Optimal. Leaf size=208 \[ -\frac {(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac {(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{10/3}}+\frac {d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {d^2 x^4 (3 b c-a d)}{4 b^2}+\frac {d^3 x^7}{7 b} \]
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Rubi [A] time = 0.15, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {390, 200, 31, 634, 617, 204, 628} \[ \frac {d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}-\frac {(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac {(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{10/3}}+\frac {d^2 x^4 (3 b c-a d)}{4 b^2}+\frac {d^3 x^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 390
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^3}{a+b x^3} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3}+\frac {d^2 (3 b c-a d) x^3}{b^2}+\frac {d^3 x^6}{b}+\frac {b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3}{b^3 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^7}{7 b}+\frac {(b c-a d)^3 \int \frac {1}{a+b x^3} \, dx}{b^3}\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^7}{7 b}+\frac {(b c-a d)^3 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b^3}+\frac {(b c-a d)^3 \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^3}\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^7}{7 b}+\frac {(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac {(b c-a d)^3 \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}{6 a^{2/3} b^{10/3}}+\frac {(b c-a d)^3 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^3}\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^7}{7 b}+\frac {(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac {(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac {(b c-a d)^3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{10/3}}\\ &=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x}{b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^7}{7 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{10/3}}+\frac {(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac {(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 203, normalized size = 0.98 \[ \frac {\frac {14 (a d-b c)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac {28 (b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {28 \sqrt {3} (b c-a d)^3 \tan ^{-1}\left (\frac {2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3}}+84 \sqrt [3]{b} d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+21 b^{4/3} d^2 x^4 (3 b c-a d)+12 b^{7/3} d^3 x^7}{84 b^{10/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 700, normalized size = 3.37 \[ \left [\frac {12 \, a^{2} b^{3} d^{3} x^{7} + 21 \, {\left (3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{4} - 42 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac {1}{3}} a x - a^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 14 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) + 28 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) + 84 \, {\left (3 \, a^{2} b^{3} c^{2} d - 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x}{84 \, a^{2} b^{4}}, \frac {12 \, a^{2} b^{3} d^{3} x^{7} + 21 \, {\left (3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{4} + 84 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 14 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) + 28 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) + 84 \, {\left (3 \, a^{2} b^{3} c^{2} d - 3 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x}{84 \, a^{2} b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 296, normalized size = 1.42 \[ -\frac {\sqrt {3} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac {4 \, b^{6} d^{3} x^{7} + 21 \, b^{6} c d^{2} x^{4} - 7 \, a b^{5} d^{3} x^{4} + 84 \, b^{6} c^{2} d x - 84 \, a b^{5} c d^{2} x + 28 \, a^{2} b^{4} d^{3} x}{28 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 486, normalized size = 2.34 \[ \frac {d^{3} x^{7}}{7 b}-\frac {a \,d^{3} x^{4}}{4 b^{2}}+\frac {3 c \,d^{2} x^{4}}{4 b}-\frac {\sqrt {3}\, a^{3} d^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {a^{3} d^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {a^{3} d^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {\sqrt {3}\, a^{2} c \,d^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {a^{2} c \,d^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {a^{2} c \,d^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {a^{2} d^{3} x}{b^{3}}-\frac {\sqrt {3}\, a \,c^{2} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {a \,c^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {a \,c^{2} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {3 a c \,d^{2} x}{b^{2}}+\frac {\sqrt {3}\, c^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {c^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {c^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {3 c^{2} d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 273, normalized size = 1.31 \[ \frac {4 \, b^{2} d^{3} x^{7} + 7 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 28 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{28 \, b^{3}} + \frac {\sqrt {3} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 192, normalized size = 0.92 \[ x\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )+\frac {d^3\,x^7}{7\,b}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^3}{3\,a^{2/3}\,b^{10/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^3}{3\,a^{2/3}\,b^{10/3}}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{2/3}\,b^{10/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 257, normalized size = 1.24 \[ x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{2} b^{10} + a^{9} d^{9} - 9 a^{8} b c d^{8} + 36 a^{7} b^{2} c^{2} d^{7} - 84 a^{6} b^{3} c^{3} d^{6} + 126 a^{5} b^{4} c^{4} d^{5} - 126 a^{4} b^{5} c^{5} d^{4} + 84 a^{3} b^{6} c^{6} d^{3} - 36 a^{2} b^{7} c^{7} d^{2} + 9 a b^{8} c^{8} d - b^{9} c^{9}, \left (t \mapsto t \log {\left (- \frac {3 t a b^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )} \right )\right )} + \frac {d^{3} x^{7}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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