Optimal. Leaf size=197 \[ -\frac{(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac{x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
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Rubi [A] time = 0.106395, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {385, 199, 200, 31, 634, 617, 204, 628} \[ -\frac{(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac{x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 385
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{a+b x^3}{\left (c+d x^3\right )^3} \, dx &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) \int \frac{1}{\left (c+d x^3\right )^2} \, dx}{6 c d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \int \frac{1}{c+d x^3} \, dx}{9 c^2 d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{27 c^{8/3} d}+\frac{(b c+5 a d) \int \frac{2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{27 c^{8/3} d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(b c+5 a d) \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{54 c^{8/3} d^{4/3}}+\frac{(b c+5 a d) \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{18 c^{7/3} d}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac{(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac{(b c+5 a d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{9 c^{8/3} d^{4/3}}\\ &=-\frac{(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac{(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}-\frac{(b c+5 a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{4/3}}+\frac{(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac{(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.131517, size = 175, normalized size = 0.89 \[ \frac{-(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-\frac{9 c^{5/3} \sqrt [3]{d} x (b c-a d)}{\left (c+d x^3\right )^2}+\frac{3 c^{2/3} \sqrt [3]{d} x (5 a d+b c)}{c+d x^3}+2 (5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} (5 a d+b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{54 c^{8/3} d^{4/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 249, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ({\frac{ \left ( 5\,ad+bc \right ){x}^{4}}{18\,{c}^{2}}}+{\frac{ \left ( 4\,ad-bc \right ) x}{9\,cd}} \right ) }+{\frac{5\,a}{27\,{c}^{2}d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{27\,c{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,a}{54\,{c}^{2}d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{54\,c{d}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}a}{27\,{c}^{2}d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{27\,c{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \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 = 1.72292, size = 1648, normalized size = 8.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36349, size = 133, normalized size = 0.68 \begin{align*} \frac{x^{4} \left (5 a d^{2} + b c d\right ) + x \left (8 a c d - 2 b c^{2}\right )}{18 c^{4} d + 36 c^{3} d^{2} x^{3} + 18 c^{2} d^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} c^{8} d^{4} - 125 a^{3} d^{3} - 75 a^{2} b c d^{2} - 15 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (\frac{27 t c^{3} d}{5 a d + b c} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14816, size = 273, normalized size = 1.39 \begin{align*} -\frac{{\left (b c + 5 \, a d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{27 \, c^{3} d} + \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{27 \, c^{3} d^{2}} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{54 \, c^{3} d^{2}} + \frac{b c d x^{4} + 5 \, a d^{2} x^{4} - 2 \, b c^{2} x + 8 \, a c d x}{18 \,{\left (d x^{3} + c\right )}^{2} c^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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