Optimal. Leaf size=258 \[ -\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{7/3}}-\frac {x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
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Rubi [A] time = 0.23, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {413, 385, 200, 31, 634, 617, 204, 628} \[ -\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{7/3}}-\frac {x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 385
Rule 413
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2}{\left (c+d x^3\right )^3} \, dx &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}+\frac {\int \frac {a (b c+5 a d)+2 b (2 b c+a d) x^3}{\left (c+d x^3\right )^2} \, dx}{6 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{c+d x^3} \, dx}{9 c^2 d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{27 c^{8/3} d^2}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \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^2}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \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^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{18 c^{7/3} d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \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^{7/3}}\\ &=-\frac {(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \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^{7/3}}+\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac {\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 234, normalized size = 0.91 \[ \frac {2 \left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt {3} \left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )-\frac {3 c^{2/3} \sqrt [3]{d} x \left (-a^2 d^2 \left (8 c+5 d x^3\right )+2 a b c d \left (2 c-d x^3\right )+b^2 c^2 \left (4 c+7 d x^3\right )\right )}{\left (c+d x^3\right )^2}-\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1067, normalized size = 4.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 264, normalized size = 1.02 \[ -\frac {\sqrt {3} {\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-c d^{2}\right )^{\frac {2}{3}} c^{2} d} - \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{54 \, \left (-c d^{2}\right )^{\frac {2}{3}} c^{2} d} - \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{27 \, c^{3} d^{2}} - \frac {7 \, b^{2} c^{2} d x^{4} - 2 \, a b c d^{2} x^{4} - 5 \, a^{2} d^{3} x^{4} + 4 \, b^{2} c^{3} x + 4 \, a b c^{2} d x - 8 \, a^{2} c d^{2} x}{18 \, {\left (d x^{3} + c\right )}^{2} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \text {hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 267, normalized size = 1.03 \[ -\frac {{\left (7 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 4 \, {\left (b^{2} c^{3} + a b c^{2} d - 2 \, a^{2} c d^{2}\right )} x}{18 \, {\left (c^{2} d^{4} x^{6} + 2 \, c^{3} d^{3} x^{3} + c^{4} d^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{27 \, c^{2} d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{54 \, c^{2} d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{27 \, c^{2} d^{3} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 249, normalized size = 0.97 \[ \frac {\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{27\,c^{8/3}\,d^{7/3}}-\frac {\frac {2\,x\,\left (-2\,a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{9\,c\,d^2}-\frac {x^4\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d-7\,b^2\,c^2\right )}{18\,c^2\,d}}{c^2+2\,c\,d\,x^3+d^2\,x^6}+\frac {\ln \left (2\,d^{1/3}\,x-c^{1/3}+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{27\,c^{8/3}\,d^{7/3}}-\frac {\ln \left (c^{1/3}-2\,d^{1/3}\,x+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,a^2\,d^2+2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{27\,c^{8/3}\,d^{7/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 233, normalized size = 0.90 \[ \frac {x^{4} \left (5 a^{2} d^{3} + 2 a b c d^{2} - 7 b^{2} c^{2} d\right ) + x \left (8 a^{2} c d^{2} - 4 a b c^{2} d - 4 b^{2} c^{3}\right )}{18 c^{4} d^{2} + 36 c^{3} d^{3} x^{3} + 18 c^{2} d^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} c^{8} d^{7} - 125 a^{6} d^{6} - 150 a^{5} b c d^{5} - 210 a^{4} b^{2} c^{2} d^{4} - 128 a^{3} b^{3} c^{3} d^{3} - 84 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left (t \mapsto t \log {\left (\frac {27 t c^{3} d^{2}}{5 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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