Optimal. Leaf size=161 \[ \frac{3 e^2 (a+b x) (b d-a e) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0940398, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{3 e^2 (a+b x) (b d-a e) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\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{(d+e x)^3}{\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{e^3}{b^6}+\frac{(b d-a e)^3}{b^6 (a+b x)^3}+\frac{3 e (b d-a e)^2}{b^6 (a+b x)^2}+\frac{3 e^2 (b d-a e)}{b^6 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{3 e (b d-a e)^2}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3}{2 b^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 e^2 (b d-a e) (a+b x) \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.056599, size = 125, normalized size = 0.78 \[ \frac{a^2 b e^2 (9 d-4 e x)-5 a^3 e^3+a b^2 e \left (-3 d^2+12 d e x+4 e^2 x^2\right )-6 e^2 (a+b x)^2 (a e-b d) \log (a+b x)+b^3 \left (-\left (6 d^2 e x+d^3-2 e^3 x^3\right )\right )}{2 b^4 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 209, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}{e}^{3}-6\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}-2\,{x}^{3}{b}^{3}{e}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{2}b{e}^{3}-12\,\ln \left ( bx+a \right ) xa{b}^{2}d{e}^{2}-4\,{x}^{2}a{b}^{2}{e}^{3}+6\,\ln \left ( bx+a \right ){a}^{3}{e}^{3}-6\,\ln \left ( bx+a \right ){a}^{2}bd{e}^{2}+4\,x{a}^{2}b{e}^{3}-12\,xa{b}^{2}d{e}^{2}+6\,x{b}^{3}{d}^{2}e+5\,{a}^{3}{e}^{3}-9\,d{e}^{2}{a}^{2}b+3\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) }{2\,{b}^{4}} \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 = 1.05174, size = 390, normalized size = 2.42 \begin{align*} \frac{e^{3} x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{3 \, d e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{3 \, a e^{3} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{9 \, a^{2} b^{2} d e^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{9 \, a^{3} b e^{3}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{6 \, a b d e^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, a^{2} e^{3} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{3 \, d^{2} e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{2 \, a^{2} e^{3}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{d^{3}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{3 \, a d^{2} e}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{a^{3} e^{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 = 1.6106, size = 375, normalized size = 2.33 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 4 \, a b^{2} e^{3} x^{2} - b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} - 2 \,{\left (3 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 2 \, a^{2} b e^{3}\right )} x + 6 \,{\left (a^{2} b d e^{2} - a^{3} e^{3} +{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{2} d e^{2} - a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\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{\left (d + e x\right )^{3}}{\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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