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

Optimal. Leaf size=94 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]

[Out]

(-2*x^4)/(3*b*(a*x + b*x^2)^(3/2)) - (10*x^2)/(3*b^2*Sqrt[a*x + b*x^2]) + (5*Sqr
t[a*x + b*x^2])/b^3 - (5*a*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/b^(7/2)

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Rubi [A]  time = 0.130669, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{b^{7/2}}+\frac{5 \sqrt{a x+b x^2}}{b^3}-\frac{10 x^2}{3 b^2 \sqrt{a x+b x^2}}-\frac{2 x^4}{3 b \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a*x + b*x^2)^(5/2),x]

[Out]

(-2*x^4)/(3*b*(a*x + b*x^2)^(3/2)) - (10*x^2)/(3*b^2*Sqrt[a*x + b*x^2]) + (5*Sqr
t[a*x + b*x^2])/b^3 - (5*a*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/b^(7/2)

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Rubi in Sympy [A]  time = 15.0416, size = 87, normalized size = 0.93 \[ - \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{b^{\frac{7}{2}}} - \frac{2 x^{4}}{3 b \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{10 x^{2}}{3 b^{2} \sqrt{a x + b x^{2}}} + \frac{5 \sqrt{a x + b x^{2}}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/b**(7/2) - 2*x**4/(3*b*(a*x + b*x**2)**
(3/2)) - 10*x**2/(3*b**2*sqrt(a*x + b*x**2)) + 5*sqrt(a*x + b*x**2)/b**3

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Mathematica [A]  time = 0.0778944, size = 89, normalized size = 0.95 \[ \frac{x \left (\sqrt{b} x \left (15 a^2+20 a b x+3 b^2 x^2\right )-15 a \sqrt{x} (a+b x)^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )\right )}{3 b^{7/2} (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[b]*x*(15*a^2 + 20*a*b*x + 3*b^2*x^2) - 15*a*Sqrt[x]*(a + b*x)^(3/2)*Log
[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]]))/(3*b^(7/2)*(x*(a + b*x))^(3/2))

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Maple [A]  time = 0.008, size = 149, normalized size = 1.6 \[{\frac{{x}^{4}}{b} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{5\,a{x}^{3}}{6\,{b}^{2}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{2}{x}^{2}}{4\,{b}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,{a}^{3}x}{12\,{b}^{4}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,ax}{6\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}+{\frac{5\,{a}^{2}}{12\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+ax}}}}-{\frac{5\,a}{2}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a*x)^(5/2),x)

[Out]

x^4/b/(b*x^2+a*x)^(3/2)+5/6*a/b^2*x^3/(b*x^2+a*x)^(3/2)-5/4*a^2/b^3*x^2/(b*x^2+a
*x)^(3/2)-5/12*a^3/b^4/(b*x^2+a*x)^(3/2)*x+35/6*a/b^3/(b*x^2+a*x)^(1/2)*x+5/12*a
^2/b^4/(b*x^2+a*x)^(1/2)-5/2*a/b^(7/2)*ln((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^5/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240203, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{b x^{2} + a x} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (3 \, b^{2} x^{3} + 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{b}}{6 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}, -\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{b x^{2} + a x} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (3 \, b^{2} x^{3} + 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{-b}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*(a*b*x + a^2)*sqrt(b*x^2 + a*x)*log((2*b*x + a)*sqrt(b) - 2*sqrt(b*x^2
+ a*x)*b) + 2*(3*b^2*x^3 + 20*a*b*x^2 + 15*a^2*x)*sqrt(b))/((b^4*x + a*b^3)*sqrt
(b*x^2 + a*x)*sqrt(b)), -1/3*(15*(a*b*x + a^2)*sqrt(b*x^2 + a*x)*arctan(sqrt(b*x
^2 + a*x)*sqrt(-b)/(b*x)) - (3*b^2*x^3 + 20*a*b*x^2 + 15*a^2*x)*sqrt(-b))/((b^4*
x + a*b^3)*sqrt(b*x^2 + a*x)*sqrt(-b))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError