Optimal. Leaf size=22 \[ \frac{x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]
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Rubi [A] time = 0.0172871, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{x^{2 (n+1)} (a+b x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
[In] Int[x^(1 + 2*n)*(a + b*x)^n*(2*a + 3*b*x),x]
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Rubi in Sympy [A] time = 3.97618, size = 17, normalized size = 0.77 \[ \frac{x^{2 n + 2} \left (a + b x\right )^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1+2*n)*(b*x+a)**n*(3*b*x+2*a),x)
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Mathematica [A] time = 0.0389279, size = 22, normalized size = 1. \[ \frac{x^{2 n+2} (a+b x)^{n+1}}{n+1} \]
Antiderivative was successfully verified.
[In] Integrate[x^(1 + 2*n)*(a + b*x)^n*(2*a + 3*b*x),x]
[Out]
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Maple [A] time = 0.006, size = 23, normalized size = 1.1 \[{\frac{{x}^{2+2\,n} \left ( bx+a \right ) ^{1+n}}{1+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1+2*n)*(b*x+a)^n*(3*b*x+2*a),x)
[Out]
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Maxima [A] time = 1.53661, size = 43, normalized size = 1.95 \[ \frac{{\left (b x^{3} + a x^{2}\right )} e^{\left (n \log \left (b x + a\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*b*x + 2*a)*(b*x + a)^n*x^(2*n + 1),x, algorithm="maxima")
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Fricas [A] time = 0.246743, size = 39, normalized size = 1.77 \[ \frac{{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{n} x^{2 \, n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*b*x + 2*a)*(b*x + a)^n*x^(2*n + 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1+2*n)*(b*x+a)**n*(3*b*x+2*a),x)
[Out]
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GIAC/XCAS [A] time = 0.238431, size = 66, normalized size = 3. \[ \frac{b x^{2} e^{\left (n{\rm ln}\left (b x + a\right ) + 2 \, n{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )} + a x e^{\left (n{\rm ln}\left (b x + a\right ) + 2 \, n{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*b*x + 2*a)*(b*x + a)^n*x^(2*n + 1),x, algorithm="giac")
[Out]