Optimal. Leaf size=312 \[ -\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}-\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}} \]
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Rubi [A]
time = 0.35, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {468, 327, 335,
215, 648, 632, 210, 642, 211} \begin {gather*} -\frac {(A b-7 a B) \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {ArcTan}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {ArcTan}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}-\frac {\sqrt {x} (A b-7 a B)}{3 a b^2}+\frac {x^{7/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 215
Rule 327
Rule 335
Rule 468
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {\left (-\frac {A b}{2}+\frac {7 a B}{2}\right ) \int \frac {x^{5/2}}{a+b x^3} \, dx}{3 a b}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{6 b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{3 b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{5/6} b^2}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{5/6} b^2}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{2/3} b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{36 a^{2/3} b^2}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{36 a^{2/3} b^2}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{18 \sqrt {3} a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{18 \sqrt {3} a^{5/6} b^{13/6}}\\ &=-\frac {(A b-7 a B) \sqrt {x}}{3 a b^2}+\frac {(A b-a B) x^{7/2}}{3 a b \left (a+b x^3\right )}-\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac {(A b-7 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}+\frac {(A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{5/6} b^{13/6}}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 181, normalized size = 0.58 \begin {gather*} \frac {\frac {6 \sqrt [6]{b} \sqrt {x} \left (-A b+7 a B+6 b B x^3\right )}{a+b x^3}+\frac {2 (A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac {(-A b+7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{a^{5/6}}+\frac {\sqrt {3} (A b-7 a B) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{a^{5/6}}}{18 b^{13/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 216, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{6}+\frac {B a}{6}\right ) \sqrt {x}}{b \,x^{3}+a}+\frac {\left (A b -7 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{3}}{b^{2}}\) | \(216\) |
default | \(\frac {2 B \sqrt {x}}{b^{2}}+\frac {\frac {2 \left (-\frac {A b}{6}+\frac {B a}{6}\right ) \sqrt {x}}{b \,x^{3}+a}+\frac {\left (A b -7 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{3}}{b^{2}}\) | \(216\) |
risch | \(\frac {2 B \sqrt {x}}{b^{2}}-\frac {\sqrt {x}\, A}{3 b \left (b \,x^{3}+a \right )}+\frac {\sqrt {x}\, B a}{3 b^{2} \left (b \,x^{3}+a \right )}+\frac {A \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b a}+\frac {A \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 b a}-\frac {A \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b a}+\frac {A \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b a}+\frac {A \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 b a}-\frac {7 B \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b^{2}}-\frac {7 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 b^{2}}+\frac {7 B \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 b^{2}}-\frac {7 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b^{2}}-\frac {7 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 b^{2}}\) | \(405\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 311, normalized size = 1.00 \begin {gather*} \frac {{\left (B a - A b\right )} \sqrt {x}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} + \frac {2 \, B \sqrt {x}}{b^{2}} - \frac {\frac {\sqrt {3} {\left (7 \, B a - A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (7 \, B a - A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (7 \, B a b^{\frac {1}{3}} - A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (7 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{36 \, b^{2}} \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. 2566 vs.
\(2 (232) = 464\).
time = 1.69, size = 2566, normalized size = 8.22 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1658 vs.
\(2 (299) = 598\).
time = 185.02, size = 1658, normalized size = 5.31 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {13}{2}}}{13}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} + 2 B \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\- \frac {12 A a b \sqrt {x}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {2 A a b \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 A a b \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {A a b \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {A a b \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A a b \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A a b \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {2 A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {2 \sqrt {3} A b^{2} x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {84 B a^{2} \sqrt {x}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {14 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {7 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {7 B a^{2} \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a^{2} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a^{2} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {72 B a b x^{\frac {7}{2}}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {14 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} + \frac {7 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {7 B a b x^{3} \sqrt [6]{- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a b x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} - \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} - \frac {14 \sqrt {3} B a b x^{3} \sqrt [6]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt {x}}{3 \sqrt [6]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{36 a^{2} b^{2} + 36 a b^{3} x^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 313, normalized size = 1.00 \begin {gather*} \frac {2 \, B \sqrt {x}}{b^{2}} - \frac {\sqrt {3} {\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a b^{3}} + \frac {\sqrt {3} {\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a b^{3}} + \frac {B a \sqrt {x} - A b \sqrt {x}}{3 \, {\left (b x^{3} + a\right )} b^{2}} - \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a b^{3}} - \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a b^{3}} - \frac {{\left (7 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 \, a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.89, size = 1884, normalized size = 6.04 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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