Optimal. Leaf size=222 \[ \frac{e x (a+b x) (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^4}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0803956, antiderivative size = 222, 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} \[ \frac{e x (a+b x) (b d-a e)^3}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^2 (b d-a e)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^3 (b d-a e)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^4}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (b d-a e)^4 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\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)^4}{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)^3}{b^5}+\frac{(b d-a e)^4}{b^4 \left (a b+b^2 x\right )}+\frac{e (b d-a e)^2 (d+e x)}{b^4}+\frac{e (b d-a e) (d+e x)^2}{b^3}+\frac{e (d+e x)^3}{b^2}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e (b d-a e)^3 x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(b d-a e)^2 (a+b x) (d+e x)^2}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(b d-a e) (a+b x) (d+e x)^3}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (d+e x)^4}{4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(b d-a e)^4 (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 = 0.0806139, size = 130, normalized size = 0.59 \[ \frac{(a+b x) \left (b e x \left (6 a^2 b e^2 (8 d+e x)-12 a^3 e^3-4 a b^2 e \left (18 d^2+6 d e x+e^2 x^2\right )+b^3 \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (a+b x)\right )}{12 b^5 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 223, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) \left ( 3\,{x}^{4}{b}^{4}{e}^{4}-4\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,{x}^{2}a{b}^{3}d{e}^{3}+36\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-48\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}+72\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}-48\,\ln \left ( bx+a \right ) a{b}^{3}{d}^{3}e+12\,\ln \left ( bx+a \right ){b}^{4}{d}^{4}-12\,x{a}^{3}b{e}^{4}+48\,x{a}^{2}{b}^{2}d{e}^{3}-72\,xa{b}^{3}{d}^{2}{e}^{2}+48\,x{b}^{4}{d}^{3}e \right ) }{12\,{b}^{5}}{\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 = 1.1538, size = 590, normalized size = 2.66 \begin{align*} \frac{6 \, a^{2} b^{2} d^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{20 \, a^{3} b d e^{3} \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{13 \, a^{4} e^{4} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{6 \, a b d^{2} e^{2} x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{20 \, a^{2} d e^{3} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{13 \, a^{3} e^{4} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{3 \, d^{2} e^{2} x^{2}}{\sqrt{b^{2}}} - \frac{10 \, a d e^{3} x^{2}}{3 \, \sqrt{b^{2}} b} + \frac{13 \, a^{2} e^{4} x^{2}}{12 \, \sqrt{b^{2}} b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{4} x^{3}}{4 \, b^{2}} + \sqrt{\frac{1}{b^{2}}} d^{4} \log \left (x + \frac{a}{b}\right ) - \frac{4 \, a \sqrt{\frac{1}{b^{2}}} d^{3} e \log \left (x + \frac{a}{b}\right )}{b} + \frac{8 \, a^{3} \sqrt{\frac{1}{b^{2}}} d e^{3} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} - \frac{7 \, a^{4} \sqrt{\frac{1}{b^{2}}} e^{4} \log \left (x + \frac{a}{b}\right )}{6 \, b^{4}} + \frac{4 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d e^{3} x^{2}}{3 \, b^{2}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{4} x^{2}}{12 \, b^{3}} + \frac{4 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} d^{3} e}{b^{2}} - \frac{8 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} d e^{3}}{3 \, b^{4}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{4}}{6 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5845, size = 369, normalized size = 1.66 \begin{align*} \frac{3 \, b^{4} e^{4} x^{4} + 4 \,{\left (4 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (6 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (4 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} + 4 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (b x + a\right )}{12 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.647496, size = 134, normalized size = 0.6 \begin{align*} \frac{e^{4} x^{4}}{4 b} - \frac{x^{3} \left (a e^{4} - 4 b d e^{3}\right )}{3 b^{2}} + \frac{x^{2} \left (a^{2} e^{4} - 4 a b d e^{3} + 6 b^{2} d^{2} e^{2}\right )}{2 b^{3}} - \frac{x \left (a^{3} e^{4} - 4 a^{2} b d e^{3} + 6 a b^{2} d^{2} e^{2} - 4 b^{3} d^{3} e\right )}{b^{4}} + \frac{\left (a e - b d\right )^{4} \log{\left (a + b x \right )}}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22037, size = 356, normalized size = 1.6 \begin{align*} \frac{3 \, b^{3} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 16 \, b^{3} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 36 \, b^{3} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 48 \, b^{3} d^{3} x e \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) - 24 \, a b^{2} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 72 \, a b^{2} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 48 \, a^{2} b d x e^{3} \mathrm{sgn}\left (b x + a\right ) - 12 \, a^{3} x e^{4} \mathrm{sgn}\left (b x + a\right )}{12 \, b^{4}} + \frac{{\left (b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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