3.941 \(\int \frac{\sqrt [4]{a-b x^2}}{(c x)^{11/2}} \, dx\)

Optimal. Leaf size=59 \[ \frac{8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]

[Out]

(-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(9/2)) + (8*(a - b*x^2)^(9/4))/(45*a^2*c*(c*
x)^(9/2))

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Rubi [A]  time = 0.0573557, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(1/4)/(c*x)^(11/2),x]

[Out]

(-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(9/2)) + (8*(a - b*x^2)^(9/4))/(45*a^2*c*(c*
x)^(9/2))

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Rubi in Sympy [A]  time = 7.26311, size = 48, normalized size = 0.81 \[ - \frac{2 \left (a - b x^{2}\right )^{\frac{5}{4}}}{5 a c \left (c x\right )^{\frac{9}{2}}} + \frac{8 \left (a - b x^{2}\right )^{\frac{9}{4}}}{45 a^{2} c \left (c x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(1/4)/(c*x)**(11/2),x)

[Out]

-2*(a - b*x**2)**(5/4)/(5*a*c*(c*x)**(9/2)) + 8*(a - b*x**2)**(9/4)/(45*a**2*c*(
c*x)**(9/2))

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Mathematica [A]  time = 0.0351386, size = 52, normalized size = 0.88 \[ \frac{2 \sqrt{c x} \sqrt [4]{a-b x^2} \left (-5 a^2+a b x^2+4 b^2 x^4\right )}{45 a^2 c^6 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(1/4)/(c*x)^(11/2),x]

[Out]

(2*Sqrt[c*x]*(a - b*x^2)^(1/4)*(-5*a^2 + a*b*x^2 + 4*b^2*x^4))/(45*a^2*c^6*x^5)

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Maple [A]  time = 0.008, size = 32, normalized size = 0.5 \[ -{\frac{2\,x \left ( 4\,b{x}^{2}+5\,a \right ) }{45\,{a}^{2}} \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(1/4)/(c*x)^(11/2),x)

[Out]

-2/45*x*(-b*x^2+a)^(5/4)*(4*b*x^2+5*a)/a^2/(c*x)^(11/2)

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Maxima [A]  time = 1.40597, size = 54, normalized size = 0.92 \[ -\frac{2 \,{\left (\frac{9 \,{\left (-b x^{2} + a\right )}^{\frac{5}{4}} b}{x^{\frac{5}{2}}} + \frac{5 \,{\left (-b x^{2} + a\right )}^{\frac{9}{4}}}{x^{\frac{9}{2}}}\right )}}{45 \, a^{2} c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(11/2),x, algorithm="maxima")

[Out]

-2/45*(9*(-b*x^2 + a)^(5/4)*b/x^(5/2) + 5*(-b*x^2 + a)^(9/4)/x^(9/2))/(a^2*c^(11
/2))

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Fricas [A]  time = 0.224112, size = 62, normalized size = 1.05 \[ \frac{2 \,{\left (4 \, b^{2} x^{4} + a b x^{2} - 5 \, a^{2}\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{45 \, a^{2} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(11/2),x, algorithm="fricas")

[Out]

2/45*(4*b^2*x^4 + a*b*x^2 - 5*a^2)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a^2*c^6*x^5)

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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)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232338, size = 149, normalized size = 2.53 \[ \frac{2 \,{\left (\frac{9 \,{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} - \frac{a c^{2}}{x^{2}}\right )} b c^{2}}{\sqrt{c x}} - \frac{5 \,{\left (b^{2} c^{8} x^{4} - 2 \, a b c^{8} x^{2} + a^{2} c^{8}\right )}{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{4} x^{4}}\right )}}{45 \, a^{2} c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(1/4)/(c*x)^(11/2),x, algorithm="giac")

[Out]

2/45*(9*(-b*c^4*x^2 + a*c^4)^(1/4)*(b*c^2 - a*c^2/x^2)*b*c^2/sqrt(c*x) - 5*(b^2*
c^8*x^4 - 2*a*b*c^8*x^2 + a^2*c^8)*(-b*c^4*x^2 + a*c^4)^(1/4)/(sqrt(c*x)*c^4*x^4
))/(a^2*c^10)