Optimal. Leaf size=53 \[ \frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{3/2}}+\frac {2 B x^{3/2}}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {459, 329, 275, 205} \[ \frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{3/2}}+\frac {2 B x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \frac {\sqrt {x} \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {2 B x^{3/2}}{3 b}-\frac {\left (2 \left (-\frac {3 A b}{2}+\frac {3 a B}{2}\right )\right ) \int \frac {\sqrt {x}}{a+b x^3} \, dx}{3 b}\\ &=\frac {2 B x^{3/2}}{3 b}-\frac {\left (4 \left (-\frac {3 A b}{2}+\frac {3 a B}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^6} \, dx,x,\sqrt {x}\right )}{3 b}\\ &=\frac {2 B x^{3/2}}{3 b}+\frac {(2 (A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^{3/2}\right )}{3 b}\\ &=\frac {2 B x^{3/2}}{3 b}+\frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 \sqrt {a} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 52, normalized size = 0.98 \[ \frac {2}{3} \left (\frac {B x^{3/2}}{b}-\frac {(a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 108, normalized size = 2.04 \[ \left [\frac {2 \, B a b x^{\frac {3}{2}} + {\left (B a - A b\right )} \sqrt {-a b} \log \left (\frac {b x^{3} - 2 \, \sqrt {-a b} x^{\frac {3}{2}} - a}{b x^{3} + a}\right )}{3 \, a b^{2}}, \frac {2 \, {\left (B a b x^{\frac {3}{2}} - {\left (B a - A b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{\frac {3}{2}}}{a}\right )\right )}}{3 \, a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 39, normalized size = 0.74 \[ \frac {2 \, B x^{\frac {3}{2}}}{3 \, b} - \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 53, normalized size = 1.00 \[ \frac {2 A \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \sqrt {a b}}-\frac {2 B a \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \sqrt {a b}\, b}+\frac {2 B \,x^{\frac {3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 39, normalized size = 0.74 \[ \frac {2 \, B x^{\frac {3}{2}}}{3 \, b} - \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 93, normalized size = 1.75 \[ \frac {2\,B\,x^{3/2}}{3\,b}-\frac {2\,\mathrm {atan}\left (\frac {3\,\sqrt {a}\,b^{3/2}\,x^{3/2}\,\left (24\,A^2\,b^3-48\,A\,B\,a\,b^2+24\,B^2\,a^2\,b\right )}{\left (72\,B\,a^2\,b^2-72\,A\,a\,b^3\right )\,\left (A\,b-B\,a\right )}\right )\,\left (A\,b-B\,a\right )}{3\,\sqrt {a}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.39, size = 537, normalized size = 10.13 \[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} + \frac {2 B x^{\frac {3}{2}}}{3}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} + \frac {2 B x^{\frac {3}{2}}}{3}}{b} & \text {for}\: a = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {9}{2}}}{9}}{a} & \text {for}\: b = 0 \\- \frac {i A \log {\left (- \sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 \sqrt {a} b \sqrt {\frac {1}{b}}} + \frac {i A \log {\left (\sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 \sqrt {a} b \sqrt {\frac {1}{b}}} + \frac {i A \log {\left (- 4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{3 \sqrt {a} b \sqrt {\frac {1}{b}}} - \frac {i A \log {\left (4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{3 \sqrt {a} b \sqrt {\frac {1}{b}}} + \frac {i B \sqrt {a} \log {\left (- \sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 b^{2} \sqrt {\frac {1}{b}}} - \frac {i B \sqrt {a} \log {\left (\sqrt [6]{-1} \sqrt [6]{a} \sqrt [6]{\frac {1}{b}} + \sqrt {x} \right )}}{3 b^{2} \sqrt {\frac {1}{b}}} - \frac {i B \sqrt {a} \log {\left (- 4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{3 b^{2} \sqrt {\frac {1}{b}}} + \frac {i B \sqrt {a} \log {\left (4 \sqrt [6]{-1} \sqrt [6]{a} \sqrt {x} \sqrt [6]{\frac {1}{b}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac {1}{b}} + 4 x \right )}}{3 b^{2} \sqrt {\frac {1}{b}}} + \frac {2 B x^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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