3.1044 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^{14}} \, dx\)

Optimal. Leaf size=150 \[ \frac{4 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{65 a^{5/2} \sqrt [4]{a+b x^4}}-\frac{4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}+\frac{2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}-\frac{\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac{b \left (a+b x^4\right )^{3/4}}{39 a x^9} \]

[Out]

(-4*b^3)/(65*a^2*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(13*x^13) - (b*(a + b*
x^4)^(3/4))/(39*a*x^9) + (2*b^2*(a + b*x^4)^(3/4))/(65*a^2*x^5) + (4*b^(7/2)*(1
+ a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(65*a^(5/2)*
(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.214469, 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 \[ \frac{4 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{65 a^{5/2} \sqrt [4]{a+b x^4}}-\frac{4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}+\frac{2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}-\frac{\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac{b \left (a+b x^4\right )^{3/4}}{39 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^14,x]

[Out]

(-4*b^3)/(65*a^2*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(13*x^13) - (b*(a + b*
x^4)^(3/4))/(39*a*x^9) + (2*b^2*(a + b*x^4)^(3/4))/(65*a^2*x^5) + (4*b^(7/2)*(1
+ a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(65*a^(5/2)*
(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{13 x^{13}} - \frac{b \left (a + b x^{4}\right )^{\frac{3}{4}}}{39 a x^{9}} + \frac{2 b^{3} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{65 a^{2} \sqrt [4]{a + b x^{4}}} - \frac{4 b^{3}}{65 a^{2} x \sqrt [4]{a + b x^{4}}} + \frac{2 b^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{65 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**14,x)

[Out]

-(a + b*x**4)**(3/4)/(13*x**13) - b*(a + b*x**4)**(3/4)/(39*a*x**9) + 2*b**3*x*(
a/(b*x**4) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-5/4), (x, x**(-2)))/(65*a**2*(
a + b*x**4)**(1/4)) - 4*b**3/(65*a**2*x*(a + b*x**4)**(1/4)) + 2*b**2*(a + b*x**
4)**(3/4)/(65*a**2*x**5)

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Mathematica [C]  time = 0.0641991, size = 104, normalized size = 0.69 \[ \frac{-15 a^4-20 a^3 b x^4+a^2 b^2 x^8+8 b^4 x^{16} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-6 a b^3 x^{12}-12 b^4 x^{16}}{195 a^3 x^{13} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^14,x]

[Out]

(-15*a^4 - 20*a^3*b*x^4 + a^2*b^2*x^8 - 6*a*b^3*x^12 - 12*b^4*x^16 + 8*b^4*x^16*
(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(195*a^3*x
^13*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{14}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^14,x)

[Out]

int((b*x^4+a)^(3/4)/x^14,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^14,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^14, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{14}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^14,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^14, x)

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Sympy [A]  time = 21.6565, size = 46, normalized size = 0.31 \[ \frac{a^{\frac{3}{4}} \Gamma \left (- \frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{13}{4}, - \frac{3}{4} \\ - \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{13} \Gamma \left (- \frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**14,x)

[Out]

a**(3/4)*gamma(-13/4)*hyper((-13/4, -3/4), (-9/4,), b*x**4*exp_polar(I*pi)/a)/(4
*x**13*gamma(-9/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^14,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^14, x)