Optimal. Leaf size=146 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^3 (a+b x)}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{6 e^3 (a+b x)} \]
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Rubi [A] time = 0.152057, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8}{8 e^3 (a+b x)}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)}{7 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^2}{6 e^3 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^5 \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 (a+b x) \left (a b+b^2 x\right ) (d+e x)^5 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^5 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^2 (d+e x)^5}{e^2}-\frac{2 b (b d-a e) (d+e x)^6}{e^2}+\frac{b^2 (d+e x)^7}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^2 (d+e x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^3 (a+b x)}-\frac{2 b (b d-a e) (d+e x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^3 (a+b x)}+\frac{b^2 (d+e x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^3 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0680088, size = 196, normalized size = 1.34 \[ \frac{x \sqrt{(a+b x)^2} \left (28 a^2 \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+8 a b x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+b^2 x^2 \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )\right )}{168 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 230, normalized size = 1.6 \begin{align*}{\frac{x \left ( 21\,{b}^{2}{e}^{5}{x}^{7}+48\,{x}^{6}ab{e}^{5}+120\,{x}^{6}{b}^{2}d{e}^{4}+28\,{x}^{5}{a}^{2}{e}^{5}+280\,{x}^{5}abd{e}^{4}+280\,{x}^{5}{b}^{2}{d}^{2}{e}^{3}+168\,{a}^{2}d{e}^{4}{x}^{4}+672\,ab{d}^{2}{e}^{3}{x}^{4}+336\,{b}^{2}{d}^{3}{e}^{2}{x}^{4}+420\,{x}^{3}{a}^{2}{d}^{2}{e}^{3}+840\,{x}^{3}ab{d}^{3}{e}^{2}+210\,{x}^{3}{b}^{2}{d}^{4}e+560\,{x}^{2}{a}^{2}{d}^{3}{e}^{2}+560\,{x}^{2}ab{d}^{4}e+56\,{x}^{2}{b}^{2}{d}^{5}+420\,x{a}^{2}{d}^{4}e+168\,xab{d}^{5}+168\,{a}^{2}{d}^{5} \right ) }{168\,bx+168\,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 = 1.50166, size = 417, normalized size = 2.86 \begin{align*} \frac{1}{8} \, b^{2} e^{5} x^{8} + a^{2} d^{5} x + \frac{1}{7} \,{\left (5 \, b^{2} d e^{4} + 2 \, a b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (10 \, b^{2} d^{2} e^{3} + 10 \, a b d e^{4} + a^{2} e^{5}\right )} x^{6} +{\left (2 \, b^{2} d^{3} e^{2} + 4 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{5} + \frac{5}{4} \,{\left (b^{2} d^{4} e + 4 \, a b d^{3} e^{2} + 2 \, a^{2} d^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{5} + 10 \, a b d^{4} e + 10 \, a^{2} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d^{5} + 5 \, a^{2} d^{4} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.143769, size = 218, normalized size = 1.49 \begin{align*} a^{2} d^{5} x + \frac{b^{2} e^{5} x^{8}}{8} + x^{7} \left (\frac{2 a b e^{5}}{7} + \frac{5 b^{2} d e^{4}}{7}\right ) + x^{6} \left (\frac{a^{2} e^{5}}{6} + \frac{5 a b d e^{4}}{3} + \frac{5 b^{2} d^{2} e^{3}}{3}\right ) + x^{5} \left (a^{2} d e^{4} + 4 a b d^{2} e^{3} + 2 b^{2} d^{3} e^{2}\right ) + x^{4} \left (\frac{5 a^{2} d^{2} e^{3}}{2} + 5 a b d^{3} e^{2} + \frac{5 b^{2} d^{4} e}{4}\right ) + x^{3} \left (\frac{10 a^{2} d^{3} e^{2}}{3} + \frac{10 a b d^{4} e}{3} + \frac{b^{2} d^{5}}{3}\right ) + x^{2} \left (\frac{5 a^{2} d^{4} e}{2} + a b d^{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15787, size = 420, normalized size = 2.88 \begin{align*} \frac{1}{8} \, b^{2} x^{8} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{7} \, b^{2} d x^{7} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, b^{2} d^{2} x^{6} e^{3} \mathrm{sgn}\left (b x + a\right ) + 2 \, b^{2} d^{3} x^{5} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{4} \, b^{2} d^{4} x^{4} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d^{5} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{7} \, a b x^{7} e^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{3} \, a b d x^{6} e^{4} \mathrm{sgn}\left (b x + a\right ) + 4 \, a b d^{2} x^{5} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a b d^{3} x^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a b d^{4} x^{3} e \mathrm{sgn}\left (b x + a\right ) + a b d^{5} x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{6} \, a^{2} x^{6} e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{2} d x^{5} e^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} d^{2} x^{4} e^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{2} d^{3} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} d^{4} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{2} d^{5} x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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