3.2962 \(\int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^3} \, dx\)

Optimal. Leaf size=810 \[ -\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{8 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3^{3/4} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{2 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3 c^{2/3} \sqrt{a+b \sqrt{c x^3}} b^{4/3}}{4 a \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}-\frac{3 c x \sqrt{a+b \sqrt{c x^3}} b}{4 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2} \]

[Out]

-Sqrt[a + b*Sqrt[c*x^3]]/(2*x^2) - (3*b*c*x*Sqrt[a + b*Sqrt[c*x^3]])/(4*a*Sqrt[c
*x^3]) + (3*b^(4/3)*c^(2/3)*Sqrt[a + b*Sqrt[c*x^3]])/(4*a*((1 + Sqrt[3])*a^(1/3)
+ (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(4/3)*c^
(2/3)*(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]*EllipticE[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)/Sq
rt[c*x^3])], -7 - 4*Sqrt[3]])/(8*a^(2/3)*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]]) + (3^(3/4)*b^(4/3)*c^(2/3)*(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]])
/(2*Sqrt[2]*a^(2/3)*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 = 1.06614, antiderivative size = 810, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{8 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3^{3/4} c^{2/3} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right ) \sqrt{\frac{-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x+a^{2/3}}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) b^{4/3}}{2 \sqrt{2} a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\sqrt [3]{a}\right )}{\left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}+\frac{3 c^{2/3} \sqrt{a+b \sqrt{c x^3}} b^{4/3}}{4 a \left (\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}\right )}-\frac{3 c x \sqrt{a+b \sqrt{c x^3}} b}{4 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a + b*Sqrt[c*x^3]]/(2*x^2) - (3*b*c*x*Sqrt[a + b*Sqrt[c*x^3]])/(4*a*Sqrt[c
*x^3]) + (3*b^(4/3)*c^(2/3)*Sqrt[a + b*Sqrt[c*x^3]])/(4*a*((1 + Sqrt[3])*a^(1/3)
+ (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(4/3)*c^
(2/3)*(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]*EllipticE[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)/Sq
rt[c*x^3])], -7 - 4*Sqrt[3]])/(8*a^(2/3)*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]]) + (3^(3/4)*b^(4/3)*c^(2/3)*(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]])
/(2*Sqrt[2]*a^(2/3)*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. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0326453, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.183, size = 869, normalized size = 1.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(3*I*(-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)*EllipticE(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))*3^(
1/2)*2^(1/2)*(-a*c*b^2)^(2/3)*x^2-2*I*(-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))*3^(1/2)*2^(1/2)*(-a*c*b^2)^(2/3)*x^2+3*(-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)*EllipticE(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))*2^(1/2)*(-a*c*b^2)^(2/3)*x^2-
12*x^3*b^2*c-20*(c*x^3)^(1/2)*a*b-8*a^2)/x^2/a/(a+b*(c*x^3)^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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