Optimal. Leaf size=49 \[ -\frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {2 A}{a \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {78, 63, 205} \[ -\frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {2 A}{a \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 205
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} (a+b x)} \, dx &=-\frac {2 A}{a \sqrt {x}}+\frac {\left (2 \left (-\frac {A b}{2}+\frac {a B}{2}\right )\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{a}\\ &=-\frac {2 A}{a \sqrt {x}}+\frac {\left (4 \left (-\frac {A b}{2}+\frac {a B}{2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a}\\ &=-\frac {2 A}{a \sqrt {x}}-\frac {2 (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 1.00 \[ \frac {2 (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {2 A}{a \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 112, normalized size = 2.29 \[ \left [-\frac {2 \, A a b \sqrt {x} - {\left (B a - A b\right )} \sqrt {-a b} x \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right )}{a^{2} b x}, -\frac {2 \, {\left (A a b \sqrt {x} + {\left (B a - A b\right )} \sqrt {a b} x \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right )\right )}}{a^{2} b x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 39, normalized size = 0.80 \[ \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {2 \, A}{a \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 1.08 \[ -\frac {2 A b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {2 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {2 A}{a \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 39, normalized size = 0.80 \[ \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {2 \, A}{a \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 50, normalized size = 1.02 \[ \frac {2\,B\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}}-\frac {2\,A\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2\,A}{a\,\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.29, size = 216, normalized size = 4.41 \[ \begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{b} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a} & \text {for}\: b = 0 \\- \frac {2 A}{a \sqrt {x}} + \frac {i A \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {i A \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {i B \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{\sqrt {a} b \sqrt {\frac {1}{b}}} + \frac {i B \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{\sqrt {a} b \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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