Optimal. Leaf size=92 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)} \]
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Rubi [A] time = 0.0489773, 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} \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^2 \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)^2 \, 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)^2}{e}+\frac{b^2 (d+e x)^3}{e}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e) (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x)}+\frac{b (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0219973, size = 67, normalized size = 0.73 \[ \frac{x \sqrt{(a+b x)^2} \left (4 a \left (3 d^2+3 d e x+e^2 x^2\right )+b x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )}{12 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 66, normalized size = 0.7 \begin{align*}{\frac{x \left ( 3\,b{e}^{2}{x}^{3}+4\,{x}^{2}a{e}^{2}+8\,{x}^{2}bde+12\,adex+6\,b{d}^{2}x+12\,a{d}^{2} \right ) }{12\,bx+12\,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 = 1.64003, size = 109, normalized size = 1.18 \begin{align*} \frac{1}{4} \, b e^{2} x^{4} + a d^{2} x + \frac{1}{3} \,{\left (2 \, b d e + a e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111704, size = 49, normalized size = 0.53 \begin{align*} a d^{2} x + \frac{b e^{2} x^{4}}{4} + x^{3} \left (\frac{a e^{2}}{3} + \frac{2 b d e}{3}\right ) + x^{2} \left (a d e + \frac{b d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16699, size = 115, normalized size = 1.25 \begin{align*} \frac{1}{4} \, b x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, b d x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, b d^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, a x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + a d x^{2} e \mathrm{sgn}\left (b x + a\right ) + a d^{2} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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