Optimal. Leaf size=98 \[ \frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {5 A b-2 a B}{a^3 \sqrt {a+b x}}+\frac {2 a B-5 A b}{3 a^2 (a+b x)^{3/2}}-\frac {A}{a x (a+b x)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {5 A b-2 a B}{a^3 \sqrt {a+b x}}-\frac {5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}+\frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {A}{a x (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^2 (a+b x)^{5/2}} \, dx &=-\frac {A}{a x (a+b x)^{3/2}}+\frac {\left (-\frac {5 A b}{2}+a B\right ) \int \frac {1}{x (a+b x)^{5/2}} \, dx}{a}\\ &=-\frac {5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}-\frac {A}{a x (a+b x)^{3/2}}-\frac {(5 A b-2 a B) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{2 a^2}\\ &=-\frac {5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}-\frac {A}{a x (a+b x)^{3/2}}-\frac {5 A b-2 a B}{a^3 \sqrt {a+b x}}-\frac {(5 A b-2 a B) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a^3}\\ &=-\frac {5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}-\frac {A}{a x (a+b x)^{3/2}}-\frac {5 A b-2 a B}{a^3 \sqrt {a+b x}}-\frac {(5 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^3 b}\\ &=-\frac {5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}-\frac {A}{a x (a+b x)^{3/2}}-\frac {5 A b-2 a B}{a^3 \sqrt {a+b x}}+\frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.53 \begin {gather*} \frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b x}{a}+1\right ) (2 a B x-5 A b x)-3 a A}{3 a^2 x (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 111, normalized size = 1.13 \begin {gather*} \frac {(5 A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {2 a^3 B-2 a^2 A b+4 a^2 B (a+b x)-10 a A b (a+b x)+15 A b (a+b x)^2-6 a B (a+b x)^2}{3 a^3 b x (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.47, size = 330, normalized size = 3.37 \begin {gather*} \left [-\frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, A a^{3} - 3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} - 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{6 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}, \frac {3 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (3 \, A a^{3} - 3 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} - 4 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{3 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 90, normalized size = 0.92 \begin {gather*} \frac {{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} - \frac {\sqrt {b x + a} A}{a^{3} x} + \frac {2 \, {\left (3 \, {\left (b x + a\right )} B a + B a^{2} - 6 \, {\left (b x + a\right )} A b - A a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 88, normalized size = 0.90 \begin {gather*} -\frac {2 \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} a^{2}}-\frac {2 \left (2 A b -B a \right )}{\sqrt {b x +a}\, a^{3}}-\frac {2 \left (-\frac {\left (5 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^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.94, size = 130, normalized size = 1.33 \begin {gather*} -\frac {1}{6} \, b {\left (\frac {2 \, {\left (2 \, B a^{3} - 2 \, A a^{2} b - 3 \, {\left (2 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{2} + 2 \, {\left (2 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {5}{2}} a^{3} b - {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b} - \frac {3 \, {\left (2 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 103, normalized size = 1.05 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-2\,B\,a\right )}{a^{7/2}}-\frac {\frac {2\,\left (A\,b-B\,a\right )}{3\,a}+\frac {2\,\left (5\,A\,b-2\,B\,a\right )\,\left (a+b\,x\right )}{3\,a^2}-\frac {\left (5\,A\,b-2\,B\,a\right )\,{\left (a+b\,x\right )}^2}{a^3}}{a\,{\left (a+b\,x\right )}^{3/2}-{\left (a+b\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 60.63, size = 1520, normalized size = 15.51
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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