Optimal. Leaf size=154 \[ \frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}+\frac{2 a^2 \sqrt{c+d x^3} (b c-a d)}{3 b^4}-\frac{2 a^2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}-\frac{2 \left (c+d x^3\right )^{5/2} (a d+b c)}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2} \]
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Rubi [A] time = 0.15477, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 88, 50, 63, 208} \[ \frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}+\frac{2 a^2 \sqrt{c+d x^3} (b c-a d)}{3 b^4}-\frac{2 a^2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}-\frac{2 \left (c+d x^3\right )^{5/2} (a d+b c)}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^8 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 (c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{(-b c-a d) (c+d x)^{3/2}}{b^2 d}+\frac{a^2 (c+d x)^{3/2}}{b^2 (a+b x)}+\frac{(c+d x)^{5/2}}{b d}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 (b c+a d) \left (c+d x^3\right )^{5/2}}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2}+\frac{a^2 \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )}{3 b^2}\\ &=\frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}-\frac{2 (b c+a d) \left (c+d x^3\right )^{5/2}}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2}+\frac{\left (a^2 (b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{a+b x} \, dx,x,x^3\right )}{3 b^3}\\ &=\frac{2 a^2 (b c-a d) \sqrt{c+d x^3}}{3 b^4}+\frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}-\frac{2 (b c+a d) \left (c+d x^3\right )^{5/2}}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2}+\frac{\left (a^2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{3 b^4}\\ &=\frac{2 a^2 (b c-a d) \sqrt{c+d x^3}}{3 b^4}+\frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}-\frac{2 (b c+a d) \left (c+d x^3\right )^{5/2}}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2}+\frac{\left (2 a^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^4 d}\\ &=\frac{2 a^2 (b c-a d) \sqrt{c+d x^3}}{3 b^4}+\frac{2 a^2 \left (c+d x^3\right )^{3/2}}{9 b^3}-\frac{2 (b c+a d) \left (c+d x^3\right )^{5/2}}{15 b^2 d^2}+\frac{2 \left (c+d x^3\right )^{7/2}}{21 b d^2}-\frac{2 a^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^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.151854, size = 145, normalized size = 0.94 \[ \frac{2 \left (105 a^2 (b c-a d) \left (\frac{\sqrt{c+d x^3}}{b}-\frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{3/2}}\right )+35 a^2 \left (c+d x^3\right )^{3/2}-\frac{21 b \left (c+d x^3\right )^{5/2} (a d+b c)}{d^2}+\frac{15 b^2 \left (c+d x^3\right )^{7/2}}{d^2}\right )}{315 b^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 605, normalized size = 3.9 \begin{align*} \text{result too large to display} \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 = 2.71122, size = 883, normalized size = 5.73 \begin{align*} \left [-\frac{105 \,{\left (a^{2} b c d^{2} - a^{3} d^{3}\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 (15 \, b^{3} d^{3} x^{9} + 3 \,{\left (8 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{6} - 6 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 140 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} +{\left (3 \, b^{3} c^{2} d - 42 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{315 \, b^{4} d^{2}}, -\frac{2 \,{\left (105 \,{\left (a^{2} b c d^{2} - a^{3} d^{3}\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 (15 \, b^{3} d^{3} x^{9} + 3 \,{\left (8 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{6} - 6 \, b^{3} c^{3} - 21 \, a b^{2} c^{2} d + 140 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} +{\left (3 \, b^{3} c^{2} d - 42 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{3}\right )} \sqrt{d x^{3} + c}\right )}}{315 \, b^{4} d^{2}}\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 = 1.13043, size = 261, normalized size = 1.69 \begin{align*} \frac{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} 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^{4}} + \frac{2 \,{\left (15 \,{\left (d x^{3} + c\right )}^{\frac{7}{2}} b^{6} d^{12} - 21 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} b^{6} c d^{12} - 21 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} a b^{5} d^{13} + 35 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} a^{2} b^{4} d^{14} + 105 \, \sqrt{d x^{3} + c} a^{2} b^{4} c d^{14} - 105 \, \sqrt{d x^{3} + c} a^{3} b^{3} d^{15}\right )}}{315 \, b^{7} d^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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