Optimal. Leaf size=250 \[ -\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{11/3} b^{2/3}}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3} \]
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Rubi [A] time = 0.22, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1854, 1855, 1860, 31, 634, 617, 204, 628} \[ -\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {2 \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{11/3} b^{2/3}}+\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 617
Rule 628
Rule 634
Rule 1854
Rule 1855
Rule 1860
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2}{\left (a+b x^3\right )^4} \, dx &=-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac {\int \frac {-8 c-7 d x}{\left (a+b x^3\right )^3} \, dx}{9 a}\\ &=\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}+\frac {\int \frac {40 c+28 d x}{\left (a+b x^3\right )^2} \, dx}{54 a^2}\\ &=\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac {\int \frac {-80 c-28 d x}{a+b x^3} \, dx}{162 a^3}\\ &=\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac {\int \frac {\sqrt [3]{a} \left (-160 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right )+\sqrt [3]{b} \left (80 \sqrt [3]{b} c-28 \sqrt [3]{a} d\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{486 a^{11/3} \sqrt [3]{b}}+\frac {\left (2 \left (20 c-\frac {7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{11/3}}\\ &=\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{11/3} b^{2/3}}+\frac {\left (20 c+\frac {7 \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{81 a^{10/3}}\\ &=\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac {\left (2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{11/3} b^{2/3}}\\ &=\frac {x (8 c+7 d x)}{54 a^2 \left (a+b x^3\right )^2}+\frac {2 x (10 c+7 d x)}{81 a^3 \left (a+b x^3\right )}-\frac {a e-b x (c+d x)}{9 a b \left (a+b x^3\right )^3}-\frac {2 \left (20 \sqrt [3]{b} c+7 \sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{11/3} b^{2/3}}+\frac {2 \left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac {\left (20 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 239, normalized size = 0.96 \[ \frac {\frac {2 \left (7 a^{2/3} d-20 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {4 \left (20 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {54 a^3 (a e-b x (c+d x))}{b \left (a+b x^3\right )^3}+\frac {9 a^2 x (8 c+7 d x)}{\left (a+b x^3\right )^2}-\frac {4 \sqrt {3} \sqrt [3]{a} \left (7 \sqrt [3]{a} d+20 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {12 a x (10 c+7 d x)}{a+b x^3}}{486 a^4} \]
Antiderivative was successfully verified.
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fricas [C] time = 2.67, size = 2344, normalized size = 9.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 234, normalized size = 0.94 \[ -\frac {2 \, \sqrt {3} {\left (20 \, b c - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {{\left (20 \, b c + 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {2 \, {\left (7 \, d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{4}} + \frac {28 \, b^{3} d x^{8} + 40 \, b^{3} c x^{7} + 77 \, a b^{2} d x^{5} + 104 \, a b^{2} c x^{4} + 67 \, a^{2} b d x^{2} + 82 \, a^{2} b c x - 18 \, a^{3} e}{162 \, {\left (b x^{3} + a\right )}^{3} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 360, normalized size = 1.44 \[ \frac {e \,x^{3}}{9 \left (b \,x^{3}+a \right )^{3} a}+\frac {d \,x^{2}}{9 \left (b \,x^{3}+a \right )^{3} a}+\frac {e \,x^{3}}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {c x}{9 \left (b \,x^{3}+a \right )^{3} a}+\frac {7 d \,x^{2}}{54 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {4 c x}{27 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {14 d \,x^{2}}{81 \left (b \,x^{3}+a \right ) a^{3}}+\frac {20 c x}{81 \left (b \,x^{3}+a \right ) a^{3}}-\frac {e}{9 \left (b \,x^{3}+a \right ) a^{2} b}+\frac {40 \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b}+\frac {40 c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b}-\frac {20 c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b}+\frac {14 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b}-\frac {14 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b}+\frac {7 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 254, normalized size = 1.02 \[ \frac {28 \, b^{3} d x^{8} + 40 \, b^{3} c x^{7} + 77 \, a b^{2} d x^{5} + 104 \, a b^{2} c x^{4} + 67 \, a^{2} b d x^{2} + 82 \, a^{2} b c x - 18 \, a^{3} e}{162 \, {\left (a^{3} b^{4} x^{9} + 3 \, a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{3} + a^{6} b\right )}} + \frac {2 \, \sqrt {3} {\left (7 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (7 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 247, normalized size = 0.99 \[ \frac {\frac {67\,d\,x^2}{162\,a}-\frac {e}{9\,b}+\frac {41\,c\,x}{81\,a}+\frac {20\,b^2\,c\,x^7}{81\,a^3}+\frac {14\,b^2\,d\,x^8}{81\,a^3}+\frac {52\,b\,c\,x^4}{81\,a^2}+\frac {77\,b\,d\,x^5}{162\,a^2}}{a^3+3\,a^2\,b\,x^3+3\,a\,b^2\,x^6+b^3\,x^9}+\left (\sum _{k=1}^3\ln \left (\frac {b\,\left (560\,c\,d+196\,d^2\,x+{\mathrm {root}\left (14348907\,a^{11}\,b^2\,z^3+408240\,a^4\,b\,c\,d\,z-64000\,b\,c^3+2744\,a\,d^3,z,k\right )}^2\,a^7\,b\,59049+\mathrm {root}\left (14348907\,a^{11}\,b^2\,z^3+408240\,a^4\,b\,c\,d\,z-64000\,b\,c^3+2744\,a\,d^3,z,k\right )\,a^3\,b\,c\,x\,9720\right )}{a^6\,6561}\right )\,\mathrm {root}\left (14348907\,a^{11}\,b^2\,z^3+408240\,a^4\,b\,c\,d\,z-64000\,b\,c^3+2744\,a\,d^3,z,k\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.47, size = 202, normalized size = 0.81 \[ \operatorname {RootSum} {\left (14348907 t^{3} a^{11} b^{2} + 408240 t a^{4} b c d + 2744 a d^{3} - 64000 b c^{3}, \left (t \mapsto t \log {\left (x + \frac {413343 t^{2} a^{8} b d + 194400 t a^{4} b c^{2} + 7840 a c d^{2}}{1372 a d^{3} + 32000 b c^{3}} \right )} \right )\right )} + \frac {- 18 a^{3} e + 82 a^{2} b c x + 67 a^{2} b d x^{2} + 104 a b^{2} c x^{4} + 77 a b^{2} d x^{5} + 40 b^{3} c x^{7} + 28 b^{3} d x^{8}}{162 a^{6} b + 486 a^{5} b^{2} x^{3} + 486 a^{4} b^{3} x^{6} + 162 a^{3} b^{4} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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