3.2943 \(\int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^6} \, dx\)

Optimal. Leaf size=352 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (c x^2\right )^{5/2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{20 a x^5 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 b \left (c x^2\right )^{5/2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{20 a c x^7}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5} \]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(5*x^5) - (3*b*(c*x^2)^(5/2)*Sqrt[a + b*(c*x^2)^(3/2)
])/(20*a*c*x^7) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(5/3)*(c*x^2)^(5/2)*(a^(1/3) + b^
(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/
((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3])
*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])],
-7 - 4*Sqrt[3]])/(20*a*x^5*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)])

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Rubi [A]  time = 0.43923, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (c x^2\right )^{5/2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{20 a x^5 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 b \left (c x^2\right )^{5/2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{20 a c x^7}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^2)^(3/2)]/x^6,x]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(5*x^5) - (3*b*(c*x^2)^(5/2)*Sqrt[a + b*(c*x^2)^(3/2)
])/(20*a*c*x^7) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(5/3)*(c*x^2)^(5/2)*(a^(1/3) + b^
(1/3)*Sqrt[c*x^2])*Sqrt[(a^(2/3) + b^(2/3)*c*x^2 - a^(1/3)*b^(1/3)*Sqrt[c*x^2])/
((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*EllipticF[ArcSin[((1 - Sqrt[3])
*a^(1/3) + b^(1/3)*Sqrt[c*x^2])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])],
-7 - 4*Sqrt[3]])/(20*a*x^5*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*Sqrt[c*x^2]))/((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])^2]*Sqrt[a + b*(c*x^2)^(3/2)])

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Rubi in Sympy [A]  time = 23.2697, size = 308, normalized size = 0.88 \[ - \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{5 x^{5}} - \frac{3^{\frac{3}{4}} b^{\frac{5}{3}} \left (c x^{2}\right )^{\frac{5}{2}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{20 a x^{5} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} - \frac{3 b \left (c x^{2}\right )^{\frac{5}{2}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{20 a c x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**6,x)

[Out]

-sqrt(a + b*(c*x**2)**(3/2))/(5*x**5) - 3**(3/4)*b**(5/3)*(c*x**2)**(5/2)*sqrt((
a**(2/3) - a**(1/3)*b**(1/3)*sqrt(c*x**2) + b**(2/3)*c*x**2)/(a**(1/3)*(1 + sqrt
(3)) + b**(1/3)*sqrt(c*x**2))**2)*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/3)*sqrt(c*
x**2))*elliptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))/(a**(1
/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))), -7 - 4*sqrt(3))/(20*a*x**5*sqrt(a**
(1/3)*(a**(1/3) + b**(1/3)*sqrt(c*x**2))/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt
(c*x**2))**2)*sqrt(a + b*(c*x**2)**(3/2))) - 3*b*(c*x**2)**(5/2)*sqrt(a + b*(c*x
**2)**(3/2))/(20*a*c*x**7)

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Mathematica [A]  time = 0.0433417, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^6} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^6,x]

[Out]

Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^6, x]

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(3/2))^(1/2)/x^6,x)

[Out]

int((a+b*(c*x^2)^(3/2))^(1/2)/x^6,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^6,x, algorithm="maxima")

[Out]

integrate(sqrt((c*x^2)^(3/2)*b + a)/x^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**6,x)

[Out]

Integral(sqrt(a + b*(c*x**2)**(3/2))/x**6, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^6,x, algorithm="giac")

[Out]

integrate(sqrt((c*x^2)^(3/2)*b + a)/x^6, x)