Optimal. Leaf size=210 \[ \frac{A b-a B}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.120337, antiderivative size = 210, normalized size of antiderivative = 1., 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 770
Rule 77
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*}
Mathematica [A] time = 0.0514607, size = 104, normalized size = 0.5 \[ \frac{a \left (52 a^2 A b^2 x+25 a^3 A b-3 a^4 B+42 a A b^3 x^2+12 A b^4 x^3\right )+12 A b \log (x) (a+b x)^4-12 A b (a+b x)^4 \log (a+b x)}{12 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 205, normalized size = 1. \begin{align*}{\frac{ \left ( 12\,A\ln \left ( x \right ){x}^{4}{b}^{5}-12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{5}+48\,A\ln \left ( x \right ){x}^{3}a{b}^{4}-48\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{4}+72\,A\ln \left ( x \right ){x}^{2}{a}^{2}{b}^{3}-72\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}+12\,A{x}^{3}a{b}^{4}+48\,A\ln \left ( x \right ) x{a}^{3}{b}^{2}-48\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}+42\,A{x}^{2}{a}^{2}{b}^{3}+12\,A\ln \left ( x \right ){a}^{4}b-12\,A\ln \left ( bx+a \right ){a}^{4}b+52\,A{a}^{3}{b}^{2}x+25\,A{a}^{4}b-3\,B{a}^{5} \right ) \left ( bx+a \right ) }{12\,b{a}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66049, size = 437, normalized size = 2.08 \begin{align*} \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 \left (x\right )}{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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{x \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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