Optimal. Leaf size=203 \[ \frac{(b c-a d) (a d+2 b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}}-\frac{2 (b c-a d) (a d+2 b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac{2 (b c-a d) (a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} d^{7/3}}+\frac{x (b c-a d)^2}{3 c d^2 \left (c+d x^3\right )}+\frac{b^2 x}{d^2} \]
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Rubi [A] time = 0.243648, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {390, 385, 200, 31, 634, 617, 204, 628} \[ \frac{(b c-a d) (a d+2 b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}}-\frac{2 (b c-a d) (a d+2 b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac{2 (b c-a d) (a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} d^{7/3}}+\frac{x (b c-a d)^2}{3 c d^2 \left (c+d x^3\right )}+\frac{b^2 x}{d^2} \]
Antiderivative was successfully verified.
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Rule 390
Rule 385
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^2}{\left (c+d x^3\right )^2} \, dx &=\int \left (\frac{b^2}{d^2}-\frac{b^2 c^2-a^2 d^2+2 b d (b c-a d) x^3}{d^2 \left (c+d x^3\right )^2}\right ) \, dx\\ &=\frac{b^2 x}{d^2}-\frac{\int \frac{b^2 c^2-a^2 d^2+2 b d (b c-a d) x^3}{\left (c+d x^3\right )^2} \, dx}{d^2}\\ &=\frac{b^2 x}{d^2}+\frac{(b c-a d)^2 x}{3 c d^2 \left (c+d x^3\right )}-\frac{(2 (b c-a d) (2 b c+a d)) \int \frac{1}{c+d x^3} \, dx}{3 c d^2}\\ &=\frac{b^2 x}{d^2}+\frac{(b c-a d)^2 x}{3 c d^2 \left (c+d x^3\right )}-\frac{(2 (b c-a d) (2 b c+a d)) \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{9 c^{5/3} d^2}-\frac{(2 (b c-a d) (2 b c+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}{9 c^{5/3} d^2}\\ &=\frac{b^2 x}{d^2}+\frac{(b c-a d)^2 x}{3 c d^2 \left (c+d x^3\right )}-\frac{2 (b c-a d) (2 b c+a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac{((b c-a d) (2 b c+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}{9 c^{5/3} d^{7/3}}-\frac{((b c-a d) (2 b c+a d)) \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{4/3} d^2}\\ &=\frac{b^2 x}{d^2}+\frac{(b c-a d)^2 x}{3 c d^2 \left (c+d x^3\right )}-\frac{2 (b c-a d) (2 b c+a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac{(b c-a d) (2 b c+a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}}-\frac{(2 (b c-a d) (2 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{3 c^{5/3} d^{7/3}}\\ &=\frac{b^2 x}{d^2}+\frac{(b c-a d)^2 x}{3 c d^2 \left (c+d x^3\right )}+\frac{2 (b c-a d) (2 b c+a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} d^{7/3}}-\frac{2 (b c-a d) (2 b c+a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{7/3}}+\frac{(b c-a d) (2 b c+a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} d^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.194235, size = 210, normalized size = 1.03 \[ \frac{\frac{\left (-a^2 d^2-a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3}}-\frac{2 \left (-a^2 d^2-a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3}}+\frac{2 \sqrt{3} \left (-a^2 d^2-a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{5/3}}+\frac{3 \sqrt [3]{d} x (b c-a d)^2}{c \left (c+d x^3\right )}+9 b^2 \sqrt [3]{d} x}{9 d^{7/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 367, normalized size = 1.8 \begin{align*}{\frac{{b}^{2}x}{{d}^{2}}}+{\frac{{a}^{2}x}{3\,c \left ( d{x}^{3}+c \right ) }}-{\frac{2\,xab}{3\,d \left ( d{x}^{3}+c \right ) }}+{\frac{cx{b}^{2}}{3\,{d}^{2} \left ( d{x}^{3}+c \right ) }}+{\frac{2\,{a}^{2}}{9\,cd}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,ab}{9\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,{b}^{2}c}{9\,{d}^{3}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}}{9\,cd}\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{ab}{9\,{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{2\,{b}^{2}c}{9\,{d}^{3}}\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{2\,\sqrt{3}{a}^{2}}{9\,cd}\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{2\,\sqrt{3}ab}{9\,{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}}}}-{\frac{4\,c\sqrt{3}{b}^{2}}{9\,{d}^{3}}\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 = 1.72141, size = 1635, normalized size = 8.05 \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 = 1.75829, size = 189, normalized size = 0.93 \begin{align*} \frac{b^{2} x}{d^{2}} + \frac{x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 c^{2} d^{2} + 3 c d^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} c^{5} d^{7} - 8 a^{6} d^{6} - 24 a^{5} b c d^{5} + 24 a^{4} b^{2} c^{2} d^{4} + 88 a^{3} b^{3} c^{3} d^{3} - 48 a^{2} b^{4} c^{4} d^{2} - 96 a b^{5} c^{5} d + 64 b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{9 t c^{2} d^{2}}{2 a^{2} d^{2} + 2 a b c d - 4 b^{2} c^{2}} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11978, size = 358, normalized size = 1.76 \begin{align*} \frac{b^{2} x}{d^{2}} + \frac{2 \,{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{9 \, c^{2} d^{2}} - \frac{2 \, \sqrt{3}{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - \left (-c d^{2}\right )^{\frac{1}{3}} a b c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\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 )}{9 \, c^{2} d^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{3 \,{\left (d x^{3} + c\right )} c d^{2}} - \frac{{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} - \left (-c d^{2}\right )^{\frac{1}{3}} a b c d - \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{9 \, c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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