Optimal. Leaf size=49 \[ -\frac {2 \sqrt {x} (2 A b-a B)}{a^2 \sqrt {a+b x}}-\frac {2 A}{a \sqrt {x} \sqrt {a+b x}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \[ -\frac {2 \sqrt {x} (2 A b-a B)}{a^2 \sqrt {a+b x}}-\frac {2 A}{a \sqrt {x} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} (a+b x)^{3/2}} \, dx &=-\frac {2 A}{a \sqrt {x} \sqrt {a+b x}}+\frac {\left (2 \left (-A b+\frac {a B}{2}\right )\right ) \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx}{a}\\ &=-\frac {2 A}{a \sqrt {x} \sqrt {a+b x}}-\frac {2 (2 A b-a B) \sqrt {x}}{a^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.67 \[ \frac {2 (-a A+a B x-2 A b x)}{a^2 \sqrt {x} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 43, normalized size = 0.88 \[ -\frac {2 \, {\left (A a - {\left (B a - 2 \, A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{a^{2} b x^{2} + a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 160, normalized size = 3.27 \[ -\frac {2 \, \sqrt {b x + a} A b^{2}}{\sqrt {{\left (b x + a\right )} b - a b} a^{2} {\left | b \right |}} + \frac {4 \, {\left (B^{2} a^{2} b^{3} - 2 \, A B a b^{4} + A^{2} b^{5}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} + B a^{2} b^{\frac {5}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} - A a b^{\frac {7}{2}}\right )} a {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.61 \[ -\frac {2 \left (2 A x b -B a x +A a \right )}{\sqrt {b x +a}\, a^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 55, normalized size = 1.12 \[ \frac {2 \, B x}{\sqrt {b x^{2} + a x} a} - \frac {4 \, A b x}{\sqrt {b x^{2} + a x} a^{2}} - \frac {2 \, A}{\sqrt {b x^{2} + a x} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 50, normalized size = 1.02 \[ -\frac {\left (\frac {2\,A}{a\,b}+\frac {x\,\left (4\,A\,b-2\,B\,a\right )}{a^2\,b}\right )\,\sqrt {a+b\,x}}{x^{3/2}+\frac {a\,\sqrt {x}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.16, size = 63, normalized size = 1.29 \[ A \left (- \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} + 1}} - \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} + 1}}\right ) + \frac {2 B}{a \sqrt {b} \sqrt {\frac {a}{b x} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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