Optimal. Leaf size=48 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]
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Rubi [A] time = 0.0190633, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 37} \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
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)^5} \, 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)^5} \, 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}}{4 (b d-a e) (d+e x)^4}\\ \end{align*}
Mathematica [B] time = 0.0424282, size = 109, normalized size = 2.27 \[ -\frac{\sqrt{(a+b x)^2} \left (a^2 b e^2 (d+4 e x)+a^3 e^3+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )}{4 e^4 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.157, size = 128, normalized size = 2.7 \begin{align*} -{\frac{4\,{x}^{3}{b}^{3}{e}^{3}+6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+4\,x{a}^{2}b{e}^{3}+4\,xa{b}^{2}d{e}^{2}+4\,x{b}^{3}{d}^{2}e+{a}^{3}{e}^{3}+d{e}^{2}{a}^{2}b+a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{4\, \left ( ex+d \right ) ^{4}{e}^{4} \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 = 1.53743, size = 284, normalized size = 5.92 \begin{align*} -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\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 )}} \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 = 1.18077, size = 224, normalized size = 4.67 \begin{align*} -\frac{{\left (4 \, b^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, b^{3} d x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, b^{3} d^{2} x e \mathrm{sgn}\left (b x + a\right ) + b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{2} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 4 \, a b^{2} d x e^{2} \mathrm{sgn}\left (b x + a\right ) + a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 4 \, a^{2} b x e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) + a^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{4 \,{\left (x e + d\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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