Optimal. Leaf size=87 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^3}}{3 \left (a+b x^3\right ) (b c-a d)} \]
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Rubi [A] time = 0.210226, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^3}}{3 \left (a+b x^3\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]
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Rubi in Sympy [A] time = 20.3996, size = 70, normalized size = 0.8 \[ \frac{\sqrt{c + d x^{3}}}{3 \left (a + b x^{3}\right ) \left (a d - b c\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 \sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
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Mathematica [A] time = 0.143089, size = 84, normalized size = 0.97 \[ \frac{\frac{\sqrt{c+d x^3}}{a+b x^3}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{3 a d-3 b c} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x^3)^2*Sqrt[c + d*x^3]),x]
[Out]
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Maple [C] time = 0.01, size = 457, normalized size = 5.3 \[{\frac{1}{ \left ( 3\,ad-3\,bc \right ) \left ( b{x}^{3}+a \right ) }\sqrt{d{x}^{3}+c}}-{\frac{{\frac{i}{6}}\sqrt{2}}{d}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ \left ( ad-bc \right ) ^{2}}\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x^3+a)^2/(d*x^3+c)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="maxima")
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Fricas [A] time = 0.224804, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d x^{3} + a d\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right ) + 2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}{6 \,{\left ({\left (b^{2} c - a b d\right )} x^{3} + a b c - a^{2} d\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (b d x^{3} + a d\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right ) - \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}{3 \,{\left ({\left (b^{2} c - a b d\right )} x^{3} + a b c - a^{2} d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.217921, size = 124, normalized size = 1.43 \[ -\frac{1}{3} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{\sqrt{d x^{3} + c}}{{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^3 + a)^2*sqrt(d*x^3 + c)),x, algorithm="giac")
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