Optimal. Leaf size=47 \[ \frac{b (c+d x)^{n+2}}{d^2 (n+2)}-\frac{(b c-a d) (c+d x)^{n+1}}{d^2 (n+1)} \]
[Out]
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Rubi [A] time = 0.0493715, antiderivative size = 47, 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 \[ \frac{b (c+d x)^{n+2}}{d^2 (n+2)}-\frac{(b c-a d) (c+d x)^{n+1}}{d^2 (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(c + d*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 9.48273, size = 37, normalized size = 0.79 \[ \frac{b \left (c + d x\right )^{n + 2}}{d^{2} \left (n + 2\right )} + \frac{\left (c + d x\right )^{n + 1} \left (a d - b c\right )}{d^{2} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(d*x+c)**n,x)
[Out]
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Mathematica [A] time = 0.0352039, size = 41, normalized size = 0.87 \[ \frac{(c+d x)^{n+1} (a d (n+2)-b c+b d (n+1) x)}{d^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(c + d*x)^n,x]
[Out]
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Maple [A] time = 0.005, size = 46, normalized size = 1. \[{\frac{ \left ( dx+c \right ) ^{1+n} \left ( bdnx+adn+bdx+2\,ad-bc \right ) }{{d}^{2} \left ({n}^{2}+3\,n+2 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(d*x+c)^n,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((b*x + a)*(d*x + c)^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22024, size = 112, normalized size = 2.38 \[ \frac{{\left (a c d n - b c^{2} + 2 \, a c d +{\left (b d^{2} n + b d^{2}\right )} x^{2} +{\left (2 \, a d^{2} +{\left (b c d + a d^{2}\right )} n\right )} x\right )}{\left (d x + c\right )}^{n}}{d^{2} n^{2} + 3 \, d^{2} n + 2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^n,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.17724, size = 377, normalized size = 8.02 \[ \begin{cases} c^{n} \left (a x + \frac{b x^{2}}{2}\right ) & \text{for}\: d = 0 \\- \frac{a d}{c d^{2} + d^{3} x} + \frac{b c \log{\left (\frac{c}{d} + x \right )}}{c d^{2} + d^{3} x} + \frac{b c}{c d^{2} + d^{3} x} + \frac{b d x \log{\left (\frac{c}{d} + x \right )}}{c d^{2} + d^{3} x} & \text{for}\: n = -2 \\\frac{a \log{\left (\frac{c}{d} + x \right )}}{d} - \frac{b c \log{\left (\frac{c}{d} + x \right )}}{d^{2}} + \frac{b x}{d} & \text{for}\: n = -1 \\\frac{a c d n \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{2 a c d \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{a d^{2} n x \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{2 a d^{2} x \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} - \frac{b c^{2} \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{b c d n x \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{b d^{2} n x^{2} \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} + \frac{b d^{2} x^{2} \left (c + d x\right )^{n}}{d^{2} n^{2} + 3 d^{2} n + 2 d^{2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(d*x+c)**n,x)
[Out]
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GIAC/XCAS [A] time = 0.230104, size = 200, normalized size = 4.26 \[ \frac{b d^{2} n x^{2} e^{\left (n{\rm ln}\left (d x + c\right )\right )} + b c d n x e^{\left (n{\rm ln}\left (d x + c\right )\right )} + a d^{2} n x e^{\left (n{\rm ln}\left (d x + c\right )\right )} + b d^{2} x^{2} e^{\left (n{\rm ln}\left (d x + c\right )\right )} + a c d n e^{\left (n{\rm ln}\left (d x + c\right )\right )} + 2 \, a d^{2} x e^{\left (n{\rm ln}\left (d x + c\right )\right )} - b c^{2} e^{\left (n{\rm ln}\left (d x + c\right )\right )} + 2 \, a c d e^{\left (n{\rm ln}\left (d x + c\right )\right )}}{d^{2} n^{2} + 3 \, d^{2} n + 2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x + c)^n,x, algorithm="giac")
[Out]