3.52 \(\int \frac{x^4}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=97 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{7/2}}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}-\frac{2 x^3}{c \sqrt{b x+c x^2}} \]

[Out]

(-2*x^3)/(c*Sqrt[b*x + c*x^2]) - (15*b*Sqrt[b*x + c*x^2])/(4*c^3) + (5*x*Sqrt[b*
x + c*x^2])/(2*c^2) + (15*b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(7/2)
)

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Rubi [A]  time = 0.12834, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{7/2}}-\frac{15 b \sqrt{b x+c x^2}}{4 c^3}+\frac{5 x \sqrt{b x+c x^2}}{2 c^2}-\frac{2 x^3}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*x^3)/(c*Sqrt[b*x + c*x^2]) - (15*b*Sqrt[b*x + c*x^2])/(4*c^3) + (5*x*Sqrt[b*
x + c*x^2])/(2*c^2) + (15*b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*c^(7/2)
)

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Rubi in Sympy [A]  time = 15.1391, size = 90, normalized size = 0.93 \[ \frac{15 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{7}{2}}} - \frac{15 b \sqrt{b x + c x^{2}}}{4 c^{3}} - \frac{2 x^{3}}{c \sqrt{b x + c x^{2}}} + \frac{5 x \sqrt{b x + c x^{2}}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(c*x**2+b*x)**(3/2),x)

[Out]

15*b**2*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(4*c**(7/2)) - 15*b*sqrt(b*x + c*x**
2)/(4*c**3) - 2*x**3/(c*sqrt(b*x + c*x**2)) + 5*x*sqrt(b*x + c*x**2)/(2*c**2)

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Mathematica [A]  time = 0.0761915, size = 90, normalized size = 0.93 \[ \frac{\sqrt{c} x \left (-15 b^2-5 b c x+2 c^2 x^2\right )+15 b^2 \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{4 c^{7/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(b*x + c*x^2)^(3/2),x]

[Out]

(Sqrt[c]*x*(-15*b^2 - 5*b*c*x + 2*c^2*x^2) + 15*b^2*Sqrt[x]*Sqrt[b + c*x]*Log[c*
Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(4*c^(7/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.008, size = 93, normalized size = 1. \[{\frac{{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{5\,b{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{15\,{b}^{2}x}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{15\,{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^2+b*x)^(3/2),x)

[Out]

1/2*x^3/c/(c*x^2+b*x)^(1/2)-5/4*b/c^2*x^2/(c*x^2+b*x)^(1/2)-15/4*b^2/c^3/(c*x^2+
b*x)^(1/2)*x+15/8*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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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(x^4/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241064, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{c x^{2} + b x} b^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (2 \, c^{2} x^{3} - 5 \, b c x^{2} - 15 \, b^{2} x\right )} \sqrt{c}}{8 \, \sqrt{c x^{2} + b x} c^{\frac{7}{2}}}, \frac{15 \, \sqrt{c x^{2} + b x} b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (2 \, c^{2} x^{3} - 5 \, b c x^{2} - 15 \, b^{2} x\right )} \sqrt{-c}}{4 \, \sqrt{c x^{2} + b x} \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(15*sqrt(c*x^2 + b*x)*b^2*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c)
+ 2*(2*c^2*x^3 - 5*b*c*x^2 - 15*b^2*x)*sqrt(c))/(sqrt(c*x^2 + b*x)*c^(7/2)), 1/4
*(15*sqrt(c*x^2 + b*x)*b^2*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (2*c^2*x^3
- 5*b*c*x^2 - 15*b^2*x)*sqrt(-c))/(sqrt(c*x^2 + b*x)*sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(x**4/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError