Optimal. Leaf size=202 \[ \frac{a^3 (A b-a B)}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^2 (3 A b-4 a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x) (A b-3 a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) (A b-2 a B) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.137825, antiderivative size = 202, 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} \[ \frac{a^3 (A b-a B)}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^2 (3 A b-4 a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x) (A b-3 a B)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (a+b x) (A b-2 a B) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{x^3 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{x^3 (A+B x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac{A b-3 a B}{b^7}+\frac{B x}{b^6}+\frac{a^3 (-A b+a B)}{b^7 (a+b x)^3}-\frac{a^2 (-3 A b+4 a B)}{b^7 (a+b x)^2}+\frac{3 a (-A b+2 a B)}{b^7 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{a^2 (3 A b-4 a B)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^3 (A b-a B)}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-3 a B) x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 a (A b-2 a B) (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0546608, size = 117, normalized size = 0.58 \[ \frac{-a^2 b^2 x (4 A+11 B x)+a^3 (2 b B x-5 A b)+7 a^4 B+4 a b^3 x^2 (A-B x)+6 a (a+b x)^2 (2 a B-A b) \log (a+b x)+b^4 x^3 (2 A+B x)}{2 b^5 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 191, normalized size = 1. \begin{align*} -{\frac{ \left ( -{b}^{4}B{x}^{4}+6\,A\ln \left ( bx+a \right ){x}^{2}a{b}^{3}-2\,A{x}^{3}{b}^{4}-12\,B\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+4\,B{x}^{3}a{b}^{3}+12\,A\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}-4\,A{x}^{2}a{b}^{3}-24\,B\ln \left ( bx+a \right ) x{a}^{3}b+11\,B{x}^{2}{a}^{2}{b}^{2}+6\,A\ln \left ( bx+a \right ){a}^{3}b+4\,A{a}^{2}{b}^{2}x-12\,B\ln \left ( bx+a \right ){a}^{4}-2\,B{a}^{3}bx+5\,A{a}^{3}b-7\,B{a}^{4} \right ) \left ( bx+a \right ) }{2\,{b}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02563, size = 419, normalized size = 2.07 \begin{align*} \frac{B x^{3}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{5 \, B a x^{2}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac{A x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{6 \, B a^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{3 \, A a \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{9 \, B a^{4}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{9 \, A a^{3} b}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, A a^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, B a^{3} x}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{5 \, B a^{3}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac{2 \, A a^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} + \frac{5 \, B a^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{4}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{A a^{3}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57715, size = 350, normalized size = 1.73 \begin{align*} \frac{B b^{4} x^{4} + 7 \, B a^{4} - 5 \, A a^{3} b - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{3} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x + 6 \,{\left (2 \, B a^{4} - A a^{3} b +{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\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{x^{3} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{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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