Optimal. Leaf size=288 \[ \frac {\sqrt [6]{a} (A b-a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}+\frac {\sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{3 b^{13/6}}-\frac {2 \sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}+\frac {2 \sqrt {x} (A b-a B)}{b^2}+\frac {2 B x^{7/2}}{7 b} \]
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Rubi [A] time = 0.52, antiderivative size = 288, 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 = {459, 321, 329, 209, 634, 618, 204, 628, 205} \[ \frac {2 \sqrt {x} (A b-a B)}{b^2}+\frac {\sqrt [6]{a} (A b-a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}+\frac {\sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{3 b^{13/6}}-\frac {2 \sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}+\frac {2 B x^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 209
Rule 321
Rule 329
Rule 459
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {2 B x^{7/2}}{7 b}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {7 a B}{2}\right )\right ) \int \frac {x^{5/2}}{a+b x^3} \, dx}{7 b}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(a (A b-a B)) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{b^2}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {(2 a (A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {\left (2 \sqrt [6]{a} (A b-a B)\right ) \operatorname {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 )}{3 b^2}-\frac {\left (2 \sqrt [6]{a} (A b-a B)\right ) \operatorname {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 )}{3 b^2}-\frac {\left (2 \sqrt [3]{a} (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{3 b^2}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {2 \sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}+\frac {\left (\sqrt [6]{a} (A b-a B)\right ) \operatorname {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 )}{2 \sqrt {3} b^{13/6}}-\frac {\left (\sqrt [6]{a} (A b-a B)\right ) \operatorname {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 )}{2 \sqrt {3} b^{13/6}}-\frac {\left (\sqrt [3]{a} (A b-a B)\right ) \operatorname {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 )}{6 b^2}-\frac {\left (\sqrt [3]{a} (A b-a B)\right ) \operatorname {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 )}{6 b^2}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}-\frac {2 \sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}+\frac {\sqrt [6]{a} (A b-a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}-\frac {\left (\sqrt [6]{a} (A b-a B)\right ) \operatorname {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 )}{3 \sqrt {3} b^{13/6}}+\frac {\left (\sqrt [6]{a} (A b-a B)\right ) \operatorname {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 )}{3 \sqrt {3} b^{13/6}}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{7/2}}{7 b}+\frac {\sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}-\frac {2 \sqrt [6]{a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 b^{13/6}}+\frac {\sqrt [6]{a} (A b-a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}-\frac {\sqrt [6]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{2 \sqrt {3} b^{13/6}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 54, normalized size = 0.19 \[ \frac {2 \sqrt {x} \left ((7 a B-7 A b) \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-\frac {b x^3}{a}\right )-7 a B+7 A b+b B x^3\right )}{7 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.31, size = 2433, normalized size = 8.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 289, normalized size = 1.00 \[ \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{3}} - \frac {\sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b^{3}} + \frac {{\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{3}} + \frac {{\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{3}} + \frac {2 \, {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a - \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{3}} + \frac {2 \, {\left (B b^{6} x^{\frac {7}{2}} - 7 \, B a b^{5} \sqrt {x} + 7 \, A b^{6} \sqrt {x}\right )}}{7 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 377, normalized size = 1.31 \[ \frac {2 B \,x^{\frac {7}{2}}}{7 b}-\frac {2 \left (\frac {a}{b}\right )^{\frac {1}{6}} A \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} A \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{3 b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} A \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{3 b}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} A \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 b}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} A \ln \left (-x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 b}+\frac {2 \left (\frac {a}{b}\right )^{\frac {1}{6}} B a \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} B a \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{3 b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} B a \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{3 b^{2}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} B a \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 b^{2}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} B a \ln \left (-x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 b^{2}}+\frac {2 A \sqrt {x}}{b}-\frac {2 B a \sqrt {x}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 295, normalized size = 1.02 \[ \frac {{\left (\frac {\sqrt {3} {\left (B a - A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (B a - A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (B a b^{\frac {1}{3}} - A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (B a^{\frac {4}{3}} b^{\frac {1}{3}} - A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )} a}{6 \, b^{2}} + \frac {2 \, {\left (B b x^{\frac {7}{2}} - 7 \, {\left (B a - A b\right )} \sqrt {x}\right )}}{7 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.89, size = 1933, normalized size = 6.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 177.80, size = 881, normalized size = 3.06 \[ \begin {cases} \tilde {\infty } \left (2 A \sqrt {x} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 A \sqrt {x} + \frac {2 B x^{\frac {7}{2}}}{7}}{b} & \text {for}\: a = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {13}{2}}}{13}}{a} & \text {for}\: b = 0 \\\frac {\sqrt [6]{-1} A \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} \log {\left (- \sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 b} - \frac {\sqrt [6]{-1} A \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} \log {\left (\sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 b} + \frac {\sqrt [6]{-1} A \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} \log {\left (- 4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{6 b} - \frac {\sqrt [6]{-1} A \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} \log {\left (4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{6 b} - \frac {\sqrt [6]{-1} \sqrt {3} A \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} \sqrt {x}}{3 \sqrt [6]{a} \sqrt [6]{\frac {1}{b}}} \right )}}{3 b} + \frac {\sqrt [6]{-1} \sqrt {3} A \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} \sqrt {x}}{3 \sqrt [6]{a} \sqrt [6]{\frac {1}{b}}} \right )}}{3 b} + \frac {2 A \sqrt {x}}{b} - \frac {\sqrt [6]{-1} B a^{\frac {7}{6}} \sqrt [6]{\frac {1}{b}} \log {\left (- \sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 b^{2}} + \frac {\sqrt [6]{-1} B a^{\frac {7}{6}} \sqrt [6]{\frac {1}{b}} \log {\left (\sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 b^{2}} - \frac {\sqrt [6]{-1} B a^{\frac {7}{6}} \sqrt [6]{\frac {1}{b}} \log {\left (- 4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{6 b^{2}} + \frac {\sqrt [6]{-1} B a^{\frac {7}{6}} \sqrt [6]{\frac {1}{b}} \log {\left (4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{6 b^{2}} + \frac {\sqrt [6]{-1} \sqrt {3} B a^{\frac {7}{6}} \sqrt [6]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} \sqrt {x}}{3 \sqrt [6]{a} \sqrt [6]{\frac {1}{b}}} \right )}}{3 b^{2}} - \frac {\sqrt [6]{-1} \sqrt {3} B a^{\frac {7}{6}} \sqrt [6]{\frac {1}{b}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} + \frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} \sqrt {x}}{3 \sqrt [6]{a} \sqrt [6]{\frac {1}{b}}} \right )}}{3 b^{2}} - \frac {2 B a \sqrt {x}}{b^{2}} + \frac {2 B x^{\frac {7}{2}}}{7 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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