Optimal. Leaf size=201 \[ \frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+5 a B) x^2}{18 a b^2 \left (a+b x^3\right )}-\frac {(A b+5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{8/3}}-\frac {(A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}+\frac {(A b+5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {468, 294, 298,
31, 648, 631, 210, 642} \begin {gather*} -\frac {(5 a B+A b) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{8/3}}+\frac {(5 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}-\frac {(5 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}-\frac {x^2 (5 a B+A b)}{18 a b^2 \left (a+b x^3\right )}+\frac {x^5 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 294
Rule 298
Rule 468
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}+\frac {(A b+5 a B) \int \frac {x^4}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+5 a B) x^2}{18 a b^2 \left (a+b x^3\right )}+\frac {(A b+5 a B) \int \frac {x}{a+b x^3} \, dx}{9 a b^2}\\ &=\frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+5 a B) x^2}{18 a b^2 \left (a+b x^3\right )}-\frac {(A b+5 a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{4/3} b^{7/3}}+\frac {(A b+5 a B) \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{4/3} b^{7/3}}\\ &=\frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+5 a B) x^2}{18 a b^2 \left (a+b x^3\right )}-\frac {(A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}+\frac {(A b+5 a B) \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}{54 a^{4/3} b^{8/3}}+\frac {(A b+5 a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a b^{7/3}}\\ &=\frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+5 a B) x^2}{18 a b^2 \left (a+b x^3\right )}-\frac {(A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}+\frac {(A b+5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}+\frac {(A b+5 a B) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{4/3} b^{8/3}}\\ &=\frac {(A b-a B) x^5}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+5 a B) x^2}{18 a b^2 \left (a+b x^3\right )}-\frac {(A b+5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{8/3}}-\frac {(A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{8/3}}+\frac {(A b+5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{8/3}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 181, normalized size = 0.90 \begin {gather*} \frac {-\frac {9 b^{2/3} (A b-a B) x^2}{\left (a+b x^3\right )^2}+\frac {6 b^{2/3} (A b-4 a B) x^2}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} (A b+5 a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}-\frac {2 (A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {(A b+5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}}{54 b^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 154, normalized size = 0.77
method | result | size |
risch | \(\frac {\frac {\left (A b -4 B a \right ) x^{5}}{9 a b}-\frac {\left (A b +5 B a \right ) x^{2}}{18 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (A b +5 B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{27 b^{3} a}\) | \(85\) |
default | \(\frac {\frac {\left (A b -4 B a \right ) x^{5}}{9 a b}-\frac {\left (A b +5 B a \right ) x^{2}}{18 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (A b +5 B a \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 b^{2} a}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 195, normalized size = 0.97 \begin {gather*} -\frac {2 \, {\left (4 \, B a b - A b^{2}\right )} x^{5} + {\left (5 \, B a^{2} + A a b\right )} x^{2}}{18 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}} + \frac {\sqrt {3} {\left (5 \, B a + A b\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 )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (5 \, B a + A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (5 \, B a + A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {1}{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. 355 vs.
\(2 (160) = 320\).
time = 1.49, size = 756, normalized size = 3.76 \begin {gather*} \left [-\frac {6 \, {\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} x^{5} + 3 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left ({\left (5 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 5 \, B a^{4} b + A a^{3} b^{2} + 2 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - {\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \, {\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \, {\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{2} b^{6} x^{6} + 2 \, a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}, -\frac {6 \, {\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} x^{5} + 3 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left ({\left (5 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 5 \, B a^{4} b + A a^{3} b^{2} + 2 \, {\left (5 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \, {\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (5 \, B a b^{2} + A b^{3}\right )} x^{6} + 5 \, B a^{3} + A a^{2} b + 2 \, {\left (5 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{2} b^{6} x^{6} + 2 \, a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 6.23, size = 155, normalized size = 0.77 \begin {gather*} \frac {x^{5} \cdot \left (2 A b^{2} - 8 B a b\right ) + x^{2} \left (- A a b - 5 B a^{2}\right )}{18 a^{3} b^{2} + 36 a^{2} b^{3} x^{3} + 18 a b^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{4} b^{8} + A^{3} b^{3} + 15 A^{2} B a b^{2} + 75 A B^{2} a^{2} b + 125 B^{3} a^{3}, \left ( t \mapsto t \log {\left (\frac {729 t^{2} a^{3} b^{5}}{A^{2} b^{2} + 10 A B a b + 25 B^{2} a^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 206, normalized size = 1.02 \begin {gather*} \frac {\sqrt {3} {\left (5 \, B a + A b\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 )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2}} - \frac {{\left (5 \, B a + A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b^{2}} - \frac {{\left (5 \, B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} + A b \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} - \frac {8 \, B a b x^{5} - 2 \, A b^{2} x^{5} + 5 \, B a^{2} x^{2} + A a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 175, normalized size = 0.87 \begin {gather*} -\frac {\frac {x^2\,\left (A\,b+5\,B\,a\right )}{18\,b^2}-\frac {x^5\,\left (A\,b-4\,B\,a\right )}{9\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b+5\,B\,a\right )}{27\,a^{4/3}\,b^{8/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b+5\,B\,a\right )}{27\,a^{4/3}\,b^{8/3}}+\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\,b+5\,B\,a\right )}{27\,a^{4/3}\,b^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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