3.1027 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^7} \, dx\)

Optimal. Leaf size=125 \[ -\frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{a} \sqrt [4]{a+b x^4}}+\frac{b^2 x^2}{4 a \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{6 x^6}-\frac{b \left (a+b x^4\right )^{3/4}}{4 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.16385, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{a} \sqrt [4]{a+b x^4}}+\frac{b^2 x^2}{4 a \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{6 x^6}-\frac{b \left (a+b x^4\right )^{3/4}}{4 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^7,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{6 x^{6}} + \frac{b^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{8 a} - \frac{b \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**7,x)

[Out]

-(a + b*x**4)**(3/4)/(6*x**6) + b**2*Integral((a + b*x**2)**(-1/4), (x, x**2))/(
8*a) - b*(a + b*x**4)**(3/4)/(4*a*x**2)

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Mathematica [C]  time = 0.0518424, size = 86, normalized size = 0.69 \[ \frac{3 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (2 a^2+5 a b x^4+3 b^2 x^8\right )}{24 a x^6 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^7,x]

[Out]

(-2*(2*a^2 + 5*a*b*x^4 + 3*b^2*x^8) + 3*b^2*x^8*(1 + (b*x^4)/a)^(1/4)*Hypergeome
tric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)])/(24*a*x^6*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^7,x)

[Out]

int((b*x^4+a)^(3/4)/x^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^7,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^7,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^7, x)

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Sympy [A]  time = 6.16067, size = 34, normalized size = 0.27 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**7,x)

[Out]

-a**(3/4)*hyper((-3/2, -3/4), (-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. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^7,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^7, x)