Optimal. Leaf size=26 \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]
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Rubi [A] time = 0.176762, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]
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Rubi in Sympy [A] time = 8.79665, size = 20, normalized size = 0.77 \[ \frac{2 \log{\left (c \sqrt{a + b x^{3}} + d \right )}}{3 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
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Mathematica [A] time = 0.0160763, size = 26, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]
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Maple [C] time = 0.02, size = 455, normalized size = 17.5 \[{\frac{-{\frac{i}{3}}\sqrt{2}}{{b}^{3}d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}b{c}^{2}+a{c}^{2}-{d}^{2} \right ) }{1\sqrt [3]{-a{b}^{2}}\sqrt{{{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}-i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{b \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -3\,\sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}b \left ( 2\,x+{\frac{1}{b} \left ( \sqrt [3]{-a{b}^{2}}+i\sqrt{3}\sqrt [3]{-a{b}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( i\sqrt [3]{-a{b}^{2}}\sqrt{3}{\it \_alpha}\,b-i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+2\,{{\it \_alpha}}^{2}{b}^{2}-\sqrt [3]{-a{b}^{2}}{\it \_alpha}\,b- \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},-{\frac{{c}^{2}}{2\,b{d}^{2}} \left ( 2\,i\sqrt{3}\sqrt [3]{-a{b}^{2}}{{\it \_alpha}}^{2}b-i\sqrt{3} \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}{\it \_alpha}+i\sqrt{3}ab-3\, \left ( -a{b}^{2} \right ) ^{2/3}{\it \_alpha}-3\,ab \right ) },\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}}}+{\frac{\ln \left ( b{c}^{2}{x}^{3}+a{c}^{2}-{d}^{2} \right ) }{3\,bc}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)
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Maxima [A] time = 0.690906, size = 30, normalized size = 1.15 \[ \frac{2 \, \log \left (\sqrt{b x^{3} + a} c + d\right )}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="maxima")
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Fricas [A] time = 0.273522, size = 82, normalized size = 3.15 \[ \frac{\log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + \log \left (\sqrt{b x^{3} + a} c + d\right ) - \log \left (\sqrt{b x^{3} + a} c - d\right )}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),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/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
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GIAC/XCAS [A] time = 0.273732, size = 31, normalized size = 1.19 \[ \frac{2 \,{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="giac")
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