Optimal. Leaf size=210 \[ \frac {A b-a B}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A \log (x) (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 77} \[ \frac {A b-a B}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A \log (x) (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{x \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {A}{a^5 b^5 x}+\frac {-A b+a B}{a b^5 (a+b x)^5}-\frac {A}{a^2 b^4 (a+b x)^4}-\frac {A}{a^3 b^4 (a+b x)^3}-\frac {A}{a^4 b^4 (a+b x)^2}-\frac {A}{a^5 b^4 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {A}{a^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{4 a b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{3 a^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A}{2 a^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 104, normalized size = 0.50 \[ \frac {a \left (-3 a^4 B+25 a^3 A b+52 a^2 A b^2 x+42 a A b^3 x^2+12 A b^4 x^3\right )+12 A b \log (x) (a+b x)^4-12 A b (a+b x)^4 \log (a+b x)}{12 a^5 b (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 203, normalized size = 0.97 \[ \frac {12 \, A a b^{4} x^{3} + 42 \, A a^{2} b^{3} x^{2} + 52 \, A a^{3} b^{2} x - 3 \, B a^{5} + 25 \, A a^{4} b - 12 \, {\left (A b^{5} x^{4} + 4 \, A a b^{4} x^{3} + 6 \, A a^{2} b^{3} x^{2} + 4 \, A a^{3} b^{2} x + A a^{4} b\right )} \log \left (b x + a\right ) + 12 \, {\left (A b^{5} x^{4} + 4 \, A a b^{4} x^{3} + 6 \, A a^{2} b^{3} x^{2} + 4 \, A a^{3} b^{2} x + A a^{4} b\right )} \log \relax (x)}{12 \, {\left (a^{5} b^{5} x^{4} + 4 \, a^{6} b^{4} x^{3} + 6 \, a^{7} b^{3} x^{2} + 4 \, a^{8} b^{2} x + a^{9} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 205, normalized size = 0.98 \[ -\frac {\left (-12 A \,b^{5} x^{4} \ln \relax (x )+12 A \,b^{5} x^{4} \ln \left (b x +a \right )-48 A a \,b^{4} x^{3} \ln \relax (x )+48 A a \,b^{4} x^{3} \ln \left (b x +a \right )-72 A \,a^{2} b^{3} x^{2} \ln \relax (x )+72 A \,a^{2} b^{3} x^{2} \ln \left (b x +a \right )-12 A a \,b^{4} x^{3}-48 A \,a^{3} b^{2} x \ln \relax (x )+48 A \,a^{3} b^{2} x \ln \left (b x +a \right )-42 A \,a^{2} b^{3} x^{2}-12 A \,a^{4} b \ln \relax (x )+12 A \,a^{4} b \ln \left (b x +a \right )-52 A \,a^{3} b^{2} x -25 A \,a^{4} b +3 B \,a^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{5} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 138, normalized size = 0.66 \[ -\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} + \frac {A}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}} + \frac {A}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}} + \frac {A}{2 \, a^{3} b^{2} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {B}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} + \frac {A}{4 \, a b^{4} {\left (x + \frac {a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B x}{x \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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