Optimal. Leaf size=345 \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]
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Rubi [A] time = 0.250281, 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} \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}{e^7 (a+b x)}+\frac{15 b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x)}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^3}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^7 (a+b x)} \]
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)^4} \, 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)^4} \, 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)^4} \, 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^4 (b d-a e)^2}{e^6}+\frac{(-b d+a e)^6}{e^6 (d+e x)^4}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^3}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^2}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)}-\frac{6 b^5 (b d-a e) (d+e x)}{e^6}+\frac{b^6 (d+e x)^2}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{15 b^4 (b d-a e)^2 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}}{3 e^7 (a+b x) (d+e x)^3}+\frac{3 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^2}-\frac{15 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}-\frac{3 b^5 (b d-a e) (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{b^6 (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{20 b^3 (b d-a e)^3 \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 = 0.158884, size = 320, normalized size = 0.93 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 b^4 e^2 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )-10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )+3 a^5 b e^5 (d+3 e x)+a^6 e^6-3 a b^5 e \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )+60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+51 d^5 e x+37 d^6+3 d e^5 x^5-e^6 x^6\right )\right )}{3 e^7 (a+b x) (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 692, normalized size = 2. \begin{align*}{\frac{-{a}^{6}{e}^{6}-37\,{b}^{6}{d}^{6}-135\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-189\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+180\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+135\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+243\,xa{b}^{5}{d}^{4}{e}^{2}-27\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-45\,{x}^{4}a{b}^{5}d{e}^{5}+270\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-405\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-45\,x{a}^{4}{b}^{2}d{e}^{5}+60\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}-180\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}+180\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e+180\,\ln \left ( ex+d \right ){x}^{2}{a}^{3}{b}^{3}d{e}^{5}-540\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+540\,\ln \left ( ex+d \right ){x}^{2}a{b}^{5}{d}^{3}{e}^{3}+180\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-540\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+540\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}-180\,\ln \left ( ex+d \right ){x}^{3}{a}^{2}{b}^{4}d{e}^{5}+180\,\ln \left ( ex+d \right ){x}^{3}a{b}^{5}{d}^{2}{e}^{4}+39\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-9\,x{a}^{5}b{e}^{6}-51\,x{b}^{6}{d}^{5}e+9\,{x}^{5}a{b}^{5}{e}^{6}-3\,{x}^{5}{b}^{6}d{e}^{5}+45\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+15\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+73\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-45\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-3\,d{e}^{5}{a}^{5}b+110\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-195\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+141\,a{b}^{5}{d}^{5}e-15\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+{x}^{6}{b}^{6}{e}^{6}-60\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}+60\,\ln \left ( ex+d \right ){x}^{3}{a}^{3}{b}^{3}{e}^{6}-180\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e-180\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}}{3\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{3}} \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 [B] time = 1.55695, size = 1161, normalized size = 3.37 \begin{align*} \frac{b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} +{\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \,{\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} +{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} 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 = 1.15168, size = 679, normalized size = 1.97 \begin{align*} -20 \,{\left (b^{6} d^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{5} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d e^{2} \mathrm{sgn}\left (b x + a\right ) - a^{3} b^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (b^{6} x^{3} e^{8} \mathrm{sgn}\left (b x + a\right ) - 6 \, b^{6} d x^{2} e^{7} \mathrm{sgn}\left (b x + a\right ) + 30 \, b^{6} d^{2} x e^{6} \mathrm{sgn}\left (b x + a\right ) + 9 \, a b^{5} x^{2} e^{8} \mathrm{sgn}\left (b x + a\right ) - 72 \, a b^{5} d x e^{7} \mathrm{sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} x e^{8} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-12\right )} - \frac{{\left (37 \, b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 141 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 195 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 110 \, 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 ) + 3 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 45 \,{\left (b^{6} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a b^{5} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{4} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{4} b^{2} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 9 \,{\left (9 \, b^{6} d^{5} e \mathrm{sgn}\left (b x + a\right ) - 35 \, a b^{5} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 50 \, a^{2} b^{4} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 30 \, a^{3} b^{3} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{5} b e^{6} \mathrm{sgn}\left (b x + a\right )\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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