Optimal. Leaf size=149 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)} \]
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Rubi [A] time = 0.0510391, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {646, 45, 37} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{168 (d+e x)^6 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{28 (d+e x)^7 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 (d+e x)^8 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^9} \, 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)^9} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\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)^8} \, dx}{4 b^3 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 (b d-a e)^2 (d+e x)^7}+\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)^7} \, dx}{28 b^2 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{8 (b d-a e) (d+e x)^8}+\frac{b (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{28 (b d-a e)^2 (d+e x)^7}+\frac{b^2 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{168 (b d-a e)^3 (d+e x)^6}\\ \end{align*}
Mathematica [A] time = 0.0803123, size = 223, normalized size = 1.5 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 b^3 e^2 \left (8 d^2 e x+d^3+28 d e^2 x^2+56 e^3 x^3\right )+10 a^3 b^2 e^3 \left (d^2+8 d e x+28 e^2 x^2\right )+15 a^4 b e^4 (d+8 e x)+21 a^5 e^5+3 a b^4 e \left (28 d^2 e^2 x^2+8 d^3 e x+d^4+56 d e^3 x^3+70 e^4 x^4\right )+b^5 \left (28 d^3 e^2 x^2+56 d^2 e^3 x^3+8 d^4 e x+d^5+70 d e^4 x^4+56 e^5 x^5\right )\right )}{168 e^6 (a+b x) (d+e x)^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.158, size = 288, normalized size = 1.9 \begin{align*} -{\frac{56\,{x}^{5}{b}^{5}{e}^{5}+210\,{x}^{4}a{b}^{4}{e}^{5}+70\,{x}^{4}{b}^{5}d{e}^{4}+336\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+168\,{x}^{3}a{b}^{4}d{e}^{4}+56\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+280\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+168\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+84\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+28\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+120\,x{a}^{4}b{e}^{5}+80\,x{a}^{3}{b}^{2}d{e}^{4}+48\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+24\,xa{b}^{4}{d}^{3}{e}^{2}+8\,x{b}^{5}{d}^{4}e+21\,{a}^{5}{e}^{5}+15\,d{e}^{4}{a}^{4}b+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}+6\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+3\,a{b}^{4}{d}^{4}e+{b}^{5}{d}^{5}}{168\,{e}^{6} \left ( ex+d \right ) ^{8} \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 [B] time = 1.62102, size = 697, normalized size = 4.68 \begin{align*} -\frac{56 \, b^{5} e^{5} x^{5} + b^{5} d^{5} + 3 \, a b^{4} d^{4} e + 6 \, a^{2} b^{3} d^{3} e^{2} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} + 21 \, a^{5} e^{5} + 70 \,{\left (b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 56 \,{\left (b^{5} d^{2} e^{3} + 3 \, a b^{4} d e^{4} + 6 \, a^{2} b^{3} e^{5}\right )} x^{3} + 28 \,{\left (b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 10 \, a^{3} b^{2} e^{5}\right )} x^{2} + 8 \,{\left (b^{5} d^{4} e + 3 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 10 \, a^{3} b^{2} d e^{4} + 15 \, a^{4} b e^{5}\right )} x}{168 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} 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 [B] time = 1.17226, size = 514, normalized size = 3.45 \begin{align*} -\frac{{\left (56 \, b^{5} x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 70 \, b^{5} d x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, b^{5} d^{2} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, b^{5} d^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 8 \, b^{5} d^{4} x e \mathrm{sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm{sgn}\left (b x + a\right ) + 210 \, a b^{4} x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 168 \, a b^{4} d x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, a b^{4} d^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 24 \, a b^{4} d^{3} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{4} d^{4} e \mathrm{sgn}\left (b x + a\right ) + 336 \, a^{2} b^{3} x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 168 \, a^{2} b^{3} d x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 48 \, a^{2} b^{3} d^{2} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + 280 \, a^{3} b^{2} x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 80 \, a^{3} b^{2} d x e^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 120 \, a^{4} b x e^{5} \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{4} b d e^{4} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{168 \,{\left (x e + d\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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