3.1038 \(\int x^{10} \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=150 \[ \frac{3 a^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{80 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b} \]

[Out]

(3*a^3*x^3)/(80*b^2*(a + b*x^4)^(1/4)) - (a^2*x^3*(a + b*x^4)^(3/4))/(40*b^2) +
(3*a*x^7*(a + b*x^4)^(3/4))/(140*b) + (x^11*(a + b*x^4)^(3/4))/14 + (3*a^(7/2)*(
1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(80*b^(5/2
)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.21566, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{3 a^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{80 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{3 a^3 x^3}{80 b^2 \sqrt [4]{a+b x^4}}-\frac{a^2 x^3 \left (a+b x^4\right )^{3/4}}{40 b^2}+\frac{1}{14} x^{11} \left (a+b x^4\right )^{3/4}+\frac{3 a x^7 \left (a+b x^4\right )^{3/4}}{140 b} \]

Antiderivative was successfully verified.

[In]  Int[x^10*(a + b*x^4)^(3/4),x]

[Out]

(3*a^3*x^3)/(80*b^2*(a + b*x^4)^(1/4)) - (a^2*x^3*(a + b*x^4)^(3/4))/(40*b^2) +
(3*a*x^7*(a + b*x^4)^(3/4))/(140*b) + (x^11*(a + b*x^4)^(3/4))/14 + (3*a^(7/2)*(
1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(80*b^(5/2
)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{4} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{160 b^{3} \sqrt [4]{a + b x^{4}}} + \frac{3 a^{3} x^{3}}{80 b^{2} \sqrt [4]{a + b x^{4}}} - \frac{a^{2} x^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{40 b^{2}} + \frac{3 a x^{7} \left (a + b x^{4}\right )^{\frac{3}{4}}}{140 b} + \frac{x^{11} \left (a + b x^{4}\right )^{\frac{3}{4}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(b*x**4+a)**(3/4),x)

[Out]

3*a**4*x*(a/(b*x**4) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-5/4), (x, x**(-2)))/
(160*b**3*(a + b*x**4)**(1/4)) + 3*a**3*x**3/(80*b**2*(a + b*x**4)**(1/4)) - a**
2*x**3*(a + b*x**4)**(3/4)/(40*b**2) + 3*a*x**7*(a + b*x**4)**(3/4)/(140*b) + x*
*11*(a + b*x**4)**(3/4)/14

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Mathematica [C]  time = 0.0688725, size = 91, normalized size = 0.61 \[ \frac{x^3 \left (7 a^3 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-7 a^3-a^2 b x^4+26 a b^2 x^8+20 b^3 x^{12}\right )}{280 b^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10*(a + b*x^4)^(3/4),x]

[Out]

(x^3*(-7*a^3 - a^2*b*x^4 + 26*a*b^2*x^8 + 20*b^3*x^12 + 7*a^3*(1 + (b*x^4)/a)^(1
/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)]))/(280*b^2*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{x}^{10} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(b*x^4+a)^(3/4),x)

[Out]

int(x^10*(b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{10}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^10,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)*x^10, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{10}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^10,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)*x^10, x)

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Sympy [A]  time = 14.1734, size = 39, normalized size = 0.26 \[ \frac{a^{\frac{3}{4}} x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(b*x**4+a)**(3/4),x)

[Out]

a**(3/4)*x**11*gamma(11/4)*hyper((-3/4, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a
)/(4*gamma(15/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{10}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^10,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)*x^10, x)