3.1057 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^7} \, dx\)

Optimal. Leaf size=98 \[ \frac{5 \sqrt{a} b^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6}-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2} \]

[Out]

(-5*b*(a + b*x^4)^(1/4))/(12*x^2) - (a + b*x^4)^(5/4)/(6*x^6) + (5*Sqrt[a]*b^(3/
2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*(a +
b*x^4)^(3/4))

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Rubi [A]  time = 0.14064, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 \sqrt{a} b^{3/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}-\frac{\left (a+b x^4\right )^{5/4}}{6 x^6}-\frac{5 b \sqrt [4]{a+b x^4}}{12 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^7,x]

[Out]

(-5*b*(a + b*x^4)^(1/4))/(12*x^2) - (a + b*x^4)^(5/4)/(6*x^6) + (5*Sqrt[a]*b^(3/
2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*(a +
b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 13.7797, size = 87, normalized size = 0.89 \[ \frac{5 \sqrt{a} b^{\frac{3}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{12 \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{5 b \sqrt [4]{a + b x^{4}}}{12 x^{2}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**7,x)

[Out]

5*sqrt(a)*b**(3/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2
, 2)/(12*(a + b*x**4)**(3/4)) - 5*b*(a + b*x**4)**(1/4)/(12*x**2) - (a + b*x**4)
**(5/4)/(6*x**6)

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Mathematica [C]  time = 0.0571416, size = 85, normalized size = 0.87 \[ \frac{5 b^2 x^2 \left (\frac{a+b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )}{24 \left (a+b x^4\right )^{3/4}}+\left (-\frac{a}{6 x^6}-\frac{7 b}{12 x^2}\right ) \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^7,x]

[Out]

(-a/(6*x^6) - (7*b)/(12*x^2))*(a + b*x^4)^(1/4) + (5*b^2*x^2*((a + b*x^4)/a)^(3/
4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)])/(24*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^7,x)

[Out]

int((b*x^4+a)^(5/4)/x^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^7,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)/x^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^7,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(5/4)/x^7, x)

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Sympy [A]  time = 8.95094, size = 34, normalized size = 0.35 \[ - \frac{a^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**7,x)

[Out]

-a**(5/4)*hyper((-3/2, -5/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*x**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^7,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^7, x)