Optimal. Leaf size=93 \[ -\frac{2 c \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )}+\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2} \]
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Rubi [A] time = 0.222704, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2155, 706, 31, 635, 207, 260} \[ -\frac{2 c \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )}+\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2} \]
Antiderivative was successfully verified.
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Rule 2155
Rule 706
Rule 31
Rule 635
Rule 207
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{x \left (a c+b c x^3+d \sqrt{a+b x^3}\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x \left (a c+b c x+d \sqrt{a+b x}\right )} \, dx,x,x^3\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{(d+c x) \left (-a+x^2\right )} \, dx,x,\sqrt{a+b x^3}\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{d-c x}{-a+x^2} \, dx,x,\sqrt{a+b x^3}\right )}{3 \left (a c^2-d^2\right )}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{d+c x} \, dx,x,\sqrt{a+b x^3}\right )}{3 \left (a c^2-d^2\right )}\\ &=-\frac{2 c \log \left (d+c \sqrt{a+b x^3}\right )}{3 \left (a c^2-d^2\right )}+\frac{(2 c) \operatorname{Subst}\left (\int \frac{x}{-a+x^2} \, dx,x,\sqrt{a+b x^3}\right )}{3 \left (a c^2-d^2\right )}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b x^3}\right )}{3 \left (a c^2-d^2\right )}\\ &=\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2}-\frac{2 c \log \left (d+c \sqrt{a+b x^3}\right )}{3 \left (a c^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 0.126702, size = 107, normalized size = 1.15 \[ \frac{\left (\sqrt{a} c-d\right ) \log \left (\sqrt{a}-\sqrt{a+b x^3}\right )+\left (\sqrt{a} c+d\right ) \log \left (\sqrt{a+b x^3}+\sqrt{a}\right )-2 \sqrt{a} c \log \left (c \sqrt{a+b x^3}+d\right )}{3 \sqrt{a} \left (a c^2-d^2\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 636, normalized size = 6.8 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b c x^{3} + a c + \sqrt{b x^{3} + a} d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55523, size = 536, normalized size = 5.76 \begin{align*} \left [-\frac{a c \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + a c \log \left (\sqrt{b x^{3} + a} c + d\right ) - a c \log \left (\sqrt{b x^{3} + a} c - d\right ) - 3 \, a c \log \left (x\right ) - \sqrt{a} d \log \left (\frac{b x^{3} + 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right )}{3 \,{\left (a^{2} c^{2} - a d^{2}\right )}}, -\frac{a c \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) + a c \log \left (\sqrt{b x^{3} + a} c + d\right ) - a c \log \left (\sqrt{b x^{3} + a} c - d\right ) - 3 \, a c \log \left (x\right ) + 2 \, \sqrt{-a} d \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right )}{3 \,{\left (a^{2} c^{2} - a d^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.01097, size = 97, normalized size = 1.04 \begin{align*} - \frac{2 c^{2} \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 \left (a c^{2} - d^{2}\right )} - \frac{2 \left (- \frac{c \log{\left (- b x^{3} \right )}}{2} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{- a}} \right )}}{\sqrt{- a}}\right )}{3 \left (a c^{2} - d^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13327, size = 127, normalized size = 1.37 \begin{align*} -\frac{2 \, c^{2} \log \left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \,{\left (a c^{3} - c d^{2}\right )}} + \frac{c \log \left (b x^{3}\right )}{3 \,{\left (a c^{2} - d^{2}\right )}} - \frac{2 \, d \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \,{\left (a c^{2} - d^{2}\right )} \sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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