3.62 \(\int \frac{x^4}{\left (a x+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{5/2}}-\frac{2 x}{b^2 \sqrt{a x+b x^2}}-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.09533, antiderivative size = 71, 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 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{5/2}}-\frac{2 x}{b^2 \sqrt{a x+b x^2}}-\frac{2 x^3}{3 b \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a*x + b*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7101, size = 65, normalized size = 0.92 \[ - \frac{2 x^{3}}{3 b \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{2 x}{b^{2} \sqrt{a x + b x^{2}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a*x)**(5/2),x)

[Out]

-2*x**3/(3*b*(a*x + b*x**2)**(3/2)) - 2*x/(b**2*sqrt(a*x + b*x**2)) + 2*atanh(sq
rt(b)*x/sqrt(a*x + b*x**2))/b**(5/2)

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Mathematica [A]  time = 0.067, size = 78, normalized size = 1.1 \[ \frac{x \left (6 \sqrt{x} (a+b x)^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-2 \sqrt{b} x (3 a+4 b x)\right )}{3 b^{5/2} (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a*x + b*x^2)^(5/2),x]

[Out]

(x*(-2*Sqrt[b]*x*(3*a + 4*b*x) + 6*Sqrt[x]*(a + b*x)^(3/2)*Log[b*Sqrt[x] + Sqrt[
b]*Sqrt[a + b*x]]))/(3*b^(5/2)*(x*(a + b*x))^(3/2))

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Maple [B]  time = 0.009, size = 123, normalized size = 1.7 \[ -{\frac{{x}^{3}}{3\,b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{a{x}^{2}}{2\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{{a}^{2}x}{6\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{7\,x}{3\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{a}{6\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{1\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a*x)^(5/2),x)

[Out]

-1/3*x^3/b/(b*x^2+a*x)^(3/2)+1/2*a/b^2*x^2/(b*x^2+a*x)^(3/2)+1/6*a^2/b^3/(b*x^2+
a*x)^(3/2)*x-7/3*x/b^2/(b*x^2+a*x)^(1/2)-1/6*a/b^3/(b*x^2+a*x)^(1/2)+1/b^(5/2)*l
n((1/2*a+b*x)/b^(1/2)+(b*x^2+a*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/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237237, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{b x^{2} + a x}{\left (b x + a\right )} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right ) - 2 \,{\left (4 \, b x^{2} + 3 \, a x\right )} \sqrt{b}}{3 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}, \frac{2 \,{\left (3 \, \sqrt{b x^{2} + a x}{\left (b x + a\right )} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (4 \, b x^{2} + 3 \, a x\right )} \sqrt{-b}\right )}}{3 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^2 + a*x)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*sqrt(b*x^2 + a*x)*(b*x + a)*log((2*b*x + a)*sqrt(b) + 2*sqrt(b*x^2 + a*x
)*b) - 2*(4*b*x^2 + 3*a*x)*sqrt(b))/((b^3*x + a*b^2)*sqrt(b*x^2 + a*x)*sqrt(b)),
2/3*(3*sqrt(b*x^2 + a*x)*(b*x + a)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) - (
4*b*x^2 + 3*a*x)*sqrt(-b))/((b^3*x + a*b^2)*sqrt(b*x^2 + a*x)*sqrt(-b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a*x)**(5/2),x)

[Out]

Integral(x**4/(x*(a + b*x))**(5/2), 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/(b*x^2 + a*x)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError