Optimal. Leaf size=75 \[ \frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A] time = 0.0296745, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ \frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}{e^2}+\frac{(-2 c d+b e) (d+e x)^{3/2}}{e^2}+\frac{c (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{3 e^3}-\frac{2 (2 c d-b e) (d+e x)^{5/2}}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}\\ \end{align*}
Mathematica [A] time = 0.0477906, size = 55, normalized size = 0.73 \[ \frac{2 (d+e x)^{3/2} \left (7 e (5 a e-2 b d+3 b e x)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 53, normalized size = 0.7 \begin{align*}{\frac{30\,c{e}^{2}{x}^{2}+42\,b{e}^{2}x-24\,cdex+70\,a{e}^{2}-28\,bde+16\,c{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97133, size = 80, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c - 21 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c d^{2} - b d e + a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24467, size = 193, normalized size = 2.57 \begin{align*} \frac{2 \,{\left (15 \, c e^{3} x^{3} + 8 \, c d^{3} - 14 \, b d^{2} e + 35 \, a d e^{2} + 3 \,{\left (c d e^{2} + 7 \, b e^{3}\right )} x^{2} -{\left (4 \, c d^{2} e - 7 \, b d e^{2} - 35 \, a e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.27761, size = 71, normalized size = 0.95 \begin{align*} \frac{2 \left (\frac{c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{2}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09711, size = 111, normalized size = 1.48 \begin{align*} \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b e^{\left (-1\right )} +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} c e^{\left (-2\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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