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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]