Optimal. Leaf size=67 \[ -\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 A}{a^2 \sqrt {a+b x}}+\frac {2 (A b-a B)}{3 a b (a+b x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 67, 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} \begin {gather*} \frac {2 A}{a^2 \sqrt {a+b x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 (A b-a B)}{3 a b (a+b x)^{3/2}} \end {gather*}
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 (a+b x)^{5/2}} \, dx &=\frac {2 (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac {A \int \frac {1}{x (a+b x)^{3/2}} \, dx}{a}\\ &=\frac {2 (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac {2 A}{a^2 \sqrt {a+b x}}+\frac {A \int \frac {1}{x \sqrt {a+b x}} \, dx}{a^2}\\ &=\frac {2 (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac {2 A}{a^2 \sqrt {a+b x}}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^2 b}\\ &=\frac {2 (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac {2 A}{a^2 \sqrt {a+b x}}-\frac {2 A \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 = 56, normalized size = 0.84 \begin {gather*} \frac {2 a (A b-a B)+6 A b (a+b x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x}{a}+1\right )}{3 a^2 b (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 64, normalized size = 0.96 \begin {gather*} \frac {2 \left (a^2 (-B)+3 A b (a+b x)+a A b\right )}{3 a^2 b (a+b x)^{3/2}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 221, normalized size = 3.30 \begin {gather*} \left [\frac {3 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, A a b^{2} x - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x + a}}{3 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}, \frac {2 \, {\left (3 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, A a b^{2} x - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x + a}\right )}}{3 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 61, normalized size = 0.91 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} - \frac {2 \, {\left (B a^{2} - 3 \, {\left (b x + a\right )} A b - A a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.88 \begin {gather*} \frac {-\frac {2 A b \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2 A b}{\sqrt {b x +a}\, a^{2}}-\frac {2 \left (-A b +B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} a}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.98, size = 69, normalized size = 1.03 \begin {gather*} \frac {A \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {2 \, {\left (B a^{2} - 3 \, {\left (b x + a\right )} A b - A a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 56, normalized size = 0.84 \begin {gather*} \frac {\frac {2\,\left (A\,b-B\,a\right )}{3\,a}+\frac {2\,A\,b\,\left (a+b\,x\right )}{a^2}}{b\,{\left (a+b\,x\right )}^{3/2}}-\frac {2\,A\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.77, size = 68, normalized size = 1.01 \begin {gather*} \frac {2 A}{a^{2} \sqrt {a + b x}} + \frac {2 A \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{a^{2} \sqrt {- a}} - \frac {2 \left (- A b + B a\right )}{3 a b \left (a + b x\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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