Optimal. Leaf size=98 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{20 (d+e x)^4 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{5 (d+e x)^5 (b d-a e)} \]
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Rubi [A] time = 0.0382047, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {646, 45, 37} \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{20 (d+e x)^4 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{5 (d+e x)^5 (b d-a e)} \]
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 )^{3/2}}{(d+e x)^6} \, 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)^6} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (b d-a e) (d+e x)^5}+\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)^5} \, dx}{5 b (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (b d-a e) (d+e x)^5}+\frac{b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{20 (b d-a e)^2 (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.0474105, size = 112, normalized size = 1.14 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^2 b e^2 (d+5 e x)+4 a^3 e^3+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )}{20 e^4 (a+b x) (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.155, size = 131, normalized size = 1.3 \begin{align*} -{\frac{10\,{x}^{3}{b}^{3}{e}^{3}+20\,{x}^{2}a{b}^{2}{e}^{3}+10\,{x}^{2}{b}^{3}d{e}^{2}+15\,x{a}^{2}b{e}^{3}+10\,xa{b}^{2}d{e}^{2}+5\,x{b}^{3}{d}^{2}e+4\,{a}^{3}{e}^{3}+3\,d{e}^{2}{a}^{2}b+2\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{20\,{e}^{4} \left ( ex+d \right ) ^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 1.59604, size = 328, normalized size = 3.35 \begin{align*} -\frac{10 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + 2 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 4 \, a^{3} e^{3} + 10 \,{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} + 5 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x}{20 \,{\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 [B] time = 1.18876, size = 228, normalized size = 2.33 \begin{align*} -\frac{{\left (10 \, b^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, b^{3} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, b^{3} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) + 20 \, a b^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a b^{2} d x e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 15 \, a^{2} b x e^{3} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{20 \,{\left (x e + d\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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