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

Optimal. Leaf size=69 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}+\frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{2 x^2}{c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0881086, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{5/2}}+\frac{3 \sqrt{b x+c x^2}}{c^2}-\frac{2 x^2}{c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.1309, size = 63, normalized size = 0.91 \[ - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{2 x^{2}}{c \sqrt{b x + c x^{2}}} + \frac{3 \sqrt{b x + c x^{2}}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0588593, size = 73, normalized size = 1.06 \[ \frac{\sqrt{c} x (3 b+c x)-3 b \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*x*(3*b + c*x) - 3*b*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[
b + c*x]])/(c^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.007, size = 68, normalized size = 1. \[{\frac{{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+3\,{\frac{bx}{{c}^{2}\sqrt{c{x}^{2}+bx}}}-{\frac{3\,b}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^2+b*x)^(3/2),x)

[Out]

x^2/c/(c*x^2+b*x)^(1/2)+3*b/c^2/(c*x^2+b*x)^(1/2)*x-3/2*b/c^(5/2)*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^3/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243964, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{c x^{2} + b x} b \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (c x^{2} + 3 \, b x\right )} \sqrt{c}}{2 \, \sqrt{c x^{2} + b x} c^{\frac{5}{2}}}, -\frac{3 \, \sqrt{c x^{2} + b x} b \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (c x^{2} + 3 \, b x\right )} \sqrt{-c}}{\sqrt{c x^{2} + b x} \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(3*sqrt(c*x^2 + b*x)*b*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2
*(c*x^2 + 3*b*x)*sqrt(c))/(sqrt(c*x^2 + b*x)*c^(5/2)), -(3*sqrt(c*x^2 + b*x)*b*a
rctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (c*x^2 + 3*b*x)*sqrt(-c))/(sqrt(c*x^2
+ b*x)*sqrt(-c)*c^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/(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^3/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError