Optimal. Leaf size=96 \[ \frac{2 \sqrt{c+d x^3} (b c-a d)}{3 b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b} \]
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Rubi [A] time = 0.0807584, antiderivative size = 96, 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, 50, 63, 208} \[ \frac{2 \sqrt{c+d x^3} (b c-a d)}{3 b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b} \]
Antiderivative was successfully verified.
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Rule 444
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )\\ &=\frac{2 \left (c+d x^3\right )^{3/2}}{9 b}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{a+b x} \, dx,x,x^3\right )}{3 b}\\ &=\frac{2 (b c-a d) \sqrt{c+d x^3}}{3 b^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{3 b^2}\\ &=\frac{2 (b c-a d) \sqrt{c+d x^3}}{3 b^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b}+\frac{\left (2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{3 b^2 d}\\ &=\frac{2 (b c-a d) \sqrt{c+d x^3}}{3 b^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0708423, size = 85, normalized size = 0.89 \[ \frac{2 \sqrt{c+d x^3} \left (-3 a d+4 b c+b d x^3\right )}{9 b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 507, normalized size = 5.3 \begin{align*}{\frac{2\,d{x}^{3}}{9\,b}\sqrt{d{x}^{3}+c}}+{\frac{2}{3\,d} \left ( -{\frac{d \left ( ad-2\,bc \right ) }{{b}^{2}}}-{\frac{2\,cd}{3\,b}} \right ) \sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{{b}^{2}{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{-{a}^{2}{d}^{2}+2\,abcd-{b}^{2}{c}^{2}}{ad-bc}\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 [A] time = 1.88092, size = 446, normalized size = 4.65 \begin{align*} \left [-\frac{3 \,{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \,{\left (b d x^{3} + 4 \, b c - 3 \, a d\right )} \sqrt{d x^{3} + c}}{9 \, b^{2}}, -\frac{2 \,{\left (3 \,{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x^{3} + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) -{\left (b d x^{3} + 4 \, b c - 3 \, a d\right )} \sqrt{d x^{3} + c}\right )}}{9 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 33.8455, size = 90, normalized size = 0.94 \begin{align*} \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 b} + \frac{\sqrt{c + d x^{3}} \left (- 2 a d + 2 b c\right )}{3 b^{2}} + \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{c + d x^{3}}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{3 b^{3} \sqrt{\frac{a d - b c}{b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10498, size = 153, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{2}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{2} + 3 \, \sqrt{d x^{3} + c} b^{2} c - 3 \, \sqrt{d x^{3} + c} a b d\right )}}{9 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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