Optimal. Leaf size=92 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)} \]
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Rubi [A] time = 0.0358736, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right ) (d+e x)^3 \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e) (d+e x)^3}{e}+\frac{b^2 (d+e x)^4}{e}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e) (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x)}+\frac{b (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^2 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0282263, size = 89, normalized size = 0.97 \[ \frac{x \sqrt{(a+b x)^2} \left (5 a \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 90, normalized size = 1. \begin{align*}{\frac{x \left ( 4\,b{e}^{3}{x}^{4}+5\,{x}^{3}a{e}^{3}+15\,{x}^{3}bd{e}^{2}+20\,ad{e}^{2}{x}^{2}+20\,b{d}^{2}e{x}^{2}+30\,xa{d}^{2}e+10\,xb{d}^{3}+20\,a{d}^{3} \right ) }{20\,bx+20\,a}\sqrt{ \left ( bx+a \right ) ^{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 = 1.85812, size = 150, normalized size = 1.63 \begin{align*} \frac{1}{5} \, b e^{3} x^{5} + a d^{3} x + \frac{1}{4} \,{\left (3 \, b d e^{2} + a e^{3}\right )} x^{4} +{\left (b d^{2} e + a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.131303, size = 73, normalized size = 0.79 \begin{align*} a d^{3} x + \frac{b e^{3} x^{5}}{5} + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 b d e^{2}}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e\right ) + x^{2} \left (\frac{3 a d^{2} e}{2} + \frac{b d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16381, size = 159, normalized size = 1.73 \begin{align*} \frac{1}{5} \, b x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, b d x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + b d^{2} x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, b d^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, a x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + a d x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a d^{2} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a d^{3} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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