3.2947 \(\int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^5} \, dx\)

Optimal. Leaf size=681 \[ \frac{3^{3/4} b^{4/3} c^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} c^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{3 b^{4/3} c^2 \sqrt{a+b \left (c x^2\right )^{3/2}}}{8 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}-\frac{3 b c^2 \sqrt{a+b \left (c x^2\right )^{3/2}}}{8 a \sqrt{c x^2}}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{4 x^4} \]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(4*x^4) - (3*b*c^2*Sqrt[a + b*(c*x^2)^(3/2)])/(8*a*Sq
rt[c*x^2]) + (3*b^(4/3)*c^2*Sqrt[a + b*(c*x^2)^(3/2)])/(8*a*((1 + Sqrt[3])*a^(1/
3) + b^(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(4/3)*c^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]*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]])/(16*a^(2/3)*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)]) + (3^
(3/4)*b^(4/3)*c^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]])/(4*Sqrt[2]*a^(2/3)*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 = 0.990057, antiderivative size = 681, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{3^{3/4} b^{4/3} c^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}-\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} c^2 \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^2}+b^{2/3} c x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} \sqrt{c x^2}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt{c x^2}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )^2}} \sqrt{a+b \left (c x^2\right )^{3/2}}}+\frac{3 b^{4/3} c^2 \sqrt{a+b \left (c x^2\right )^{3/2}}}{8 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt{c x^2}\right )}-\frac{3 b c^2 \sqrt{a+b \left (c x^2\right )^{3/2}}}{8 a \sqrt{c x^2}}-\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^2)^(3/2)]/x^5,x]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(4*x^4) - (3*b*c^2*Sqrt[a + b*(c*x^2)^(3/2)])/(8*a*Sq
rt[c*x^2]) + (3*b^(4/3)*c^2*Sqrt[a + b*(c*x^2)^(3/2)])/(8*a*((1 + Sqrt[3])*a^(1/
3) + b^(1/3)*Sqrt[c*x^2])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(4/3)*c^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]*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]])/(16*a^(2/3)*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)]) + (3^
(3/4)*b^(4/3)*c^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]])/(4*Sqrt[2]*a^(2/3)*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 = 58.0226, size = 597, normalized size = 0.88 \[ - \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{4 x^{4}} + \frac{3 b^{\frac{4}{3}} c^{2} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{8 a \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )} - \frac{3 b c^{2} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{8 a \sqrt{c x^{2}}} - \frac{3 \sqrt [4]{3} b^{\frac{4}{3}} c^{2} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{16 a^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} b^{\frac{4}{3}} c^{2} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} \sqrt{c x^{2}} + b^{\frac{2}{3}} c x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{8 a^{\frac{2}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} \sqrt{c x^{2}}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} \sqrt{c x^{2}}\right )^{2}}} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**5,x)

[Out]

-sqrt(a + b*(c*x**2)**(3/2))/(4*x**4) + 3*b**(4/3)*c**2*sqrt(a + b*(c*x**2)**(3/
2))/(8*a*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*sqrt(c*x**2))) - 3*b*c**2*sqrt(a + b
*(c*x**2)**(3/2))/(8*a*sqrt(c*x**2)) - 3*3**(1/4)*b**(4/3)*c**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))*el
liptic_e(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))/(16*a**(2/3)*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)*b**(4/3)*c**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)*(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))/(8*a**(2/3)*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)))

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Mathematica [A]  time = 0.0429488, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^5,x]

[Out]

Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^5, x]

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(3/2))^(1/2)/x^5,x)

[Out]

int((a+b*(c*x^2)^(3/2))^(1/2)/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt((c*x^2)^(3/2)*b + a)/x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^5,x, algorithm="fricas")

[Out]

integral(sqrt(sqrt(c*x^2)*b*c*x^2 + a)/x^5, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**5,x)

[Out]

Integral(sqrt(a + b*(c*x**2)**(3/2))/x**5, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (c x^{2}\right )^{\frac{3}{2}} b + a}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt((c*x^2)^(3/2)*b + a)/x^5, x)