3.948 \(\int \frac{1}{(c x)^{13/2} \sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]

[Out]

(-2*(a + b*x^2)^(3/4))/(3*a*c*(c*x)^(11/2)) + (16*(a + b*x^2)^(7/4))/(21*a^2*c*(
c*x)^(11/2)) - (64*(a + b*x^2)^(11/4))/(231*a^3*c*(c*x)^(11/2))

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Rubi [A]  time = 0.0874795, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{64 \left (a+b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac{16 \left (a+b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac{2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x^2)^(3/4))/(3*a*c*(c*x)^(11/2)) + (16*(a + b*x^2)^(7/4))/(21*a^2*c*(
c*x)^(11/2)) - (64*(a + b*x^2)^(11/4))/(231*a^3*c*(c*x)^(11/2))

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Rubi in Sympy [A]  time = 10.7774, size = 73, normalized size = 0.86 \[ - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{4}}}{3 a c \left (c x\right )^{\frac{11}{2}}} + \frac{16 \left (a + b x^{2}\right )^{\frac{7}{4}}}{21 a^{2} c \left (c x\right )^{\frac{11}{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{11}{4}}}{231 a^{3} c \left (c x\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a + b*x**2)**(3/4)/(3*a*c*(c*x)**(11/2)) + 16*(a + b*x**2)**(7/4)/(21*a**2*c
*(c*x)**(11/2)) - 64*(a + b*x**2)**(11/4)/(231*a**3*c*(c*x)**(11/2))

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Mathematica [A]  time = 0.0436354, size = 52, normalized size = 0.61 \[ -\frac{2 \sqrt{c x} \left (a+b x^2\right )^{3/4} \left (21 a^2-24 a b x^2+32 b^2 x^4\right )}{231 a^3 c^7 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c*x]*(a + b*x^2)^(3/4)*(21*a^2 - 24*a*b*x^2 + 32*b^2*x^4))/(231*a^3*c^7
*x^6)

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Maple [A]  time = 0.008, size = 42, normalized size = 0.5 \[ -{\frac{2\,x \left ( 32\,{b}^{2}{x}^{4}-24\,ab{x}^{2}+21\,{a}^{2} \right ) }{231\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{4}}} \left ( cx \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/231*x*(b*x^2+a)^(3/4)*(32*b^2*x^4-24*a*b*x^2+21*a^2)/a^3/(c*x)^(13/2)

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Maxima [A]  time = 1.40343, size = 74, normalized size = 0.87 \[ -\frac{2 \,{\left (\frac{77 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} b^{2}}{x^{\frac{3}{2}}} - \frac{66 \,{\left (b x^{2} + a\right )}^{\frac{7}{4}} b}{x^{\frac{7}{2}}} + \frac{21 \,{\left (b x^{2} + a\right )}^{\frac{11}{4}}}{x^{\frac{11}{2}}}\right )}}{231 \, a^{3} c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/231*(77*(b*x^2 + a)^(3/4)*b^2/x^(3/2) - 66*(b*x^2 + a)^(7/4)*b/x^(7/2) + 21*(
b*x^2 + a)^(11/4)/x^(11/2))/(a^3*c^(13/2))

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Fricas [A]  time = 0.213683, size = 77, normalized size = 0.91 \[ -\frac{2 \,{\left (32 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} - 3 \, a^{2} b x^{2} + 21 \, a^{3}\right )}}{231 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} a^{3} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/231*(32*b^3*x^6 + 8*a*b^2*x^4 - 3*a^2*b*x^2 + 21*a^3)/((b*x^2 + a)^(1/4)*sqrt
(c*x)*a^3*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(1/(c*x)**(13/2)/(b*x**2+a)**(1/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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