Optimal. Leaf size=73 \[ -\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}-\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}+\frac{x^3}{3 b c} \]
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Rubi [A] time = 0.201527, antiderivative size = 73, 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} \[ -\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}-\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}+\frac{x^3}{3 b c} \]
Antiderivative was successfully verified.
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Rule 2155
Rule 697
Rubi steps
\begin{align*} \int \frac{x^5}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{a c+b c x+d \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{-a+x^2}{d+c x} \, dx,x,\sqrt{a+b x^3}\right )}{3 b^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d}{c^2}+\frac{x}{c}+\frac{-a c^2+d^2}{c^2 (d+c x)}\right ) \, dx,x,\sqrt{a+b x^3}\right )}{3 b^2}\\ &=\frac{x^3}{3 b c}-\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left (a c^2-d^2\right ) \log \left (d+c \sqrt{a+b x^3}\right )}{3 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 0.0692534, size = 63, normalized size = 0.86 \[ \frac{\left (2 d^2-2 a c^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )+c \left (b c x^3-2 d \sqrt{a+b x^3}\right )}{3 b^2 c^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 943, normalized size = 12.9 \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 [A] time = 1.17846, size = 84, normalized size = 1.15 \begin{align*} \frac{\frac{{\left (b x^{3} + a\right )} c - 2 \, \sqrt{b x^{3} + a} d}{c^{2}} - \frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24101, size = 246, normalized size = 3.37 \begin{align*} \frac{b c^{2} x^{3} - 2 \, \sqrt{b x^{3} + a} c d -{\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) -{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right ) +{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.27612, size = 95, normalized size = 1.3 \begin{align*} \begin{cases} \frac{2 \left (\frac{a + b x^{3}}{6 b c} - \frac{d \sqrt{a + b x^{3}}}{3 b c^{2}} - \frac{\left (a c^{2} - d^{2}\right ) \left (\begin{cases} \frac{\sqrt{a + b x^{3}}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (c \sqrt{a + b x^{3}} + d \right )}}{c} & \text{otherwise} \end{cases}\right )}{3 b c^{2}}\right )}{b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{2 \left (3 \sqrt{a} d + 3 a c\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11376, size = 97, normalized size = 1.33 \begin{align*} -\frac{\frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac{{\left (b x^{3} + a\right )} b c - 2 \, \sqrt{b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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