Optimal. Leaf size=208 \[ -\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}} \]
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Rubi [A] time = 0.0659555, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ -\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}+\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^4 (a+b x) (d+e x)^{9/2}} \]
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 )^{3/2}}{(d+e x)^{11/2}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{(d+e x)^{11/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3}{e^3 (d+e x)^{11/2}}+\frac{3 b^4 (b d-a e)^2}{e^3 (d+e x)^{9/2}}-\frac{3 b^5 (b d-a e)}{e^3 (d+e x)^{7/2}}+\frac{b^6}{e^3 (d+e x)^{5/2}}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^4 (a+b x) (d+e x)^{9/2}}-\frac{6 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x) (d+e x)^{7/2}}+\frac{6 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x) (d+e x)^{5/2}}-\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x) (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0649713, size = 120, normalized size = 0.58 \[ -\frac{2 \sqrt{(a+b x)^2} \left (15 a^2 b e^2 (2 d+9 e x)+35 a^3 e^3+3 a b^2 e \left (8 d^2+36 d e x+63 e^2 x^2\right )+b^3 \left (72 d^2 e x+16 d^3+126 d e^2 x^2+105 e^3 x^3\right )\right )}{315 e^4 (a+b x) (d+e x)^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.153, size = 132, normalized size = 0.6 \begin{align*} -{\frac{210\,{x}^{3}{b}^{3}{e}^{3}+378\,{x}^{2}a{b}^{2}{e}^{3}+252\,{x}^{2}{b}^{3}d{e}^{2}+270\,x{a}^{2}b{e}^{3}+216\,xa{b}^{2}d{e}^{2}+144\,x{b}^{3}{d}^{2}e+70\,{a}^{3}{e}^{3}+60\,d{e}^{2}{a}^{2}b+48\,a{b}^{2}{d}^{2}e+32\,{b}^{3}{d}^{3}}{315\, \left ( bx+a \right ) ^{3}{e}^{4}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.254, size = 215, normalized size = 1.03 \begin{align*} -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )}}{315 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt{e x + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56084, size = 366, normalized size = 1.76 \begin{align*} -\frac{2 \,{\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \,{\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\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 = 1.17838, size = 262, normalized size = 1.26 \begin{align*} -\frac{2 \,{\left (105 \,{\left (x e + d\right )}^{3} b^{3} \mathrm{sgn}\left (b x + a\right ) - 189 \,{\left (x e + d\right )}^{2} b^{3} d \mathrm{sgn}\left (b x + a\right ) + 135 \,{\left (x e + d\right )} b^{3} d^{2} \mathrm{sgn}\left (b x + a\right ) - 35 \, b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) + 189 \,{\left (x e + d\right )}^{2} a b^{2} e \mathrm{sgn}\left (b x + a\right ) - 270 \,{\left (x e + d\right )} a b^{2} d e \mathrm{sgn}\left (b x + a\right ) + 105 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 135 \,{\left (x e + d\right )} a^{2} b e^{2} \mathrm{sgn}\left (b x + a\right ) - 105 \, a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{315 \,{\left (x e + d\right )}^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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