Optimal. Leaf size=292 \[ -\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)}+\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)} \]
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Rubi [A] time = 0.148387, antiderivative size = 292, 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} \[ -\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)}+\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^2}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{3 e^6 (a+b x) (d+e x)^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{4 e^6 (a+b x) (d+e x)^4}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^6 (a+b x)}+\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5}{(d+e x)^5} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{b^{10}}{e^5}-\frac{b^5 (b d-a e)^5}{e^5 (d+e x)^5}+\frac{5 b^6 (b d-a e)^4}{e^5 (d+e x)^4}-\frac{10 b^7 (b d-a e)^3}{e^5 (d+e x)^3}+\frac{10 b^8 (b d-a e)^2}{e^5 (d+e x)^2}-\frac{5 b^9 (b d-a e)}{e^5 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{b^5 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x)}+\frac{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^6 (a+b x) (d+e x)^4}-\frac{5 b (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^3}+\frac{5 b^2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^2}-\frac{10 b^3 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)}-\frac{5 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^6 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.128634, size = 243, normalized size = 0.83 \[ -\frac{\sqrt{(a+b x)^2} \left (30 a^2 b^3 e^2 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )+10 a^3 b^2 e^3 \left (d^2+4 d e x+6 e^2 x^2\right )+5 a^4 b e^4 (d+4 e x)+3 a^5 e^5-5 a b^4 d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )+60 b^4 (d+e x)^4 (b d-a e) \log (d+e x)+b^5 \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.232, size = 458, normalized size = 1.6 \begin{align*}{\frac{240\,\ln \left ( ex+d \right ) xa{b}^{4}{d}^{3}{e}^{2}+360\,\ln \left ( ex+d \right ){x}^{2}a{b}^{4}{d}^{2}{e}^{3}-360\,\ln \left ( ex+d \right ){x}^{2}{b}^{5}{d}^{3}{e}^{2}-3\,{a}^{5}{e}^{5}-77\,{b}^{5}{d}^{5}-5\,d{e}^{4}{a}^{4}b-248\,x{b}^{5}{d}^{4}e+48\,{x}^{4}{b}^{5}d{e}^{4}-120\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-48\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}-60\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-252\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}-20\,x{a}^{4}b{e}^{5}-240\,\ln \left ( ex+d \right ) x{b}^{5}{d}^{4}e+540\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+240\,{x}^{3}a{b}^{4}d{e}^{4}-30\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}-10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-60\,\ln \left ( ex+d \right ){b}^{5}{d}^{5}+12\,{x}^{5}{b}^{5}{e}^{5}+440\,xa{b}^{4}{d}^{3}{e}^{2}+60\,\ln \left ( ex+d \right ) a{b}^{4}{d}^{4}e-40\,x{a}^{3}{b}^{2}d{e}^{4}-120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-240\,\ln \left ( ex+d \right ){x}^{3}{b}^{5}{d}^{2}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{4}a{b}^{4}{e}^{5}-60\,\ln \left ( ex+d \right ){x}^{4}{b}^{5}d{e}^{4}+240\,\ln \left ( ex+d \right ){x}^{3}a{b}^{4}d{e}^{4}+125\,a{b}^{4}{d}^{4}e}{12\, \left ( bx+a \right ) ^{5}{e}^{6} \left ( ex+d \right ) ^{4}} \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 = 1.62253, size = 838, normalized size = 2.87 \begin{align*} \frac{12 \, b^{5} e^{5} x^{5} + 48 \, b^{5} d e^{4} x^{4} - 77 \, b^{5} d^{5} + 125 \, a b^{4} d^{4} e - 30 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} - 5 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 24 \,{\left (2 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 5 \, a^{2} b^{3} e^{5}\right )} x^{3} - 12 \,{\left (21 \, b^{5} d^{3} e^{2} - 45 \, a b^{4} d^{2} e^{3} + 15 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} - 4 \,{\left (62 \, b^{5} d^{4} e - 110 \, a b^{4} d^{3} e^{2} + 30 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 5 \, a^{4} b e^{5}\right )} x - 60 \,{\left (b^{5} d^{5} - a b^{4} d^{4} e +{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \,{\left (b^{5} d^{2} e^{3} - a b^{4} d e^{4}\right )} x^{3} + 6 \,{\left (b^{5} d^{3} e^{2} - a b^{4} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{5} d^{4} e - a b^{4} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\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 = 1.21613, size = 500, normalized size = 1.71 \begin{align*} b^{5} x e^{\left (-5\right )} \mathrm{sgn}\left (b x + a\right ) - 5 \,{\left (b^{5} d \mathrm{sgn}\left (b x + a\right ) - a b^{4} e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) - \frac{{\left (77 \, b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) - 125 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 30 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 120 \,{\left (b^{5} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{4} d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{3} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 60 \,{\left (5 \, b^{5} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 9 \, a b^{4} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{3} d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{3} b^{2} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 20 \,{\left (13 \, b^{5} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 22 \, a b^{4} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, a^{3} b^{2} d e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{4} b e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-6\right )}}{12 \,{\left (x e + d\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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