Optimal. Leaf size=197 \[ -\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}-\frac {(b c+5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{4/3}}+\frac {(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac {(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {393, 205, 206,
31, 648, 631, 210, 642} \begin {gather*} -\frac {(5 a d+b c) \text {ArcTan}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} d^{4/3}}-\frac {(5 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac {(5 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}+\frac {x (5 a d+b c)}{18 c^2 d \left (c+d x^3\right )}-\frac {x (b c-a d)}{6 c d \left (c+d x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 205
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 )^3} \, dx &=-\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) \int \frac {1}{\left (c+d x^3\right )^2} \, dx}{6 c d}\\ &=-\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac {(b c+5 a d) \int \frac {1}{c+d x^3} \, dx}{9 c^2 d}\\ &=-\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac {(b c+5 a d) \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{27 c^{8/3} d}+\frac {(b c+5 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}{27 c^{8/3} d}\\ &=-\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac {(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac {(b c+5 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}{54 c^{8/3} d^{4/3}}+\frac {(b c+5 a d) \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{18 c^{7/3} d}\\ &=-\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}+\frac {(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac {(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}+\frac {(b c+5 a d) \text {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^{4/3}}\\ &=-\frac {(b c-a d) x}{6 c d \left (c+d x^3\right )^2}+\frac {(b c+5 a d) x}{18 c^2 d \left (c+d x^3\right )}-\frac {(b c+5 a d) \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^{4/3}}+\frac {(b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{4/3}}-\frac {(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 175, normalized size = 0.89 \begin {gather*} \frac {-\frac {9 c^{5/3} \sqrt [3]{d} (b c-a d) x}{\left (c+d x^3\right )^2}+\frac {3 c^{2/3} \sqrt [3]{d} (b c+5 a d) x}{c+d x^3}-2 \sqrt {3} (b c+5 a d) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+2 (b c+5 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-(b c+5 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 153, normalized size = 0.78
method | result | size |
risch | \(\frac {\frac {\left (5 a d +b c \right ) x^{4}}{18 c^{2}}+\frac {\left (4 a d -b c \right ) x}{9 c d}}{\left (d \,x^{3}+c \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (5 a d +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 c^{2} d^{2}}\) | \(84\) |
default | \(\frac {\frac {\left (5 a d +b c \right ) x^{4}}{18 c^{2}}+\frac {\left (4 a d -b c \right ) x}{9 c d}}{\left (d \,x^{3}+c \right )^{2}}+\frac {\left (5 a d +b c \right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{9 c^{2} d}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 192, normalized size = 0.97 \begin {gather*} \frac {{\left (b c d + 5 \, a d^{2}\right )} x^{4} - 2 \, {\left (b c^{2} - 4 \, a c d\right )} x}{18 \, {\left (c^{2} d^{3} x^{6} + 2 \, c^{3} d^{2} x^{3} + c^{4} d\right )}} + \frac {\sqrt {3} {\left (b c + 5 \, 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 )}{27 \, c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {{\left (b c + 5 \, 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 )}{54 \, c^{2} d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (b c + 5 \, a d\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{27 \, c^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 351 vs.
\(2 (156) = 312\).
time = 2.54, size = 743, normalized size = 3.77 \begin {gather*} \left [\frac {3 \, {\left (b c^{3} d^{2} + 5 \, a c^{2} d^{3}\right )} x^{4} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (b c^{2} d^{3} + 5 \, a c d^{4}\right )} x^{6} + b c^{4} d + 5 \, a c^{3} d^{2} + 2 \, {\left (b c^{3} d^{2} + 5 \, a c^{2} 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} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \, {\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\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} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \, {\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\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^{4} d - 4 \, a c^{3} d^{2}\right )} x}{54 \, {\left (c^{4} d^{4} x^{6} + 2 \, c^{5} d^{3} x^{3} + c^{6} d^{2}\right )}}, \frac {3 \, {\left (b c^{3} d^{2} + 5 \, a c^{2} d^{3}\right )} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left ({\left (b c^{2} d^{3} + 5 \, a c d^{4}\right )} x^{6} + b c^{4} d + 5 \, a c^{3} d^{2} + 2 \, {\left (b c^{3} d^{2} + 5 \, a c^{2} 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} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \, {\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\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} + 5 \, a d^{3}\right )} x^{6} + b c^{3} + 5 \, a c^{2} d + 2 \, {\left (b c^{2} d + 5 \, a c d^{2}\right )} x^{3}\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^{4} d - 4 \, a c^{3} d^{2}\right )} x}{54 \, {\left (c^{4} d^{4} x^{6} + 2 \, c^{5} d^{3} x^{3} + c^{6} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 133, normalized size = 0.68 \begin {gather*} \frac {x^{4} \cdot \left (5 a d^{2} + b c d\right ) + x \left (8 a c d - 2 b c^{2}\right )}{18 c^{4} d + 36 c^{3} d^{2} x^{3} + 18 c^{2} d^{3} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} c^{8} d^{4} - 125 a^{3} d^{3} - 75 a^{2} b c d^{2} - 15 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log {\left (\frac {27 t c^{3} d}{5 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 = 0.84, size = 180, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {3} {\left (b c + 5 \, 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 )}{27 \, \left (-c d^{2}\right )^{\frac {2}{3}} c^{2}} - \frac {{\left (b c + 5 \, 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 )}{54 \, \left (-c d^{2}\right )^{\frac {2}{3}} c^{2}} - \frac {{\left (b c + 5 \, 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 )}{27 \, c^{3} d} + \frac {b c d x^{4} + 5 \, a d^{2} x^{4} - 2 \, b c^{2} x + 8 \, a c d x}{18 \, {\left (d x^{3} + c\right )}^{2} c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.40, size = 173, normalized size = 0.88 \begin {gather*} \frac {\frac {x^4\,\left (5\,a\,d+b\,c\right )}{18\,c^2}+\frac {x\,\left (4\,a\,d-b\,c\right )}{9\,c\,d}}{c^2+2\,c\,d\,x^3+d^2\,x^6}+\frac {\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (5\,a\,d+b\,c\right )}{27\,c^{8/3}\,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 (5\,a\,d+b\,c\right )}{27\,c^{8/3}\,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 (5\,a\,d+b\,c\right )}{27\,c^{8/3}\,d^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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