Optimal. Leaf size=318 \[ -\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}+\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}} \]
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Rubi [A]
time = 0.38, antiderivative size = 318, 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, 331, 335,
215, 648, 632, 210, 642, 211} \begin {gather*} \frac {(11 A b-5 a B) \text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \text {ArcTan}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \text {ArcTan}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 215
Rule 331
Rule 335
Rule 468
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^{7/2} \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}+\frac {\left (\frac {11 A b}{2}-\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{6 a^2}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \text {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{3 a^2}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 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^{17/6}}-\frac {(11 A b-5 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^{17/6}}-\frac {(11 A b-5 a B) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{8/3}}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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^{8/3}}-\frac {(11 A b-5 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^{8/3}}+\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}+\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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^{17/6} \sqrt [6]{b}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 188, normalized size = 0.59 \begin {gather*} \frac {\frac {6 a^{5/6} \left (-6 a A-11 A b x^3+5 a B x^3\right )}{x^{5/2} \left (a+b x^3\right )}+\frac {10 (-11 A b+5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac {5 (11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{\sqrt [6]{b}}+\frac {5 \sqrt {3} (-11 A b+5 a B) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{\sqrt [6]{b}}}{90 a^{17/6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 217, normalized size = 0.68
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A b}{6}-\frac {B a}{6}\right ) \sqrt {x}}{b \,x^{3}+a}+\frac {\left (11 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{6}\right )}{a^{2}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}\) | \(217\) |
default | \(-\frac {2 \left (\frac {\left (\frac {A b}{6}-\frac {B a}{6}\right ) \sqrt {x}}{b \,x^{3}+a}+\frac {\left (11 A b -5 B a \right ) \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}\right )}{6}\right )}{a^{2}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}\) | \(217\) |
risch | \(-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {\sqrt {x}\, A b}{3 a^{2} \left (b \,x^{3}+a \right )}+\frac {\sqrt {x}\, B}{3 a \left (b \,x^{3}+a \right )}-\frac {11 A b \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 a^{3}}-\frac {11 A b \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 a^{3}}+\frac {11 A b \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 a^{3}}-\frac {11 A b \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 a^{3}}-\frac {11 A b \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 a^{3}}+\frac {5 B \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 a^{2}}+\frac {5 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 a^{2}}-\frac {5 B \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 a^{2}}+\frac {5 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 a^{2}}+\frac {5 B \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 a^{2}}\) | \(395\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 312, normalized size = 0.98 \begin {gather*} \frac {{\left (5 \, B a - 11 \, A b\right )} x^{3} - 6 \, A a}{15 \, {\left (a^{2} b x^{\frac {11}{2}} + a^{3} x^{\frac {5}{2}}\right )}} + \frac {\frac {\sqrt {3} {\left (5 \, B a - 11 \, 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 (5 \, B a - 11 \, 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 (5 \, B a b^{\frac {1}{3}} - 11 \, 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 (5 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - 11 \, 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 (5 \, B a^{\frac {4}{3}} b^{\frac {1}{3}} - 11 \, 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 \, a^{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. 2584 vs.
\(2 (232) = 464\).
time = 2.46, size = 2584, normalized size = 8.13 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 313, normalized size = 0.98 \begin {gather*} \frac {\sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 11 \, \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^{3} b} - \frac {\sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 11 \, \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^{3} b} + \frac {B a \sqrt {x} - A b \sqrt {x}}{3 \, {\left (b x^{3} + a\right )} a^{2}} + \frac {{\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 11 \, \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^{3} b} + \frac {{\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 11 \, \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^{3} b} + \frac {{\left (5 \, \left (a b^{5}\right )^{\frac {1}{6}} B a - 11 \, \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^{3} b} - \frac {2 \, A}{5 \, a^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.96, size = 2080, normalized size = 6.54 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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