3.208 \(\int (a+b x) \left (1+\left (a x+\frac{b x^2}{2}\right )^n\right ) \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (a x+\frac{b x^2}{2}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]

[Out]

a*x + (b*x^2)/2 + (a*x + (b*x^2)/2)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.019103, antiderivative size = 34, 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 \[ \frac{\left (a x+\frac{b x^2}{2}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(1 + (a*x + (b*x^2)/2)^n),x]

[Out]

a*x + (b*x^2)/2 + (a*x + (b*x^2)/2)^(1 + n)/(1 + n)

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Rubi in Sympy [A]  time = 2.34797, size = 26, normalized size = 0.76 \[ a x + \frac{b x^{2}}{2} + \frac{\left (a x + \frac{b x^{2}}{2}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(1+(a*x+1/2*b*x**2)**n),x)

[Out]

a*x + b*x**2/2 + (a*x + b*x**2/2)**(n + 1)/(n + 1)

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Mathematica [A]  time = 0.0301894, size = 34, normalized size = 1. \[ \frac{x (2 a+b x) \left (\left (a x+\frac{b x^2}{2}\right )^n+n+1\right )}{2 (n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(1 + (a*x + (b*x^2)/2)^n),x]

[Out]

(x*(2*a + b*x)*(1 + n + (a*x + (b*x^2)/2)^n))/(2*(1 + n))

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Maple [A]  time = 0.002, size = 31, normalized size = 0.9 \[ ax+{\frac{b{x}^{2}}{2}}+{\frac{1}{1+n} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) ^{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(1+(a*x+1/2*b*x^2)^n),x)

[Out]

a*x+1/2*b*x^2+(a*x+1/2*b*x^2)^(1+n)/(1+n)

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Maxima [A]  time = 0.991402, size = 70, normalized size = 2.06 \[ \frac{1}{2} \, b x^{2} + a x + \frac{{\left (b x^{2} + 2 \, a x\right )} e^{\left (n \log \left (b x + 2 \, a\right ) + n \log \left (x\right )\right )}}{2^{n + 1} n + 2^{n + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*((1/2*b*x^2 + a*x)^n + 1),x, algorithm="maxima")

[Out]

1/2*b*x^2 + a*x + (b*x^2 + 2*a*x)*e^(n*log(b*x + 2*a) + n*log(x))/(2^(n + 1)*n +
2^(n + 1))

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Fricas [A]  time = 0.291046, size = 65, normalized size = 1.91 \[ \frac{{\left (b n + b\right )} x^{2} +{\left (b x^{2} + 2 \, a x\right )}{\left (\frac{1}{2} \, b x^{2} + a x\right )}^{n} + 2 \,{\left (a n + a\right )} x}{2 \,{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*((1/2*b*x^2 + a*x)^n + 1),x, algorithm="fricas")

[Out]

1/2*((b*n + b)*x^2 + (b*x^2 + 2*a*x)*(1/2*b*x^2 + a*x)^n + 2*(a*n + a)*x)/(n + 1
)

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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((b*x+a)*(1+(a*x+1/2*b*x**2)**n),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.261112, size = 41, normalized size = 1.21 \[ \frac{1}{2} \, b x^{2} + a x + \frac{{\left (\frac{1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*((1/2*b*x^2 + a*x)^n + 1),x, algorithm="giac")

[Out]

1/2*b*x^2 + a*x + (1/2*b*x^2 + a*x)^(n + 1)/(n + 1)