3.936 \(\int \frac{\sqrt [4]{a-b x^2}}{(c x)^{13/2}} \, dx\)

Optimal. Leaf size=159 \[ \frac{8 b^{7/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{5/2} c^8 \left (a-b x^2\right )^{3/4}}+\frac{4 b^2 \sqrt [4]{a-b x^2}}{77 a^2 c^5 (c x)^{3/2}}+\frac{2 b \sqrt [4]{a-b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{2 \sqrt [4]{a-b x^2}}{11 c (c x)^{11/2}} \]

[Out]

(-2*(a - b*x^2)^(1/4))/(11*c*(c*x)^(11/2)) + (2*b*(a - b*x^2)^(1/4))/(77*a*c^3*(
c*x)^(7/2)) + (4*b^2*(a - b*x^2)^(1/4))/(77*a^2*c^5*(c*x)^(3/2)) + (8*b^(7/2)*(1
- a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCsc[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77
*a^(5/2)*c^8*(a - b*x^2)^(3/4))

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Rubi [A]  time = 0.315112, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{8 b^{7/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{5/2} c^8 \left (a-b x^2\right )^{3/4}}+\frac{4 b^2 \sqrt [4]{a-b x^2}}{77 a^2 c^5 (c x)^{3/2}}+\frac{2 b \sqrt [4]{a-b x^2}}{77 a c^3 (c x)^{7/2}}-\frac{2 \sqrt [4]{a-b x^2}}{11 c (c x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(1/4)/(c*x)^(13/2),x]

[Out]

(-2*(a - b*x^2)^(1/4))/(11*c*(c*x)^(11/2)) + (2*b*(a - b*x^2)^(1/4))/(77*a*c^3*(
c*x)^(7/2)) + (4*b^2*(a - b*x^2)^(1/4))/(77*a^2*c^5*(c*x)^(3/2)) + (8*b^(7/2)*(1
- a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCsc[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(77
*a^(5/2)*c^8*(a - b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 45.5805, size = 141, normalized size = 0.89 \[ - \frac{2 \sqrt [4]{a - b x^{2}}}{11 c \left (c x\right )^{\frac{11}{2}}} + \frac{2 b \sqrt [4]{a - b x^{2}}}{77 a c^{3} \left (c x\right )^{\frac{7}{2}}} + \frac{4 b^{2} \sqrt [4]{a - b x^{2}}}{77 a^{2} c^{5} \left (c x\right )^{\frac{3}{2}}} + \frac{8 b^{\frac{7}{2}} \left (c x\right )^{\frac{3}{2}} \left (- \frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{77 a^{\frac{5}{2}} c^{8} \left (a - b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(1/4)/(c*x)**(13/2),x)

[Out]

-2*(a - b*x**2)**(1/4)/(11*c*(c*x)**(11/2)) + 2*b*(a - b*x**2)**(1/4)/(77*a*c**3
*(c*x)**(7/2)) + 4*b**2*(a - b*x**2)**(1/4)/(77*a**2*c**5*(c*x)**(3/2)) + 8*b**(
7/2)*(c*x)**(3/2)*(-a/(b*x**2) + 1)**(3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x))/
2, 2)/(77*a**(5/2)*c**8*(a - b*x**2)**(3/4))

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Mathematica [C]  time = 0.105193, size = 105, normalized size = 0.66 \[ -\frac{2 \sqrt{c x} \left (7 a^3-8 a^2 b x^2+4 b^3 x^6 \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^2}{a}\right )-a b^2 x^4+2 b^3 x^6\right )}{77 a^2 c^7 x^6 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(1/4)/(c*x)^(13/2),x]

[Out]

(-2*Sqrt[c*x]*(7*a^3 - 8*a^2*b*x^2 - a*b^2*x^4 + 2*b^3*x^6 + 4*b^3*x^6*(1 - (b*x
^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^2)/a]))/(77*a^2*c^7*x^6*(a -
b*x^2)^(3/4))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{13}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(1/4)/(c*x)^(13/2),x)

[Out]

int((-b*x^2+a)^(1/4)/(c*x)^(13/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(13/2),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{6} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(13/2),x, algorithm="fricas")

[Out]

integral((-b*x^2 + a)^(1/4)/(sqrt(c*x)*c^6*x^6), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(1/4)/(c*x)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(13/2),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(13/2), x)