Optimal. Leaf size=173 \[ \frac{e x (a+b x) (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^3 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0619336, antiderivative size = 173, 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} \[ \frac{e x (a+b x) (b d-a e)^2}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^3 \log (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{(d+e x)^3}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{e (b d-a e)^2}{b^4}+\frac{(b d-a e)^3}{b^3 \left (a b+b^2 x\right )}+\frac{e (b d-a e) (d+e x)}{b^3}+\frac{e (d+e x)^2}{b^2}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e (b d-a e)^2 x (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(b d-a e) (a+b x) (d+e x)^2}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(b d-a e)^3 (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 = 0.0526176, size = 90, normalized size = 0.52 \[ \frac{(a+b x) \left (b e x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (b d-a e)^3 \log (a+b x)\right )}{6 b^4 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 147, normalized size = 0.9 \begin{align*} -{\frac{ \left ( bx+a \right ) \left ( -2\,{x}^{3}{b}^{3}{e}^{3}+3\,{x}^{2}a{b}^{2}{e}^{3}-9\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( bx+a \right ){a}^{3}{e}^{3}-18\,\ln \left ( bx+a \right ){a}^{2}bd{e}^{2}+18\,\ln \left ( bx+a \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( bx+a \right ){b}^{3}{d}^{3}-6\,x{a}^{2}b{e}^{3}+18\,xa{b}^{2}d{e}^{2}-18\,x{b}^{3}{d}^{2}e \right ) }{6\,{b}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14639, size = 346, normalized size = 2. \begin{align*} \frac{3 \, a^{2} b^{2} d e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{5 \, a^{3} b e^{3} \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{3 \, a b d e^{2} x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{5 \, a^{2} e^{3} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, d e^{2} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{5 \, a e^{3} x^{2}}{6 \, \sqrt{b^{2}} b} + \sqrt{\frac{1}{b^{2}}} d^{3} \log \left (x + \frac{a}{b}\right ) - \frac{3 \, a \sqrt{\frac{1}{b^{2}}} d^{2} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{2 \, a^{3} \sqrt{\frac{1}{b^{2}}} e^{3} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{2}}{3 \, b^{2}} + \frac{3 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d^{2} e}{b^{2}} - \frac{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e^{3}}{3 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55755, size = 238, normalized size = 1.38 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} + 3 \,{\left (3 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (3 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.549075, size = 82, normalized size = 0.47 \begin{align*} \frac{e^{3} x^{3}}{3 b} - \frac{x^{2} \left (a e^{3} - 3 b d e^{2}\right )}{2 b^{2}} + \frac{x \left (a^{2} e^{3} - 3 a b d e^{2} + 3 b^{2} d^{2} e\right )}{b^{3}} - \frac{\left (a e - b d\right )^{3} \log{\left (a + b x \right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15909, size = 230, normalized size = 1.33 \begin{align*} \frac{2 \, b^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 9 \, b^{2} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 18 \, b^{2} d^{2} x e \mathrm{sgn}\left (b x + a\right ) - 3 \, a b x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 18 \, a b d x e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} x e^{3} \mathrm{sgn}\left (b x + a\right )}{6 \, b^{3}} + \frac{{\left (b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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