3.1025 \(\int x \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=98 \[ -\frac{3 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}} \]

[Out]

(3*a*x^2)/(5*(a + b*x^4)^(1/4)) + (x^2*(a + b*x^4)^(3/4))/5 - (3*a^(3/2)*(1 + (b
*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*Sqrt[b]*(a + b*
x^4)^(1/4))

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Rubi [A]  time = 0.122223, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{3 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^4)^(3/4),x]

[Out]

(3*a*x^2)/(5*(a + b*x^4)^(1/4)) + (x^2*(a + b*x^4)^(3/4))/5 - (3*a^(3/2)*(1 + (b
*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*Sqrt[b]*(a + b*
x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 a^{2} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{10} + \frac{3 a x^{2}}{5 \sqrt [4]{a + b x^{4}}} + \frac{x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*Integral((a + b*x**2)**(-5/4), (x, x**2))/10 + 3*a*x**2/(5*(a + b*x**4)*
*(1/4)) + x**2*(a + b*x**4)**(3/4)/5

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Mathematica [C]  time = 0.0571778, size = 64, normalized size = 0.65 \[ \frac{x^2 \left (3 a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+2 \left (a+b x^4\right )\right )}{10 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(2*(a + b*x^4) + 3*a*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2,
-((b*x^4)/a)]))/(10*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int x \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^4+a)^(3/4),x)

[Out]

int(x*(b*x^4+a)^(3/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(3/4)*x, x)

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Sympy [A]  time = 3.14084, size = 29, normalized size = 0.3 \[ \frac{a^{\frac{3}{4}} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/4)*x**2*hyper((-3/4, 1/2), (3/2,), b*x**4*exp_polar(I*pi)/a)/2

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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((b*x^4 + a)^(3/4)*x,x, algorithm="giac")

[Out]

Exception raised: TypeError