + e*x)/sqrt(b*e - c*d))/(b*
*3*c**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.29066, size = 132, normalized size = 0.83 \[ \frac{-\frac{(c d-b e)^{3/2} (b e+4 c d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{3/2}}+d^{3/2} (4 c d-5 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )-b \sqrt{d+e x} \left (\frac{(c d-b e)^2}{c (b+c x)}+\frac{d^2}{x}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(b*x + c*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.025, size = 313, normalized size = 2. \[ -{\frac{{e}^{3}}{c \left ( cex+be \right ) }\sqrt{ex+d}}+2\,{\frac{{e}^{2}\sqrt{ex+d}d}{b \left ( cex+be \right ) }}-{\frac{ce{d}^{2}}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{3}}{c}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}+2\,{\frac{d{e}^{2}}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-7\,{\frac{ce{d}^{2}}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{3}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{{d}^{2}}{{b}^{2}x}\sqrt{ex+d}}-5\,{\frac{e{d}^{3/2}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{5/2}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(c*x^2+b*x)^2,x)

[Out]

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

_______________________________________________________________________________________

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)^(5/2)/(c*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.340805, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + b*x)^2,x, algorithm="fricas")

[Out]

[-1/2*(((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e -
b^3*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sq
rt((c*d - b*e)/c))/(c*x + b)) + ((4*c^3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 -
5*b^2*c*d*e)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(b^2*c*
d^2 + (2*b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4
*c*x), -1/2*(2*((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2
*c*d*e - b^3*e^2)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)
/c)) + ((4*c^3*d^2 - 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(d)*l
og((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(b^2*c*d^2 + (2*b*c^2*d^2 - 2*b^
2*c*d*e + b^3*e^2)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), 1/2*(2*((4*c^3*d^2
- 5*b*c^2*d*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(-d)*arctan(sqrt(e*x +
d)/sqrt(-d)) - ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2
*c*d*e - b^3*e^2)*x)*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*(b^2*c*d^2 + (2*b*c^2*d^2 - 2*b^2*c*d*
e + b^3*e^2)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), (((4*c^3*d^2 - 5*b*c^2*d
*e)*x^2 + (4*b*c^2*d^2 - 5*b^2*c*d*e)*x)*sqrt(-d)*arctan(sqrt(e*x + d)/sqrt(-d))
- ((4*c^3*d^2 - 3*b*c^2*d*e - b^2*c*e^2)*x^2 + (4*b*c^2*d^2 - 3*b^2*c*d*e - b^3
*e^2)*x)*sqrt(-(c*d - b*e)/c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) - (b^2*
c*d^2 + (2*b*c^2*d^2 - 2*b^2*c*d*e + b^3*e^2)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b
^4*c*x)]

_______________________________________________________________________________________

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)**(5/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21847, size = 369, normalized size = 2.32 \[ -\frac{{\left (4 \, c d^{3} - 5 \, b d^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (4 \, c^{3} d^{3} - 7 \, b c^{2} d^{2} e + 2 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e - 2 \, \sqrt{x e + d} c^{2} d^{3} e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} b c d e^{2} + 3 \, \sqrt{x e + d} b c d^{2} e^{2} +{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{3} - \sqrt{x e + d} b^{2} d e^{3}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*x^2 + b*x)^2,x, algorithm="giac")

[Out]

-(4*c*d^3 - 5*b*d^2*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) + (4*c^3*d^
3 - 7*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d
+ b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c) - (2*(x*e + d)^(3/2)*c^2*d^2*e - 2*sqrt(x
*e + d)*c^2*d^3*e - 2*(x*e + d)^(3/2)*b*c*d*e^2 + 3*sqrt(x*e + d)*b*c*d^2*e^2 +
(x*e + d)^(3/2)*b^2*e^3 - sqrt(x*e + d)*b^2*d*e^3)/(((x*e + d)^2*c - 2*(x*e + d)
*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c)