3.963 \(\int \frac{1}{(c x)^{7/2} \sqrt [4]{a-b x^2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a-b x^2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]

[Out]

(-2*(a - b*x^2)^(3/4))/(5*a*c*(c*x)^(5/2)) - (4*b^(3/2)*(1 - a/(b*x^2))^(1/4)*Sq
rt[c*x]*EllipticE[ArcCsc[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*c^4*(a - b*x^2)^
(1/4))

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Rubi [A]  time = 0.121854, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a-b x^2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(7/2)*(a - b*x^2)^(1/4)),x]

[Out]

(-2*(a - b*x^2)^(3/4))/(5*a*c*(c*x)^(5/2)) - (4*b^(3/2)*(1 - a/(b*x^2))^(1/4)*Sq
rt[c*x]*EllipticE[ArcCsc[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*c^4*(a - b*x^2)^
(1/4))

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Rubi in Sympy [A]  time = 16.1951, size = 87, normalized size = 0.87 \[ - \frac{2 \left (a - b x^{2}\right )^{\frac{3}{4}}}{5 a c \left (c x\right )^{\frac{5}{2}}} - \frac{4 b^{\frac{3}{2}} \sqrt{c x} \sqrt [4]{- \frac{a}{b x^{2}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{5 a^{\frac{3}{2}} c^{4} \sqrt [4]{a - b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(7/2)/(-b*x**2+a)**(1/4),x)

[Out]

-2*(a - b*x**2)**(3/4)/(5*a*c*(c*x)**(5/2)) - 4*b**(3/2)*sqrt(c*x)*(-a/(b*x**2)
+ 1)**(1/4)*elliptic_e(asin(sqrt(a)/(sqrt(b)*x))/2, 2)/(5*a**(3/2)*c**4*(a - b*x
**2)**(1/4))

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Mathematica [C]  time = 0.0830733, size = 89, normalized size = 0.89 \[ \frac{x \left (-6 \left (a^2+a b x^2-2 b^2 x^4\right )-8 b^2 x^4 \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )\right )}{15 a^2 (c x)^{7/2} \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(7/2)*(a - b*x^2)^(1/4)),x]

[Out]

(x*(-6*(a^2 + a*b*x^2 - 2*b^2*x^4) - 8*b^2*x^4*(1 - (b*x^2)/a)^(1/4)*Hypergeomet
ric2F1[1/4, 3/4, 7/4, (b*x^2)/a]))/(15*a^2*(c*x)^(7/2)*(a - b*x^2)^(1/4))

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(7/2)/(-b*x^2+a)^(1/4),x)

[Out]

int(1/(c*x)^(7/2)/(-b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(7/2)),x, algorithm="fricas")

[Out]

integral(1/((-b*x^2 + a)^(1/4)*sqrt(c*x)*c^3*x^3), 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(1/(c*x)**(7/2)/(-b*x**2+a)**(1/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(7/2)),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(7/2)), x)