Optimal. Leaf size=69 \[ -\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 \sqrt {x} (A b-a B)}{b^2}+\frac {2 B x^{3/2}}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 63, 205} \[ \frac {2 \sqrt {x} (A b-a B)}{b^2}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 B x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{a+b x} \, 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} \, dx}{3 b}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {(a (A b-a B)) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{b^2}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {(2 a (A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 (A b-a B) \sqrt {x}}{b^2}+\frac {2 B x^{3/2}}{3 b}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.91 \[ \frac {2 \sqrt {a} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {2 \sqrt {x} (-3 a B+3 A b+b B x)}{3 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 129, normalized size = 1.87 \[ \left [-\frac {3 \, {\left (B a - A b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt {x}}{3 \, b^{2}}, \frac {2 \, {\left (3 \, {\left (B a - A b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt {x}\right )}}{3 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 64, normalized size = 0.93 \[ \frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (B b^{2} x^{\frac {3}{2}} - 3 \, B a b \sqrt {x} + 3 \, A b^{2} \sqrt {x}\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 78, normalized size = 1.13 \[ -\frac {2 A a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {2 B \,a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {2 B \,x^{\frac {3}{2}}}{3 b}+\frac {2 A \sqrt {x}}{b}-\frac {2 B a \sqrt {x}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.04, size = 58, normalized size = 0.84 \[ \frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {2 \, {\left (B b x^{\frac {3}{2}} - 3 \, {\left (B a - A b\right )} \sqrt {x}\right )}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 76, normalized size = 1.10 \[ \sqrt {x}\,\left (\frac {2\,A}{b}-\frac {2\,B\,a}{b^2}\right )+\frac {2\,B\,x^{3/2}}{3\,b}+\frac {2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,\sqrt {x}\,\left (A\,b-B\,a\right )}{B\,a^2-A\,a\,b}\right )\,\left (A\,b-B\,a\right )}{b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.87, size = 212, normalized size = 3.07 \[ \begin {cases} \frac {i A \sqrt {a} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{2} \sqrt {\frac {1}{b}}} - \frac {i A \sqrt {a} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{2} \sqrt {\frac {1}{b}}} + \frac {2 A \sqrt {x}}{b} - \frac {i B a^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{3} \sqrt {\frac {1}{b}}} + \frac {i B a^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{b^{3} \sqrt {\frac {1}{b}}} - \frac {2 B a \sqrt {x}}{b^{2}} + \frac {2 B x^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {5}{2}}}{5}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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