Optimal. Leaf size=46 \[ \frac{(b c-a d) (a+b x)^{m+1}}{b^2 (m+1)}+\frac{d (a+b x)^{m+2}}{b^2 (m+2)} \]
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Rubi [A] time = 0.0468932, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(b c-a d) (a+b x)^{m+1}}{b^2 (m+1)}+\frac{d (a+b x)^{m+2}}{b^2 (m+2)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x),x]
[Out]
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Rubi in Sympy [A] time = 9.42377, size = 37, normalized size = 0.8 \[ \frac{d \left (a + b x\right )^{m + 2}}{b^{2} \left (m + 2\right )} - \frac{\left (a + b x\right )^{m + 1} \left (a d - b c\right )}{b^{2} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c),x)
[Out]
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Mathematica [A] time = 0.0366205, size = 41, normalized size = 0.89 \[ \frac{(a+b x)^{m+1} (-a d+b c (m+2)+b d (m+1) x)}{b^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x),x]
[Out]
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Maple [A] time = 0.004, size = 49, normalized size = 1.1 \[ -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( -bdmx-bcm-bdx+ad-2\,bc \right ) }{{b}^{2} \left ({m}^{2}+3\,m+2 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(b*x + a)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216684, size = 112, normalized size = 2.43 \[ \frac{{\left (a b c m + 2 \, a b c - a^{2} d +{\left (b^{2} d m + b^{2} d\right )} x^{2} +{\left (2 \, b^{2} c +{\left (b^{2} c + a b d\right )} m\right )} x\right )}{\left (b x + a\right )}^{m}}{b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(b*x + a)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.18981, size = 377, normalized size = 8.2 \[ \begin{cases} a^{m} \left (c x + \frac{d x^{2}}{2}\right ) & \text{for}\: b = 0 \\\frac{a d \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a d}{a b^{2} + b^{3} x} - \frac{b c}{a b^{2} + b^{3} x} + \frac{b d x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: m = -2 \\- \frac{a d \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{c \log{\left (\frac{a}{b} + x \right )}}{b} + \frac{d x}{b} & \text{for}\: m = -1 \\- \frac{a^{2} d \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{a b c m \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{2 a b c \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{a b d m x \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{b^{2} c m x \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{2 b^{2} c x \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{b^{2} d m x^{2} \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} + \frac{b^{2} d x^{2} \left (a + b x\right )^{m}}{b^{2} m^{2} + 3 b^{2} m + 2 b^{2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**m*(d*x+c),x)
[Out]
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GIAC/XCAS [A] time = 0.337596, size = 200, normalized size = 4.35 \[ \frac{b^{2} d m x^{2} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + b^{2} c m x e^{\left (m{\rm ln}\left (b x + a\right )\right )} + a b d m x e^{\left (m{\rm ln}\left (b x + a\right )\right )} + b^{2} d x^{2} e^{\left (m{\rm ln}\left (b x + a\right )\right )} + a b c m e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 2 \, b^{2} c x e^{\left (m{\rm ln}\left (b x + a\right )\right )} + 2 \, a b c e^{\left (m{\rm ln}\left (b x + a\right )\right )} - a^{2} d e^{\left (m{\rm ln}\left (b x + a\right )\right )}}{b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(b*x + a)^m,x, algorithm="giac")
[Out]