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 = 0.110529, 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, Rules used = {2155, 31} \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]
Antiderivative was successfully verified.
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Rule 2155
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{a c+b c x+d \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{d+c x} \, dx,x,\sqrt{a+b x^3}\right )}{3 b}\\ &=\frac{2 \log \left (d+c \sqrt{a+b x^3}\right )}{3 b c}\\ \end{align*}
Mathematica [A] time = 0.0283146, size = 26, normalized size = 1. \[ \frac{2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b c} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 455, normalized size = 17.5 \begin{align*}{\frac{-{\frac{i}{3}}\sqrt{2}}{{b}^{3}d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}b{c}^{2}+{c}^{2}a-{d}^{2} \right ) }{\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]{-a{b}^{2}}\sqrt{3}{{\it \_alpha}}^{2}b-i \left ( -a{b}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{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}+{c}^{2}a-{d}^{2} \right ) }{3\,bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19344, size = 30, normalized size = 1.15 \begin{align*} \frac{2 \, \log \left (\sqrt{b x^{3} + a} c + d\right )}{3 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26852, size = 135, normalized size = 5.19 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.50954, size = 32, normalized size = 1.23 \begin{align*} \frac{2 \left (\begin{cases} \frac{\sqrt{a + b x^{3}}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (c \sqrt{a + b x^{3}} + d \right )}}{c} & \text{otherwise} \end{cases}\right )}{3 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12645, size = 31, normalized size = 1.19 \begin{align*} \frac{2 \, \log \left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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