Optimal. Leaf size=65 \[ \frac{(b d+2 c d x)^{m+3}}{8 c^2 d^3 (m+3)}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{m+1}}{8 c^2 d (m+1)} \]
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Rubi [A] time = 0.0284689, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ \frac{(b d+2 c d x)^{m+3}}{8 c^2 d^3 (m+3)}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{m+1}}{8 c^2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^m}{4 c}+\frac{(b d+2 c d x)^{2+m}}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{1+m}}{8 c^2 d (1+m)}+\frac{(b d+2 c d x)^{3+m}}{8 c^2 d^3 (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0334382, size = 64, normalized size = 0.98 \[ \frac{(b+2 c x) \left (2 c \left (a (m+3)+c (m+1) x^2\right )-b^2+2 b c (m+1) x\right ) (d (b+2 c x))^m}{4 c^2 (m+1) (m+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 76, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2\,cdx+bd \right ) ^{m} \left ( 2\,{c}^{2}m{x}^{2}+2\,bcmx+2\,{c}^{2}{x}^{2}+2\,acm+2\,bcx+6\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) }{4\,{c}^{2} \left ({m}^{2}+4\,m+3 \right ) }} \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 = 2.22908, size = 223, normalized size = 3.43 \begin{align*} \frac{{\left (2 \, a b c m + 4 \,{\left (c^{3} m + c^{3}\right )} x^{3} - b^{3} + 6 \, a b c + 6 \,{\left (b c^{2} m + b c^{2}\right )} x^{2} + 2 \,{\left (6 \, a c^{2} +{\left (b^{2} c + 2 \, a c^{2}\right )} m\right )} x\right )}{\left (2 \, c d x + b d\right )}^{m}}{4 \,{\left (c^{2} m^{2} + 4 \, c^{2} m + 3 \, c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.5383, size = 707, normalized size = 10.88 \begin{align*} \begin{cases} \left (b d\right )^{m} \left (a x + \frac{b x^{2}}{2}\right ) & \text{for}\: c = 0 \\- \frac{4 a c}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{2 b^{2} \log{\left (\frac{b}{2 c} + x \right )}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{b^{2}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{8 b c x \log{\left (\frac{b}{2 c} + x \right )}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{8 c^{2} x^{2} \log{\left (\frac{b}{2 c} + x \right )}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} & \text{for}\: m = -3 \\\frac{a \log{\left (\frac{b}{2 c} + x \right )}}{2 c d} - \frac{b^{2} \log{\left (\frac{b}{2 c} + x \right )}}{8 c^{2} d} + \frac{b x}{4 c d} + \frac{x^{2}}{4 d} & \text{for}\: m = -1 \\\frac{2 a b c m \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{6 a b c \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{4 a c^{2} m x \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{12 a c^{2} x \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} - \frac{b^{3} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{2 b^{2} c m x \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{6 b c^{2} m x^{2} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{6 b c^{2} x^{2} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{4 c^{3} m x^{3} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{4 c^{3} x^{3} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16209, size = 282, normalized size = 4.34 \begin{align*} \frac{4 \,{\left (2 \, c d x + b d\right )}^{m} c^{3} m x^{3} + 6 \,{\left (2 \, c d x + b d\right )}^{m} b c^{2} m x^{2} + 4 \,{\left (2 \, c d x + b d\right )}^{m} c^{3} x^{3} + 2 \,{\left (2 \, c d x + b d\right )}^{m} b^{2} c m x + 4 \,{\left (2 \, c d x + b d\right )}^{m} a c^{2} m x + 6 \,{\left (2 \, c d x + b d\right )}^{m} b c^{2} x^{2} + 2 \,{\left (2 \, c d x + b d\right )}^{m} a b c m + 12 \,{\left (2 \, c d x + b d\right )}^{m} a c^{2} x -{\left (2 \, c d x + b d\right )}^{m} b^{3} + 6 \,{\left (2 \, c d x + b d\right )}^{m} a b c}{4 \,{\left (c^{2} m^{2} + 4 \, c^{2} m + 3 \, c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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