Optimal. Leaf size=39 \[ \frac{(a+b x) (d+e x)^4}{4 e \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0299404, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 32} \[ \frac{(a+b x) (d+e x)^4}{4 e \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^3}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{(a+b x) (d+e x)^3}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int (d+e x)^3 \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{(a+b x) (d+e x)^4}{4 e \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0117452, size = 30, normalized size = 0.77 \[ \frac{(a+b x) (d+e x)^4}{4 e \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 47, normalized size = 1.2 \begin{align*}{\frac{x \left ({e}^{3}{x}^{3}+4\,{e}^{2}{x}^{2}d+6\,e{d}^{2}x+4\,{d}^{3} \right ) \left ( bx+a \right ) }{4}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00349, size = 713, normalized size = 18.28 \begin{align*} \frac{13 \, a^{4} b e^{3} \log \left (x + \frac{a}{b}\right )}{6 \,{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{13 \, a^{3} e^{3} x}{6 \,{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{13 \, a^{2} e^{3} x^{2}}{12 \, \sqrt{b^{2}} b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} e^{3} x^{3}}{4 \, b} + a \sqrt{\frac{1}{b^{2}}} d^{3} \log \left (x + \frac{a}{b}\right ) - \frac{7 \, a^{4} \sqrt{\frac{1}{b^{2}}} e^{3} \log \left (x + \frac{a}{b}\right )}{6 \, b^{3}} - \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a e^{3} x^{2}}{12 \, b^{2}} - \frac{5 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} b \log \left (x + \frac{a}{b}\right )}{3 \,{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{3 \,{\left (b d^{2} e + a d e^{2}\right )} a^{2} b^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} + \frac{7 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} e^{3}}{6 \, b^{4}} + \frac{5 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a^{2} x}{3 \,{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (b d^{2} e + a d e^{2}\right )} a b x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (b d^{2} e + a d e^{2}\right )} x^{2}}{2 \, \sqrt{b^{2}}} - \frac{5 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a x^{2}}{6 \, \sqrt{b^{2}} b} + \frac{2 \,{\left (3 \, b d e^{2} + a e^{3}\right )} a^{3} \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{3 \, b^{3}} - \frac{{\left (b d^{3} + 3 \, a d^{2} e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} x^{2}}{3 \, b^{2}} - \frac{2 \,{\left (3 \, b d e^{2} + a e^{3}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{3 \, b^{4}} + \frac{{\left (b d^{3} + 3 \, a d^{2} e\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44218, size = 66, normalized size = 1.69 \begin{align*} \frac{1}{4} \, e^{3} x^{4} + d e^{2} x^{3} + \frac{3}{2} \, d^{2} e x^{2} + d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102028, size = 32, normalized size = 0.82 \begin{align*} d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11084, size = 24, normalized size = 0.62 \begin{align*} \frac{1}{4} \,{\left (x e + d\right )}^{4} e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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