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

Optimal. Leaf size=253 \[ \frac{3 x \left (a+b x^2\right )}{b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{3 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a x+b x^3}}-\frac{3 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{7/4} \sqrt{a x+b x^3}}-\frac{x^2}{b \sqrt{a x+b x^3}} \]

[Out]

-(x^2/(b*Sqrt[a*x + b*x^3])) + (3*x*(a + b*x^2))/(b^(3/2)*(Sqrt[a] + Sqrt[b]*x)*
Sqrt[a*x + b*x^3]) - (3*a^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(
Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^
(7/4)*Sqrt[a*x + b*x^3]) + (3*a^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*
x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2
])/(2*b^(7/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.436303, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{3 x \left (a+b x^2\right )}{b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{3 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 b^{7/4} \sqrt{a x+b x^3}}-\frac{3 \sqrt [4]{a} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{b^{7/4} \sqrt{a x+b x^3}}-\frac{x^2}{b \sqrt{a x+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a*x + b*x^3)^(3/2),x]

[Out]

-(x^2/(b*Sqrt[a*x + b*x^3])) + (3*x*(a + b*x^2))/(b^(3/2)*(Sqrt[a] + Sqrt[b]*x)*
Sqrt[a*x + b*x^3]) - (3*a^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(
Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^
(7/4)*Sqrt[a*x + b*x^3]) + (3*a^(1/4)*Sqrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*
x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2
])/(2*b^(7/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 41.5544, size = 235, normalized size = 0.93 \[ - \frac{3 \sqrt [4]{a} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{b^{\frac{7}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{3 \sqrt [4]{a} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 b^{\frac{7}{4}} \sqrt{x} \left (a + b x^{2}\right )} - \frac{x^{2}}{b \sqrt{a x + b x^{3}}} + \frac{3 \sqrt{a x + b x^{3}}}{b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sq
rt(a*x + b*x**3)*elliptic_e(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(b**(7/4)*sq
rt(x)*(a + b*x**2)) + 3*a**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sq
rt(a) + sqrt(b)*x)*sqrt(a*x + b*x**3)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4
)), 1/2)/(2*b**(7/4)*sqrt(x)*(a + b*x**2)) - x**2/(b*sqrt(a*x + b*x**3)) + 3*sqr
t(a*x + b*x**3)/(b**(3/2)*(sqrt(a) + sqrt(b)*x))

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Mathematica [C]  time = 0.135128, size = 161, normalized size = 0.64 \[ -\frac{x \left (3 \sqrt{a} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-3 \sqrt{a} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{b^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.028, size = 182, normalized size = 0.7 \[ -{\frac{{x}^{2}}{b}{\frac{1}{\sqrt{ \left ({x}^{2}+{\frac{a}{b}} \right ) xb}}}}+{\frac{3}{2\,{b}^{2}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a*x)^(3/2),x)

[Out]

-x^2/b/((x^2+a/b)*x*b)^(1/2)+3/2/b^2*(-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)*(-2/b*(-a*b)^(1/2)*EllipticE(((x+1/b*(-a*b)^(1/2))*b/(-
a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/b*(-a*b)^(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^4/(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. \[{\rm integral}\left (\frac{x^{3}}{\sqrt{b x^{3} + a x}{\left (b x^{2} + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^3 + a*x)^(3/2), x)