Optimal. Leaf size=169 \[ -\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}-\frac {(b c+2 a d) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} d^{4/3}}+\frac {(b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac {(b c+2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {393, 206, 31,
648, 631, 210, 642} \begin {gather*} -\frac {(2 a d+b c) \text {ArcTan}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} d^{4/3}}-\frac {(2 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}}+\frac {(2 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 393
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx &=-\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}+\frac {(b c+2 a d) \int \frac {1}{c+d x^3} \, dx}{3 c d}\\ &=-\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}+\frac {(b c+2 a d) \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{9 c^{5/3} d}+\frac {(b c+2 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}\\ &=-\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}+\frac {(b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac {(b c+2 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}{18 c^{5/3} d^{4/3}}+\frac {(b c+2 a d) \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{4/3} d}\\ &=-\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}+\frac {(b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac {(b c+2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}}+\frac {(b c+2 a d) \text {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^{4/3}}\\ &=-\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}-\frac {(b c+2 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^{4/3}}+\frac {(b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac {(b c+2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 145, normalized size = 0.86 \begin {gather*} \frac {-\frac {6 c^{2/3} \sqrt [3]{d} (b c-a d) x}{c+d x^3}-2 \sqrt {3} (b c+2 a d) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+2 (b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-(b c+2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 134, normalized size = 0.79
method | result | size |
risch | \(\frac {\left (a d -b c \right ) x}{3 c d \left (d \,x^{3}+c \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (2 a d +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 c \,d^{2}}\) | \(65\) |
default | \(\frac {\left (a d -b c \right ) x}{3 c d \left (d \,x^{3}+c \right )}+\frac {\left (2 a d +b c \right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{3 c d}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 158, normalized size = 0.93 \begin {gather*} -\frac {{\left (b c - a d\right )} x}{3 \, {\left (c d^{2} x^{3} + c^{2} d\right )}} + \frac {\sqrt {3} {\left (b c + 2 \, 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 )}{9 \, c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {{\left (b c + 2 \, 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 )}{18 \, c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (b c + 2 \, a d\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \, c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.73, size = 537, normalized size = 3.18 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (b c^{3} d + 2 \, a c^{2} d^{2} + {\left (b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{3}\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, c d x^{3} - 3 \, \left (c^{2} d\right )^{\frac {1}{3}} c x - c^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d x^{2} + \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{d x^{3} + c}\right ) - {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) + 2 \, {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right ) - 6 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} x}{18 \, {\left (c^{3} d^{3} x^{3} + c^{4} d^{2}\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (b c^{3} d + 2 \, a c^{2} d^{2} + {\left (b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{3}\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{c^{2}}\right ) - {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) + 2 \, {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right ) - 6 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} x}{18 \, {\left (c^{3} d^{3} x^{3} + c^{4} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 97, normalized size = 0.57 \begin {gather*} \frac {x \left (a d - b c\right )}{3 c^{2} d + 3 c d^{2} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} c^{5} d^{4} - 8 a^{3} d^{3} - 12 a^{2} b c d^{2} - 6 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t c^{2} d}{2 a d + b c} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 160, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {3} {\left (b c + 2 \, 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 )}{9 \, \left (-c d^{2}\right )^{\frac {2}{3}} c} - \frac {{\left (b c + 2 \, 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 )}{18 \, \left (-c d^{2}\right )^{\frac {2}{3}} c} - \frac {{\left (b c + 2 \, 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 )}{9 \, c^{2} d} - \frac {b c x - a d x}{3 \, {\left (d x^{3} + c\right )} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.40, size = 143, normalized size = 0.85 \begin {gather*} \frac {\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (2\,a\,d+b\,c\right )}{9\,c^{5/3}\,d^{4/3}}-\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 (2\,a\,d+b\,c\right )}{9\,c^{5/3}\,d^{4/3}}+\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 (2\,a\,d+b\,c\right )}{9\,c^{5/3}\,d^{4/3}}+\frac {x\,\left (a\,d-b\,c\right )}{3\,c\,d\,\left (d\,x^3+c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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