Optimal. Leaf size=140 \[ -\frac{x^3 \left (2 a c^2-d^2\right )}{3 b^2 c^3}+\frac{2 d \sqrt{a+b x^3} \left (2 a c^2-d^2\right )}{3 b^3 c^4}+\frac{2 \left (a c^2-d^2\right )^2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^3 c^5}-\frac{2 d \left (a+b x^3\right )^{3/2}}{9 b^3 c^2}+\frac{\left (a+b x^3\right )^2}{6 b^3 c} \]
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Rubi [A] time = 0.299107, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2155, 697} \[ -\frac{x^3 \left (2 a c^2-d^2\right )}{3 b^2 c^3}+\frac{2 d \sqrt{a+b x^3} \left (2 a c^2-d^2\right )}{3 b^3 c^4}+\frac{2 \left (a c^2-d^2\right )^2 \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^3 c^5}-\frac{2 d \left (a+b x^3\right )^{3/2}}{9 b^3 c^2}+\frac{\left (a+b x^3\right )^2}{6 b^3 c} \]
Antiderivative was successfully verified.
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Rule 2155
Rule 697
Rubi steps
\begin{align*} \int \frac{x^8}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{a c+b c x+d \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (a-x^2\right )^2}{d+c x} \, dx,x,\sqrt{a+b x^3}\right )}{3 b^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{2 a c^2 d-d^3}{c^4}-\frac{\left (2 a c^2-d^2\right ) x}{c^3}-\frac{d x^2}{c^2}+\frac{x^3}{c}+\frac{\left (a c^2-d^2\right )^2}{c^4 (d+c x)}\right ) \, dx,x,\sqrt{a+b x^3}\right )}{3 b^3}\\ &=-\frac{\left (2 a c^2-d^2\right ) x^3}{3 b^2 c^3}+\frac{2 d \left (2 a c^2-d^2\right ) \sqrt{a+b x^3}}{3 b^3 c^4}-\frac{2 d \left (a+b x^3\right )^{3/2}}{9 b^3 c^2}+\frac{\left (a+b x^3\right )^2}{6 b^3 c}+\frac{2 \left (a c^2-d^2\right )^2 \log \left (d+c \sqrt{a+b x^3}\right )}{3 b^3 c^5}\\ \end{align*}
Mathematica [A] time = 0.181143, size = 126, normalized size = 0.9 \[ \frac{c \left (a \left (20 c^2 d \sqrt{a+b x^3}-6 b c^3 x^3\right )+2 b c d x^3 \left (3 d-2 c \sqrt{a+b x^3}\right )-12 d^3 \sqrt{a+b x^3}+3 b^2 c^3 x^6\right )+12 \left (d^2-a c^2\right )^2 \log \left (c \sqrt{a+b x^3}+d\right )}{18 b^3 c^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.099, size = 1473, normalized size = 10.5 \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 [A] time = 1.31592, size = 169, normalized size = 1.21 \begin{align*} \frac{\frac{3 \,{\left (b x^{3} + a\right )}^{2} c^{3} - 4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} c^{2} d - 6 \,{\left (2 \, a c^{3} - c d^{2}\right )}{\left (b x^{3} + a\right )} + 12 \,{\left (2 \, a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a}}{c^{4}} + \frac{12 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right )}{c^{5}}}{18 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28078, size = 409, normalized size = 2.92 \begin{align*} \frac{3 \, b^{2} c^{4} x^{6} - 6 \,{\left (a b c^{4} - b c^{2} d^{2}\right )} x^{3} + 6 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + 6 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right ) - 6 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt{b x^{3} + a} c - d\right ) - 4 \,{\left (b c^{3} d x^{3} - 5 \, a c^{3} d + 3 \, c d^{3}\right )} \sqrt{b x^{3} + a}}{18 \, b^{3} c^{5}} \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}}{a c + b c x^{3} + d \sqrt{a + b x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13094, size = 211, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \, b^{3} c^{5}} + \frac{3 \,{\left (b x^{3} + a\right )}^{2} b^{9} c^{3} - 12 \,{\left (b x^{3} + a\right )} a b^{9} c^{3} - 4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{9} c^{2} d + 24 \, \sqrt{b x^{3} + a} a b^{9} c^{2} d + 6 \,{\left (b x^{3} + a\right )} b^{9} c d^{2} - 12 \, \sqrt{b x^{3} + a} b^{9} d^{3}}{18 \, b^{12} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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