3.1059 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^{15}} \, dx\)

Optimal. Leaf size=146 \[ \frac{5 b^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{336 a^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{5 b^3 \sqrt [4]{a+b x^4}}{336 a^2 x^2}-\frac{b^2 \sqrt [4]{a+b x^4}}{168 a x^6}-\frac{\left (a+b x^4\right )^{5/4}}{14 x^{14}}-\frac{b \sqrt [4]{a+b x^4}}{28 x^{10}} \]

[Out]

-(b*(a + b*x^4)^(1/4))/(28*x^10) - (b^2*(a + b*x^4)^(1/4))/(168*a*x^6) + (5*b^3*
(a + b*x^4)^(1/4))/(336*a^2*x^2) - (a + b*x^4)^(5/4)/(14*x^14) + (5*b^(7/2)*(1 +
(b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(336*a^(3/2)*(a
+ b*x^4)^(3/4))

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Rubi [A]  time = 0.228894, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 b^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{336 a^{3/2} \left (a+b x^4\right )^{3/4}}+\frac{5 b^3 \sqrt [4]{a+b x^4}}{336 a^2 x^2}-\frac{b^2 \sqrt [4]{a+b x^4}}{168 a x^6}-\frac{\left (a+b x^4\right )^{5/4}}{14 x^{14}}-\frac{b \sqrt [4]{a+b x^4}}{28 x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^15,x]

[Out]

-(b*(a + b*x^4)^(1/4))/(28*x^10) - (b^2*(a + b*x^4)^(1/4))/(168*a*x^6) + (5*b^3*
(a + b*x^4)^(1/4))/(336*a^2*x^2) - (a + b*x^4)^(5/4)/(14*x^14) + (5*b^(7/2)*(1 +
(b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(336*a^(3/2)*(a
+ b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 23.7917, size = 129, normalized size = 0.88 \[ - \frac{b \sqrt [4]{a + b x^{4}}}{28 x^{10}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{14 x^{14}} - \frac{b^{2} \sqrt [4]{a + b x^{4}}}{168 a x^{6}} + \frac{5 b^{3} \sqrt [4]{a + b x^{4}}}{336 a^{2} x^{2}} + \frac{5 b^{\frac{7}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{336 a^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**15,x)

[Out]

-b*(a + b*x**4)**(1/4)/(28*x**10) - (a + b*x**4)**(5/4)/(14*x**14) - b**2*(a + b
*x**4)**(1/4)/(168*a*x**6) + 5*b**3*(a + b*x**4)**(1/4)/(336*a**2*x**2) + 5*b**(
7/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2, 2)/(336*a**(
3/2)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0662179, size = 105, normalized size = 0.72 \[ \frac{-48 a^4-120 a^3 b x^4-76 a^2 b^2 x^8+5 b^4 x^{16} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+6 a b^3 x^{12}+10 b^4 x^{16}}{672 a^2 x^{14} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^15,x]

[Out]

(-48*a^4 - 120*a^3*b*x^4 - 76*a^2*b^2*x^8 + 6*a*b^3*x^12 + 10*b^4*x^16 + 5*b^4*x
^16*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)])/(672*a
^2*x^14*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{15}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^15,x)

[Out]

int((b*x^4+a)^(5/4)/x^15,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{15}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^15,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)/x^15, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{15}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^15,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(5/4)/x^15, x)

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Sympy [A]  time = 31.5124, size = 34, normalized size = 0.23 \[ - \frac{a^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{2}, - \frac{5}{4} \\ - \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{14 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**15,x)

[Out]

-a**(5/4)*hyper((-7/2, -5/4), (-5/2,), b*x**4*exp_polar(I*pi)/a)/(14*x**14)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{15}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^15,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^15, x)