Optimal. Leaf size=107 \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}}+\frac{2 c^2}{3 d^2 \sqrt{c+d x^3} (b c-a d)}+\frac{2 \sqrt{c+d x^3}}{3 b d^2} \]
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Rubi [A] time = 0.120525, antiderivative size = 107, 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, 87, 63, 208} \[ -\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}}+\frac{2 c^2}{3 d^2 \sqrt{c+d x^3} (b c-a d)}+\frac{2 \sqrt{c+d x^3}}{3 b d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{c^2}{d (-b c+a d) (c+d x)^{3/2}}+\frac{1}{b d \sqrt{c+d x}}+\frac{a^2}{b (b c-a d) (a+b x) \sqrt{c+d x}}\right ) \, dx,x,x^3\right )\\ &=\frac{2 c^2}{3 d^2 (b c-a d) \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{3 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 (b c-a d)}\\ &=\frac{2 c^2}{3 d^2 (b c-a d) \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{3 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 d (b c-a d)}\\ &=\frac{2 c^2}{3 d^2 (b c-a d) \sqrt{c+d x^3}}+\frac{2 \sqrt{c+d x^3}}{3 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^{3/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0687899, size = 100, normalized size = 0.93 \[ \frac{2 \left (-a^2 d^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b \left (d x^3+c\right )}{b c-a d}\right )+a^2 d^2+a b d \left (c+d x^3\right )+b^2 (-c) \left (2 c+d x^3\right )\right )}{3 b^2 d^2 \sqrt{c+d x^3} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.037, size = 527, normalized size = 4.9 \begin{align*}{\frac{1}{{b}^{2}} \left ( b \left ({\frac{2\,c}{3\,{d}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{2}{3\,{d}^{2}}\sqrt{d{x}^{3}+c}} \right ) +{\frac{2\,a}{3\,d}{\frac{1}{\sqrt{d{x}^{3}+c}}}} \right ) }+{\frac{{a}^{2}}{{b}^{2}} \left ( -{\frac{2}{3\,ad-3\,bc}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}-{\frac{{\frac{i}{3}}b\sqrt{2}}{{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{1}{ \left ( -ad+bc \right ) \left ( ad-bc \right ) }\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}}}}} \right ) } \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 = 2.46543, size = 896, normalized size = 8.37 \begin{align*} \left [-\frac{{\left (a^{2} d^{3} x^{3} + a^{2} c d^{2}\right )} \sqrt{b^{2} c - a b d} \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^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{3 \,{\left (b^{4} c^{3} d^{2} - 2 \, a b^{3} c^{2} d^{3} + a^{2} b^{2} c d^{4} +{\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{3}\right )}}, \frac{2 \,{\left ({\left (a^{2} d^{3} x^{3} + a^{2} c d^{2}\right )} \sqrt{-b^{2} c + a b d} \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^{3} - 3 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c}\right )}}{3 \,{\left (b^{4} c^{3} d^{2} - 2 \, a b^{3} c^{2} d^{3} + a^{2} b^{2} c d^{4} +{\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{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 ) \left (c + d x^{3}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13845, size = 139, normalized size = 1.3 \begin{align*} \frac{2 \, a^{2} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \,{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \, c^{2}}{3 \,{\left (b c d^{2} - a d^{3}\right )} \sqrt{d x^{3} + c}} + \frac{2 \, \sqrt{d x^{3} + c}}{3 \, b d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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