3.2942 \(\int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx\)

Optimal. Leaf size=298 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{2 x^2} \]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(2*x^2) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/3)*c*(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 - S
qrt[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]])/(2*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 = 0.338915, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} c \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(2*x^2) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/3)*c*(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 - S
qrt[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]])/(2*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 = 15.9602, size = 258, normalized size = 0.87 \[ \frac{3^{\frac{3}{4}} b^{\frac{2}{3}} c \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} - \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(3/4)*b**(2/3)*c*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*sq
rt(3))/(2*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))) - sqrt(a + b*(c*x**
2)**(3/2))/(2*x**2)

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Mathematica [A]  time = 0.0477482, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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