Optimal. Leaf size=345 \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)}+\frac{15 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (a+b x)} \]
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Rubi [A] time = 0.299367, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)}+\frac{15 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^2} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{15 b^2 (b d-a e)^4}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^2}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)}-\frac{20 b^3 (b d-a e)^3 (d+e x)}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^2}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^3}{e^6}+\frac{b^6 (d+e x)^4}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{15 b^2 (b d-a e)^4 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac{10 b^3 (b d-a e)^3 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{5 b^4 (b d-a e)^2 (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{3 b^5 (b d-a e) (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}+\frac{b^6 (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac{6 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.212058, size = 320, normalized size = 0.93 \[ \frac{\sqrt{(a+b x)^2} \left (50 a^2 b^4 e^2 \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+100 a^3 b^3 e^3 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+60 a^5 b d e^5-10 a^6 e^6+5 a b^5 e \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left (30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4+50 d^5 e x-10 d^6-3 d e^5 x^5+2 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 601, normalized size = 1.7 \begin{align*}{\frac{-10\,{a}^{6}{e}^{6}-10\,{b}^{6}{d}^{6}+300\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+50\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-300\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-100\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-240\,xa{b}^{5}{d}^{4}{e}^{2}-150\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-25\,{x}^{4}a{b}^{5}d{e}^{5}-400\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+450\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+150\,x{a}^{4}{b}^{2}d{e}^{5}+600\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}-600\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}-300\,\ln \left ( ex+d \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}+300\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+60\,\ln \left ( ex+d \right ){a}^{5}bd{e}^{5}-300\,\ln \left ( ex+d \right ) x{a}^{4}{b}^{2}d{e}^{5}+600\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-600\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+300\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}+30\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+50\,x{b}^{6}{d}^{5}e+15\,{x}^{5}a{b}^{5}{e}^{6}-3\,{x}^{5}{b}^{6}d{e}^{5}+50\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+5\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+100\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-10\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+150\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+60\,d{e}^{5}{a}^{5}b+200\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-150\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+60\,a{b}^{5}{d}^{5}e-150\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+2\,{x}^{6}{b}^{6}{e}^{6}+60\,\ln \left ( ex+d \right ) x{a}^{5}b{e}^{6}-60\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e}{10\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) } \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 = 1.54705, size = 1018, normalized size = 2.95 \begin{align*} \frac{2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \,{\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \,{\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} - a^{5} b d e^{5} +{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13524, size = 701, normalized size = 2.03 \begin{align*} -6 \,{\left (b^{6} d^{5} \mathrm{sgn}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4} \mathrm{sgn}\left (b x + a\right ) - a^{5} b e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{10} \,{\left (2 \, b^{6} x^{5} e^{8} \mathrm{sgn}\left (b x + a\right ) - 5 \, b^{6} d x^{4} e^{7} \mathrm{sgn}\left (b x + a\right ) + 10 \, b^{6} d^{2} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) - 20 \, b^{6} d^{3} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 50 \, b^{6} d^{4} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 15 \, a b^{5} x^{4} e^{8} \mathrm{sgn}\left (b x + a\right ) - 40 \, a b^{5} d x^{3} e^{7} \mathrm{sgn}\left (b x + a\right ) + 90 \, a b^{5} d^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) - 240 \, a b^{5} d^{3} x e^{5} \mathrm{sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} x^{3} e^{8} \mathrm{sgn}\left (b x + a\right ) - 150 \, a^{2} b^{4} d x^{2} e^{7} \mathrm{sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x e^{6} \mathrm{sgn}\left (b x + a\right ) + 100 \, a^{3} b^{3} x^{2} e^{8} \mathrm{sgn}\left (b x + a\right ) - 400 \, a^{3} b^{3} d x e^{7} \mathrm{sgn}\left (b x + a\right ) + 150 \, a^{4} b^{2} x e^{8} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-10\right )} - \frac{{\left (b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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