3.29 \(\int (c x^2+d x^3)^n \, dx\)

Optimal. Leaf size=55 \[ \frac{x \left (\frac{d x}{c}+1\right )^{-n} \left (c x^2+d x^3\right )^n \text{Hypergeometric2F1}\left (-n,2 n+1,2 (n+1),-\frac{d x}{c}\right )}{2 n+1} \]

[Out]

(x*(c*x^2 + d*x^3)^n*Hypergeometric2F1[-n, 1 + 2*n, 2*(1 + n), -((d*x)/c)])/((1 + 2*n)*(1 + (d*x)/c)^n)

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Rubi [A]  time = 0.0192491, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2011, 66, 64} \[ \frac{x \left (\frac{d x}{c}+1\right )^{-n} \left (c x^2+d x^3\right )^n \text{Hypergeometric2F1}\left (-n,2 n+1,2 (n+1),-\frac{d x}{c}\right )}{2 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2 + d*x^3)^n,x]

[Out]

(x*(c*x^2 + d*x^3)^n*Hypergeometric2F1[-n, 1 + 2*n, 2*(1 + n), -((d*x)/c)])/((1 + 2*n)*(1 + (d*x)/c)^n)

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c^IntPart[n]*(c + d*x)^FracPart[n])/(1 + (d
*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0]))
||  !RationalQ[n])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int \left (c x^2+d x^3\right )^n \, dx &=\left (x^{-2 n} (c+d x)^{-n} \left (c x^2+d x^3\right )^n\right ) \int x^{2 n} (c+d x)^n \, dx\\ &=\left (x^{-2 n} \left (1+\frac{d x}{c}\right )^{-n} \left (c x^2+d x^3\right )^n\right ) \int x^{2 n} \left (1+\frac{d x}{c}\right )^n \, dx\\ &=\frac{x \left (1+\frac{d x}{c}\right )^{-n} \left (c x^2+d x^3\right )^n \, _2F_1\left (-n,1+2 n;2 (1+n);-\frac{d x}{c}\right )}{1+2 n}\\ \end{align*}

Mathematica [A]  time = 0.0121895, size = 53, normalized size = 0.96 \[ \frac{x \left (x^2 (c+d x)\right )^n \left (\frac{d x}{c}+1\right )^{-n} \text{Hypergeometric2F1}\left (-n,2 n+1,2 n+2,-\frac{d x}{c}\right )}{2 n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2 + d*x^3)^n,x]

[Out]

(x*(x^2*(c + d*x))^n*Hypergeometric2F1[-n, 1 + 2*n, 2 + 2*n, -((d*x)/c)])/((1 + 2*n)*(1 + (d*x)/c)^n)

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( d{x}^{3}+c{x}^{2} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c*x^2)^n,x)

[Out]

int((d*x^3+c*x^2)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{3} + c x^{2}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2)^n,x, algorithm="maxima")

[Out]

integrate((d*x^3 + c*x^2)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x^{3} + c x^{2}\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2)^n,x, algorithm="fricas")

[Out]

integral((d*x^3 + c*x^2)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c x^{2} + d x^{3}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c*x**2)**n,x)

[Out]

Integral((c*x**2 + d*x**3)**n, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{3} + c x^{2}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c*x^2)^n,x, algorithm="giac")

[Out]

integrate((d*x^3 + c*x^2)^n, x)