Optimal. Leaf size=88 \[ -\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 88, 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 = {78, 51, 63, 205} \begin {gather*} -\frac {3 A b-a B}{a^2 b \sqrt {x}}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx &=\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {\left (-\frac {3 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{a b}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 59, normalized size = 0.67 \begin {gather*} \frac {(a+b x) (a B-3 A b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b x}{a}\right )+a (A b-a B)}{a^2 b \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 67, normalized size = 0.76 \begin {gather*} \frac {(a B-3 A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}+\frac {-2 a A+a B x-3 A b x}{a^2 \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 215, normalized size = 2.44 \begin {gather*} \left [\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{a^{3} b^{2} x^{2} + a^{4} b x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 60, normalized size = 0.68 \begin {gather*} \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {B a x - 3 \, A b x - 2 \, A a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 87, normalized size = 0.99 \begin {gather*} -\frac {3 A b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}-\frac {A b \sqrt {x}}{\left (b x +a \right ) a^{2}}+\frac {B \sqrt {x}}{\left (b x +a \right ) a}-\frac {2 A}{a^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.93, size = 65, normalized size = 0.74 \begin {gather*} -\frac {2 \, A a - {\left (B a - 3 \, A b\right )} x}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} + \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 65, normalized size = 0.74 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {x\,\left (3\,A\,b-B\,a\right )}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b-B\,a\right )}{a^{5/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.12, size = 884, normalized size = 10.05 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {4 i A a^{\frac {3}{2}} b \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {6 i A \sqrt {a} b^{2} x \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 A a b \sqrt {x} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 A a b \sqrt {x} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {2 i B a^{\frac {3}{2}} b x \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {B a^{2} \sqrt {x} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {B a^{2} \sqrt {x} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {B a b x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {B a b x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} b \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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