3.1023 \(\int x^9 \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{4 a^{7/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{4 a^3 x^2}{65 b^2 \sqrt [4]{a+b x^4}}-\frac{2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac{1}{13} x^{10} \left (a+b x^4\right )^{3/4}+\frac{a x^6 \left (a+b x^4\right )^{3/4}}{39 b} \]

[Out]

(4*a^3*x^2)/(65*b^2*(a + b*x^4)^(1/4)) - (2*a^2*x^2*(a + b*x^4)^(3/4))/(65*b^2)
+ (a*x^6*(a + b*x^4)^(3/4))/(39*b) + (x^10*(a + b*x^4)^(3/4))/13 - (4*a^(7/2)*(1
+ (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(65*b^(5/2)*(
a + b*x^4)^(1/4))

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Rubi [A]  time = 0.220688, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{4 a^{7/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{65 b^{5/2} \sqrt [4]{a+b x^4}}+\frac{4 a^3 x^2}{65 b^2 \sqrt [4]{a+b x^4}}-\frac{2 a^2 x^2 \left (a+b x^4\right )^{3/4}}{65 b^2}+\frac{1}{13} x^{10} \left (a+b x^4\right )^{3/4}+\frac{a x^6 \left (a+b x^4\right )^{3/4}}{39 b} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a + b*x^4)^(3/4),x]

[Out]

(4*a^3*x^2)/(65*b^2*(a + b*x^4)^(1/4)) - (2*a^2*x^2*(a + b*x^4)^(3/4))/(65*b^2)
+ (a*x^6*(a + b*x^4)^(3/4))/(39*b) + (x^10*(a + b*x^4)^(3/4))/13 - (4*a^(7/2)*(1
+ (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(65*b^(5/2)*(
a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{4} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{65 b^{2}} + \frac{4 a^{3} x^{2}}{65 b^{2} \sqrt [4]{a + b x^{4}}} - \frac{2 a^{2} x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{65 b^{2}} + \frac{a x^{6} \left (a + b x^{4}\right )^{\frac{3}{4}}}{39 b} + \frac{x^{10} \left (a + b x^{4}\right )^{\frac{3}{4}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(b*x**4+a)**(3/4),x)

[Out]

-2*a**4*Integral((a + b*x**2)**(-5/4), (x, x**2))/(65*b**2) + 4*a**3*x**2/(65*b*
*2*(a + b*x**4)**(1/4)) - 2*a**2*x**2*(a + b*x**4)**(3/4)/(65*b**2) + a*x**6*(a
+ b*x**4)**(3/4)/(39*b) + x**10*(a + b*x**4)**(3/4)/13

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Mathematica [C]  time = 0.0650817, size = 91, normalized size = 0.61 \[ \frac{x^2 \left (6 a^3 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-6 a^3-a^2 b x^4+20 a b^2 x^8+15 b^3 x^{12}\right )}{195 b^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(a + b*x^4)^(3/4),x]

[Out]

(x^2*(-6*a^3 - a^2*b*x^4 + 20*a*b^2*x^8 + 15*b^3*x^12 + 6*a^3*(1 + (b*x^4)/a)^(1
/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)]))/(195*b^2*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{x}^{9} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(b*x^4+a)^(3/4),x)

[Out]

int(x^9*(b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^9,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)*x^9, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{9}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^9,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)*x^9, x)

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Sympy [A]  time = 11.822, size = 29, normalized size = 0.19 \[ \frac{a^{\frac{3}{4}} x^{10}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(b*x**4+a)**(3/4),x)

[Out]

a**(3/4)*x**10*hyper((-3/4, 5/2), (7/2,), b*x**4*exp_polar(I*pi)/a)/10

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^9,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)*x^9, x)