3.2946 \(\int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^2} \, dx\)

Optimal. Leaf size=661 \[ \frac{\sqrt{2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x}+\frac{3 \sqrt [3]{b} \sqrt{c x^2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{x \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )} \]

[Out]

-(Sqrt[a + b*(c*x^2)^(3/2)]/x) + (3*b^(1/3)*Sqrt[c*x^2]*Sqrt[a + b*(c*x^2)^(3/2)
])/(x*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[
3]]*a^(1/3)*b^(1/3)*Sqrt[c*x^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)*Sq
rt[c*x^2])^2]*EllipticE[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]])/(2*x*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)]) + (Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/3)*Sqrt[c*x^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)*Sqr
t[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]])/(x*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 = 0.813717, antiderivative size = 661, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\sqrt{2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2 x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x}+\frac{3 \sqrt [3]{b} \sqrt{c x^2} \sqrt{a+b \left (c x^2\right )^{3/2}}}{x \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*(c*x^2)^(3/2)]/x) + (3*b^(1/3)*Sqrt[c*x^2]*Sqrt[a + b*(c*x^2)^(3/2)
])/(x*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[
3]]*a^(1/3)*b^(1/3)*Sqrt[c*x^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)*Sq
rt[c*x^2])^2]*EllipticE[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]])/(2*x*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)]) + (Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/3)*Sqrt[c*x^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)*Sqr
t[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]])/(x*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 = 44.8861, size = 575, normalized size = 0.87 \[ - \frac{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{3 \sqrt [3]{b} \sqrt{c x^{2}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )} - \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*3**(1/4)*a**(1/3)*b**(1/3)*sqrt(c*x**2)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*sq
rt(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_e(asin((-a**(1
/3)*(-1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*s
qrt(c*x**2))), -7 - 4*sqrt(3))/(2*x*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(2)*3**(3/4)*a**(1/3)*b**(1/3)*sqrt(c*x**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)*s
qrt(c*x**2))**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))/(x*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**(1/3)*sqrt(c*x**2)*sqrt(a + b*(c*x**2)**(3/2))/(x*(a**(1/3)*(1 + sqrt(3))
+ b**(1/3)*sqrt(c*x**2))) - sqrt(a + b*(c*x**2)**(3/2))/x

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Mathematica [A]  time = 0.0435468, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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