3.216 \(\int \left (a+c x^2\right ) \left (1+\left (a x+\frac{c x^3}{3}\right )^n\right ) \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (a x+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{c x^3}{3} \]

[Out]

a*x + (c*x^3)/3 + (a*x + (c*x^3)/3)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0195225, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\left (a x+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)*(1 + (a*x + (c*x^3)/3)^n),x]

[Out]

a*x + (c*x^3)/3 + (a*x + (c*x^3)/3)^(1 + n)/(1 + n)

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Rubi in Sympy [A]  time = 2.50489, size = 26, normalized size = 0.76 \[ a x + \frac{c x^{3}}{3} + \frac{\left (a x + \frac{c x^{3}}{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)*(1+(a*x+1/3*c*x**3)**n),x)

[Out]

a*x + c*x**3/3 + (a*x + c*x**3/3)**(n + 1)/(n + 1)

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Mathematica [A]  time = 0.033954, size = 36, normalized size = 1.06 \[ \frac{x \left (3 a+c x^2\right ) \left (\left (a x+\frac{c x^3}{3}\right )^n+n+1\right )}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)*(1 + (a*x + (c*x^3)/3)^n),x]

[Out]

(x*(3*a + c*x^2)*(1 + n + (a*x + (c*x^3)/3)^n))/(3*(1 + n))

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Maple [A]  time = 0.003, size = 31, normalized size = 0.9 \[ ax+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ( ax+{\frac{c{x}^{3}}{3}} \right ) ^{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)*(1+(a*x+1/3*c*x^3)^n),x)

[Out]

a*x+1/3*c*x^3+(a*x+1/3*c*x^3)^(1+n)/(1+n)

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Maxima [A]  time = 0.998622, size = 73, normalized size = 2.15 \[ \frac{1}{3} \, c x^{3} + a x + \frac{{\left (c x^{3} + 3 \, a x\right )} e^{\left (n \log \left (c x^{2} + 3 \, a\right ) + n \log \left (x\right )\right )}}{3^{n + 1} n + 3^{n + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*((1/3*c*x^3 + a*x)^n + 1),x, algorithm="maxima")

[Out]

1/3*c*x^3 + a*x + (c*x^3 + 3*a*x)*e^(n*log(c*x^2 + 3*a) + n*log(x))/(3^(n + 1)*n
+ 3^(n + 1))

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Fricas [A]  time = 0.290244, size = 65, normalized size = 1.91 \[ \frac{{\left (c n + c\right )} x^{3} +{\left (c x^{3} + 3 \, a x\right )}{\left (\frac{1}{3} \, c x^{3} + a x\right )}^{n} + 3 \,{\left (a n + a\right )} x}{3 \,{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*((1/3*c*x^3 + a*x)^n + 1),x, algorithm="fricas")

[Out]

1/3*((c*n + c)*x^3 + (c*x^3 + 3*a*x)*(1/3*c*x^3 + a*x)^n + 3*(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((c*x**2+a)*(1+(a*x+1/3*c*x**3)**n),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.258519, size = 41, normalized size = 1.21 \[ \frac{1}{3} \, c x^{3} + a x + \frac{{\left (\frac{1}{3} \, c x^{3} + a x\right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*((1/3*c*x^3 + a*x)^n + 1),x, algorithm="giac")

[Out]

1/3*c*x^3 + a*x + (1/3*c*x^3 + a*x)^(n + 1)/(n + 1)