Optimal. Leaf size=130 \[ \frac{2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^4}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{6 c (d+e x)^{5/2} (2 c d-b e)}{5 e^4}+\frac{4 c^2 (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.0624168, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^4}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac{6 c (d+e x)^{5/2} (2 c d-b e)}{5 e^4}+\frac{4 c^2 (d+e x)^{7/2}}{7 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 \sqrt{d+e x}}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt{d+e x}}{e^3}-\frac{3 c (2 c d-b e) (d+e x)^{3/2}}{e^3}+\frac{2 c^2 (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}{e^4}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{3 e^4}-\frac{6 c (2 c d-b e) (d+e x)^{5/2}}{5 e^4}+\frac{4 c^2 (d+e x)^{7/2}}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.12033, size = 109, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (7 c e \left (10 a e (e x-2 d)+3 b \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+35 b e^2 (3 a e-2 b d+b e x)-6 c^2 \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right )\right )}{105 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 123, normalized size = 1. \begin{align*}{\frac{60\,{c}^{2}{x}^{3}{e}^{3}+126\,bc{e}^{3}{x}^{2}-72\,{c}^{2}d{e}^{2}{x}^{2}+140\,ac{e}^{3}x+70\,{b}^{2}{e}^{3}x-168\,bcd{e}^{2}x+96\,{c}^{2}{d}^{2}ex+210\,ab{e}^{3}-280\,acd{e}^{2}-140\,{b}^{2}d{e}^{2}+336\,b{d}^{2}ce-192\,{c}^{2}{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991059, size = 163, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (30 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 63 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20003, size = 271, normalized size = 2.08 \begin{align*} \frac{2 \,{\left (30 \, c^{2} e^{3} x^{3} - 96 \, c^{2} d^{3} + 168 \, b c d^{2} e + 105 \, a b e^{3} - 70 \,{\left (b^{2} + 2 \, a c\right )} d e^{2} - 9 \,{\left (4 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} x^{2} +{\left (48 \, c^{2} d^{2} e - 84 \, b c d e^{2} + 35 \,{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.0092, size = 427, normalized size = 3.28 \begin{align*} \begin{cases} - \frac{\frac{2 a b d}{\sqrt{d + e x}} + 2 a b \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a c d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 a c \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 b^{2} d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 b^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{6 b c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{6 b c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 c^{2} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{4 c^{2} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{\left (a + b x + c x^{2}\right )^{2}}{2 \sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42554, size = 224, normalized size = 1.72 \begin{align*} \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} b^{2} e^{\left (-1\right )} + 70 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a c e^{\left (-1\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} b c e^{\left (-2\right )} + 6 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} c^{2} e^{\left (-3\right )} + 105 \, \sqrt{x e + d} a b\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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