3.67 \(\int \frac{x^5}{\left (a x+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{5 a^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a x+b x^3}}+\frac{5 \sqrt{a x+b x^3}}{3 b^2}-\frac{x^3}{b \sqrt{a x+b x^3}} \]

[Out]

-(x^3/(b*Sqrt[a*x + b*x^3])) + (5*Sqrt[a*x + b*x^3])/(3*b^2) - (5*a^(3/4)*Sqrt[x
]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*Ar
cTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*b^(9/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.268411, antiderivative size = 137, 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 \[ -\frac{5 a^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 b^{9/4} \sqrt{a x+b x^3}}+\frac{5 \sqrt{a x+b x^3}}{3 b^2}-\frac{x^3}{b \sqrt{a x+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a*x + b*x^3)^(3/2),x]

[Out]

-(x^3/(b*Sqrt[a*x + b*x^3])) + (5*Sqrt[a*x + b*x^3])/(3*b^2) - (5*a^(3/4)*Sqrt[x
]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*Ar
cTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*b^(9/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 24.9843, size = 129, normalized size = 0.94 \[ - \frac{5 a^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{6 b^{\frac{9}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{x^{3}}{b \sqrt{a x + b x^{3}}} + \frac{5 \sqrt{a x + b x^{3}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a*x)**(3/2),x)

[Out]

-5*a**(3/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sq
rt(a*x + b*x**3)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(6*b**(9/4)*
sqrt(x)*(a + b*x**2)) - x**3/(b*sqrt(a*x + b*x**3)) + 5*sqrt(a*x + b*x**3)/(3*b*
*2)

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Mathematica [C]  time = 0.117299, size = 124, normalized size = 0.91 \[ \frac{x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (5 a+2 b x^2\right )-5 i a x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{3 b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a*x + b*x^3)^(3/2),x]

[Out]

(Sqrt[(I*Sqrt[a])/Sqrt[b]]*x*(5*a + 2*b*x^2) - (5*I)*a*Sqrt[1 + a/(b*x^2)]*x^(3/
2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/(3*Sqrt[(I*Sqrt[
a])/Sqrt[b]]*b^2*Sqrt[x*(a + b*x^2)])

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Maple [A]  time = 0.028, size = 147, normalized size = 1.1 \[{\frac{ax}{{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{2}+{\frac{a}{b}} \right ) xb}}}}+{\frac{2}{3\,{b}^{2}}\sqrt{b{x}^{3}+ax}}-{\frac{5\,a}{6\,{b}^{3}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a*x)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a*x)^(3/2),x, algorithm="fricas")

[Out]

integral(x^4/(sqrt(b*x^3 + a*x)*(b*x^2 + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a*x)**(3/2),x)

[Out]

Integral(x**5/(x*(a + b*x**2))**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^3 + a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^5/(b*x^3 + a*x)^(3/2), x)