3.2958 \(\int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^2} \, dx\)

Optimal. Leaf size=355 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{x} \]

[Out]

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

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Rubi [A]  time = 0.442541, antiderivative size = 355, 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 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0325349, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.177, size = 306, normalized size = 0.9 \[ -{\frac{1}{2\,x} \left ( i\sqrt{3}\sqrt [3]{-ac{b}^{2}}\sqrt{2}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{{\frac{1}{x \left ( i\sqrt{3}-3 \right ) } \left ( b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}+2\,b\sqrt{c{x}^{3}}+\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) x+2\,a+2\,b\sqrt{c{x}^{3}} \right ){\frac{1}{\sqrt{a+b\sqrt{c{x}^{3}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(I*3^(1/2)*(-a*c*b^2)^(1/3)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(
c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*((b*(c*x^3)^(
1/2)-(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*
x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3
)/x)^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(
c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2),2^(1/2)*(I*3^
(1/2)/(I*3^(1/2)-3))^(1/2))*x+2*a+2*b*(c*x^3)^(1/2))/x/(a+b*(c*x^3)^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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