Optimal. Leaf size=362 \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}} \]
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Rubi [A] time = 0.196803, antiderivative size = 362, 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} \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{12}}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{13 e^7 (a+b x) (d+e x)^{13}}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{14}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{15 e^7 (a+b x) (d+e x)^{15}} \]
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)^{16}} \, 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)^{16}} \, 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)^{16}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{16}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{15}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{14}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{13}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^{12}}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^{11}}+\frac{b^6}{e^6 (d+e x)^{10}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x) (d+e x)^{15}}+\frac{3 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{14}}-\frac{15 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}+\frac{5 b^3 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{12}}-\frac{15 b^4 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{3 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{10}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}\\ \end{align*}
Mathematica [A] time = 0.118651, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (45 a^2 b^4 e^2 \left (105 d^2 e^2 x^2+15 d^3 e x+d^4+455 d e^3 x^3+1365 e^4 x^4\right )+165 a^3 b^3 e^3 \left (15 d^2 e x+d^3+105 d e^2 x^2+455 e^3 x^3\right )+495 a^4 b^2 e^4 \left (d^2+15 d e x+105 e^2 x^2\right )+1287 a^5 b e^5 (d+15 e x)+3003 a^6 e^6+9 a b^5 e \left (105 d^3 e^2 x^2+455 d^2 e^3 x^3+15 d^4 e x+d^5+1365 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (105 d^4 e^2 x^2+455 d^3 e^3 x^3+1365 d^2 e^4 x^4+15 d^5 e x+d^6+3003 d e^5 x^5+5005 e^6 x^6\right )\right )}{45045 e^7 (a+b x) (d+e x)^{15}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 392, normalized size = 1.1 \begin{align*} -{\frac{5005\,{x}^{6}{b}^{6}{e}^{6}+27027\,{x}^{5}a{b}^{5}{e}^{6}+3003\,{x}^{5}{b}^{6}d{e}^{5}+61425\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+12285\,{x}^{4}a{b}^{5}d{e}^{5}+1365\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+75075\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+20475\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+4095\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+455\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+51975\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+17325\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+4725\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+945\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+105\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+19305\,x{a}^{5}b{e}^{6}+7425\,x{a}^{4}{b}^{2}d{e}^{5}+2475\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+675\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+135\,xa{b}^{5}{d}^{4}{e}^{2}+15\,x{b}^{6}{d}^{5}e+3003\,{a}^{6}{e}^{6}+1287\,d{e}^{5}{a}^{5}b+495\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+165\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+45\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+9\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{45045\,{e}^{7} \left ( ex+d \right ) ^{15} \left ( bx+a \right ) ^{5}} \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 = 1.54682, size = 1141, normalized size = 3.15 \begin{align*} -\frac{5005 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 9 \, a b^{5} d^{5} e + 45 \, a^{2} b^{4} d^{4} e^{2} + 165 \, a^{3} b^{3} d^{3} e^{3} + 495 \, a^{4} b^{2} d^{2} e^{4} + 1287 \, a^{5} b d e^{5} + 3003 \, a^{6} e^{6} + 3003 \,{\left (b^{6} d e^{5} + 9 \, a b^{5} e^{6}\right )} x^{5} + 1365 \,{\left (b^{6} d^{2} e^{4} + 9 \, a b^{5} d e^{5} + 45 \, a^{2} b^{4} e^{6}\right )} x^{4} + 455 \,{\left (b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 45 \, a^{2} b^{4} d e^{5} + 165 \, a^{3} b^{3} e^{6}\right )} x^{3} + 105 \,{\left (b^{6} d^{4} e^{2} + 9 \, a b^{5} d^{3} e^{3} + 45 \, a^{2} b^{4} d^{2} e^{4} + 165 \, a^{3} b^{3} d e^{5} + 495 \, a^{4} b^{2} e^{6}\right )} x^{2} + 15 \,{\left (b^{6} d^{5} e + 9 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} + 165 \, a^{3} b^{3} d^{2} e^{4} + 495 \, a^{4} b^{2} d e^{5} + 1287 \, a^{5} b e^{6}\right )} x}{45045 \,{\left (e^{22} x^{15} + 15 \, d e^{21} x^{14} + 105 \, d^{2} e^{20} x^{13} + 455 \, d^{3} e^{19} x^{12} + 1365 \, d^{4} e^{18} x^{11} + 3003 \, d^{5} e^{17} x^{10} + 5005 \, d^{6} e^{16} x^{9} + 6435 \, d^{7} e^{15} x^{8} + 6435 \, d^{8} e^{14} x^{7} + 5005 \, d^{9} e^{13} x^{6} + 3003 \, d^{10} e^{12} x^{5} + 1365 \, d^{11} e^{11} x^{4} + 455 \, d^{12} e^{10} x^{3} + 105 \, d^{13} e^{9} x^{2} + 15 \, d^{14} e^{8} x + d^{15} 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 = 1.14847, size = 702, normalized size = 1.94 \begin{align*} -\frac{{\left (5005 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 3003 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 1365 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 455 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 105 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 15 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 27027 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 12285 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 4095 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 945 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 135 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 9 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 61425 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 20475 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 4725 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 675 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 45 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 75075 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 17325 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 2475 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 165 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 51975 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 7425 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 495 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 19305 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 1287 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 3003 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{45045 \,{\left (x e + d\right )}^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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