Optimal. Leaf size=104 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c+d x^3} (a d+b c)}{3 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2} \]
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Rubi [A] time = 0.104588, antiderivative size = 104, 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 = {446, 88, 63, 208} \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}-\frac{2 \sqrt{c+d x^3} (a d+b c)}{3 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^3\right ) \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{-b c-a d}{b^2 d \sqrt{c+d x}}+\frac{a^2}{b^2 (a+b x) \sqrt{c+d x}}+\frac{\sqrt{c+d x}}{b d}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 (b c+a d) \sqrt{c+d x^3}}{3 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2}+\frac{a^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 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2}+\frac{\left (2 a^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 d^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d^2}-\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}}\\ \end{align*}
Mathematica [A] time = 0.14478, size = 91, normalized size = 0.88 \[ \frac{2 \sqrt{c+d x^3} \left (-3 a d-2 b c+b d x^3\right )}{9 b^2 d^2}-\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.035, size = 488, normalized size = 4.7 \begin{align*}{\frac{1}{{b}^{2}} \left ( b \left ({\frac{2\,{x}^{3}}{9\,d}\sqrt{d{x}^{3}+c}}-{\frac{4\,c}{9\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) -{\frac{2\,a}{3\,d}\sqrt{d{x}^{3}+c}} \right ) }-{\frac{{\frac{i}{3}}{a}^{2}\sqrt{2}}{{b}^{2}{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{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 \left ( -{d}^{2}c \right ) ^{{\frac{2}{3}}}\sqrt{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{{id\sqrt{3} \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\, \left ( ad-bc \right ) d} \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 = 1.84879, size = 593, normalized size = 5.7 \begin{align*} \left [\frac{3 \, \sqrt{b^{2} c - a b d} a^{2} d^{2} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d}}{b x^{3} + a}\right ) - 2 \,{\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} -{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{9 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac{2 \,{\left (3 \, \sqrt{-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}{b d x^{3} + b c}\right ) -{\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} -{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c}\right )}}{9 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\left (a + b x^{3}\right ) \sqrt{c + d x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11457, size = 143, normalized size = 1.38 \begin{align*} \frac{2 \, a^{2} \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} d^{4} - 3 \, \sqrt{d x^{3} + c} b^{2} c d^{4} - 3 \, \sqrt{d x^{3} + c} a b d^{5}\right )}}{9 \, b^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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