Optimal. Leaf size=108 \[ -\frac{d}{\sqrt{c+d x^3} (b c-a d)^2}-\frac{1}{3 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{\sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}} \]
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Rubi [A] time = 0.0913992, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {444, 51, 63, 208} \[ -\frac{d}{\sqrt{c+d x^3} (b c-a d)^2}-\frac{1}{3 \left (a+b x^3\right ) \sqrt{c+d x^3} (b c-a d)}+\frac{\sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 444
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt{c+d x^3}}-\frac{d \operatorname{Subst}\left (\int \frac{1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{2 (b c-a d)}\\ &=-\frac{d}{(b c-a d)^2 \sqrt{c+d x^3}}-\frac{1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt{c+d x^3}}-\frac{(b d) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{2 (b c-a d)^2}\\ &=-\frac{d}{(b c-a d)^2 \sqrt{c+d x^3}}-\frac{1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt{c+d x^3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{(b c-a d)^2}\\ &=-\frac{d}{(b c-a d)^2 \sqrt{c+d x^3}}-\frac{1}{3 (b c-a d) \left (a+b x^3\right ) \sqrt{c+d x^3}}+\frac{\sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0170034, size = 54, normalized size = 0.5 \[ -\frac{2 d \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{b \left (d x^3+c\right )}{a d-b c}\right )}{3 \sqrt{c+d x^3} (a d-b c)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 485, normalized size = 4.5 \begin{align*} -{\frac{2\,d}{3\, \left ( ad-bc \right ) ^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{b}{3\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{3}+a \right ) }\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{2}}b\sqrt{2}}{d}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ \left ( ad-bc \right ) ^{3}}\sqrt [3]{-{d}^{2}c}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-{d}^{2}c}} \right ) \left ( -3\,\sqrt [3]{-{d}^{2}c}+i\sqrt{3}\sqrt [3]{-{d}^{2}c} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-{d}^{2}c}+\sqrt [3]{-{d}^{2}c} \right ) } \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}} \left ( i\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,\sqrt{3}d-i\sqrt{3} \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}+2\,{{\it \_alpha}}^{2}{d}^{2}-\sqrt [3]{-{d}^{2}c}{\it \_alpha}\,d- \left ( -{d}^{2}c \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]{-{d}^{2}c}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c}} \right ){\frac{1}{\sqrt [3]{-{d}^{2}c}}}}}},{\frac{b}{2\,d \left ( ad-bc \right ) } \left ( 2\,i\sqrt [3]{-{d}^{2}c}\sqrt{3}{{\it \_alpha}}^{2}d-i \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}\sqrt{3}{\it \_alpha}+i\sqrt{3}cd-3\, \left ( -{d}^{2}c \right ) ^{2/3}{\it \_alpha}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c} \left ( -{\frac{3}{2\,d}\sqrt [3]{-{d}^{2}c}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-{d}^{2}c}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84924, size = 923, normalized size = 8.55 \begin{align*} \left [\frac{3 \,{\left (b d^{2} x^{6} +{\left (b c d + a d^{2}\right )} x^{3} + a c d\right )} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{3} + a}\right ) - 2 \,{\left (3 \, b d x^{3} + b c + 2 \, a d\right )} \sqrt{d x^{3} + c}}{6 \,{\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )}}, \frac{3 \,{\left (b d^{2} x^{6} +{\left (b c d + a d^{2}\right )} x^{3} + a c d\right )} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{\sqrt{d x^{3} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{b d x^{3} + b c}\right ) -{\left (3 \, b d x^{3} + b c + 2 \, a d\right )} \sqrt{d x^{3} + c}}{3 \,{\left ({\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{6} + a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10527, size = 203, normalized size = 1.88 \begin{align*} -\frac{1}{3} \, d{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x^{3} + c\right )} b - 2 \, b c + 2 \, a d}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{3} + c} b c + \sqrt{d x^{3} + c} a d\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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