Optimal. Leaf size=73 \[ \frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}} \]
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Rubi [A] time = 0.03, antiderivative size = 73, 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, 208} \[ -\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {A}{a x \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 (a+b x)^{3/2}} \, dx &=-\frac {A}{a x \sqrt {a+b x}}+\frac {\left (-\frac {3 A b}{2}+a B\right ) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{a}\\ &=-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}-\frac {(3 A b-2 a B) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a^2}\\ &=-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}-\frac {(3 A b-2 a B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^2 b}\\ &=-\frac {3 A b-2 a B}{a^2 \sqrt {a+b x}}-\frac {A}{a x \sqrt {a+b x}}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 49, normalized size = 0.67 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x}{a}+1\right ) (2 a B x-3 A b x)-a A}{a^2 x \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 211, normalized size = 2.89 \[ \left [-\frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b x + a}}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, \frac {{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{2} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (A a^{2} - {\left (2 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b x + a}}{a^{3} b x^{2} + a^{4} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 87, normalized size = 1.19 \[ \frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {2 \, {\left (b x + a\right )} B a - 2 \, B a^{2} - 3 \, {\left (b x + a\right )} A b + 2 \, A a b}{{\left ({\left (b x + a\right )}^{\frac {3}{2}} - \sqrt {b x + a} a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 67, normalized size = 0.92 \[ -\frac {2 \left (A b -B a \right )}{\sqrt {b x +a}\, a^{2}}-\frac {2 \left (-\frac {\left (3 A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {\sqrt {b x +a}\, A}{2 x}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 107, normalized size = 1.47 \[ -\frac {1}{2} \, b {\left (\frac {2 \, {\left (2 \, B a^{2} - 2 \, A a b - {\left (2 \, B a - 3 \, A b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} a^{2} b - \sqrt {b x + a} a^{3} b} - \frac {{\left (2 \, B a - 3 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}} b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 82, normalized size = 1.12 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (3\,A\,b-2\,B\,a\right )}{a^{5/2}}-\frac {\frac {2\,\left (A\,b-B\,a\right )}{a}-\frac {\left (3\,A\,b-2\,B\,a\right )\,\left (a+b\,x\right )}{a^2}}{a\,\sqrt {a+b\,x}-{\left (a+b\,x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 48.22, size = 224, normalized size = 3.07 \[ A \left (- \frac {1}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 \sqrt {b}}{a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x}{a}}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{3} \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {a^{2} b x \log {\left (\frac {b x}{a} \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{2} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{a^{\frac {9}{2}} + a^{\frac {7}{2}} b x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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