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

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 = 0.0090691, 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, Rules used = {1591} \[ \frac{\left (a x+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{c x^3}{3} \]

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)

Rule 1591

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[Qr, x, r]/(q*Coeff[Pq, x, q]), Subst[Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x,
r]*D[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] && PolyQ[Qr, x]

Rubi steps

\begin{align*} \int \left (a+c x^2\right ) \left (1+\left (a x+\frac{c x^3}{3}\right )^n\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^n\right ) \, dx,x,a x+\frac{c x^3}{3}\right )\\ &=a x+\frac{c x^3}{3}+\frac{\left (a x+\frac{c x^3}{3}\right )^{1+n}}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.0616451, 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 = 0.001, size = 31, normalized size = 0.9 \begin{align*} ax+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ( ax+{\frac{c{x}^{3}}{3}} \right ) ^{1+n}} \end{align*}

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 = 1.65076, size = 73, normalized size = 2.15 \begin{align*} \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}} \end{align*}

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, 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 = 1.35445, size = 112, normalized size = 3.29 \begin{align*} \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 )}} \end{align*}

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, 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 [B]  time = 175.955, size = 201, normalized size = 5.91 \begin{align*} \begin{cases} \frac{3 \cdot 3^{n} a n x}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3 \cdot 3^{n} a x}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3^{n} c n x^{3}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3^{n} c x^{3}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3 a x \left (3 a x + c x^{3}\right )^{n}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{c x^{3} \left (3 a x + c x^{3}\right )^{n}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} & \text{for}\: n \neq -1 \\a x + \frac{c x^{3}}{3} + \log{\left (x \right )} + \log{\left (- \sqrt{3} i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )} + \log{\left (\sqrt{3} i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )} & \text{otherwise} \end{cases} \end{align*}

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]

Piecewise((3*3**n*a*n*x/(3*3**n*n + 3*3**n) + 3*3**n*a*x/(3*3**n*n + 3*3**n) + 3**n*c*n*x**3/(3*3**n*n + 3*3**
n) + 3**n*c*x**3/(3*3**n*n + 3*3**n) + 3*a*x*(3*a*x + c*x**3)**n/(3*3**n*n + 3*3**n) + c*x**3*(3*a*x + c*x**3)
**n/(3*3**n*n + 3*3**n), Ne(n, -1)), (a*x + c*x**3/3 + log(x) + log(-sqrt(3)*I*sqrt(a)*sqrt(1/c) + x) + log(sq
rt(3)*I*sqrt(a)*sqrt(1/c) + x), True))

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Giac [A]  time = 1.17752, size = 41, normalized size = 1.21 \begin{align*} \frac{1}{3} \, c x^{3} + a x + \frac{{\left (\frac{1}{3} \, c x^{3} + a x\right )}^{n + 1}}{n + 1} \end{align*}

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, algorithm="giac")

[Out]

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