Optimal. Leaf size=90 \[ \frac {2 a^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}}-\frac {2 a \sqrt {x} (A b-a B)}{b^3}+\frac {2 x^{3/2} (A b-a B)}{3 b^2}+\frac {2 B x^{5/2}}{5 b} \]
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Rubi [A] time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 63, 205} \[ \frac {2 a^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}}+\frac {2 x^{3/2} (A b-a B)}{3 b^2}-\frac {2 a \sqrt {x} (A b-a B)}{b^3}+\frac {2 B x^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rubi steps
\begin {align*} \int \frac {x^{3/2} (A+B x)}{a+b x} \, dx &=\frac {2 B x^{5/2}}{5 b}+\frac {\left (2 \left (\frac {5 A b}{2}-\frac {5 a B}{2}\right )\right ) \int \frac {x^{3/2}}{a+b x} \, dx}{5 b}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{5/2}}{5 b}-\frac {(a (A b-a B)) \int \frac {\sqrt {x}}{a+b x} \, dx}{b^2}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{5/2}}{5 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^3}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{5/2}}{5 b}+\frac {\left (2 a^2 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 a (A b-a B) \sqrt {x}}{b^3}+\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{5/2}}{5 b}+\frac {2 a^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 81, normalized size = 0.90 \[ \frac {2 \sqrt {x} \left (15 a^2 B-5 a b (3 A+B x)+b^2 x (5 A+3 B x)\right )}{15 b^3}-\frac {2 a^{3/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 180, normalized size = 2.00 \[ \left [-\frac {15 \, {\left (B a^{2} - A a b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \, {\left (B a b - A b^{2}\right )} x\right )} \sqrt {x}}{15 \, b^{3}}, -\frac {2 \, {\left (15 \, {\left (B a^{2} - A a b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) - {\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \, {\left (B a b - A b^{2}\right )} x\right )} \sqrt {x}\right )}}{15 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 91, normalized size = 1.01 \[ -\frac {2 \, {\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (3 \, B b^{4} x^{\frac {5}{2}} - 5 \, B a b^{3} x^{\frac {3}{2}} + 5 \, A b^{4} x^{\frac {3}{2}} + 15 \, B a^{2} b^{2} \sqrt {x} - 15 \, A a b^{3} \sqrt {x}\right )}}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 102, normalized size = 1.13 \[ \frac {2 B \,x^{\frac {5}{2}}}{5 b}+\frac {2 A \,a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {2 B \,a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {2 A \,x^{\frac {3}{2}}}{3 b}-\frac {2 B a \,x^{\frac {3}{2}}}{3 b^{2}}-\frac {2 A a \sqrt {x}}{b^{2}}+\frac {2 B \,a^{2} \sqrt {x}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.93, size = 82, normalized size = 0.91 \[ -\frac {2 \, {\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (3 \, B b^{2} x^{\frac {5}{2}} - 5 \, {\left (B a b - A b^{2}\right )} x^{\frac {3}{2}} + 15 \, {\left (B a^{2} - A a b\right )} \sqrt {x}\right )}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 101, normalized size = 1.12 \[ x^{3/2}\,\left (\frac {2\,A}{3\,b}-\frac {2\,B\,a}{3\,b^2}\right )+\frac {2\,B\,x^{5/2}}{5\,b}-\frac {2\,a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,\sqrt {x}\,\left (A\,b-B\,a\right )}{B\,a^3-A\,a^2\,b}\right )\,\left (A\,b-B\,a\right )}{b^{7/2}}-\frac {a\,\sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.82, size = 245, normalized size = 2.72 \[ \begin {cases} - \frac {i A a^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{3} \sqrt {\frac {1}{b}}} + \frac {i A a^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{3} \sqrt {\frac {1}{b}}} - \frac {2 A a \sqrt {x}}{b^{2}} + \frac {2 A x^{\frac {3}{2}}}{3 b} + \frac {i B a^{\frac {5}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{4} \sqrt {\frac {1}{b}}} - \frac {i B a^{\frac {5}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{4} \sqrt {\frac {1}{b}}} + \frac {2 B a^{2} \sqrt {x}}{b^{3}} - \frac {2 B a x^{\frac {3}{2}}}{3 b^{2}} + \frac {2 B x^{\frac {5}{2}}}{5 b} & \text {for}\: b \neq 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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