3.30 \(\int (b x+d x^3)^n \, dx\)

Optimal. Leaf size=53 \[ \frac{x \left (b+d x^2\right ) \left (b x+d x^3\right )^n \text{Hypergeometric2F1}\left (1,\frac{3 (n+1)}{2},\frac{n+3}{2},-\frac{d x^2}{b}\right )}{b (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0227277, antiderivative size = 59, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2011, 365, 364} \[ \frac{x \left (\frac{d x^2}{b}+1\right )^{-n} \left (b x+d x^3\right )^n \text{Hypergeometric2F1}\left (-n,\frac{n+1}{2},\frac{n+3}{2},-\frac{d x^2}{b}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + d*x^3)^n,x]

[Out]

(x*(b*x + d*x^3)^n*Hypergeometric2F1[-n, (1 + n)/2, (3 + n)/2, -((d*x^2)/b)])/((1 + n)*(1 + (d*x^2)/b)^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 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0123611, size = 61, normalized size = 1.15 \[ \frac{x \left (x \left (b+d x^2\right )\right )^n \left (\frac{d x^2}{b}+1\right )^{-n} \text{Hypergeometric2F1}\left (-n,\frac{n+1}{2},\frac{n+1}{2}+1,-\frac{d x^2}{b}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + d*x^3)^n,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+b*x)^n,x)

[Out]

int((d*x^3+b*x)^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x^3 + b*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x^3 + b*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+b*x)**n,x)

[Out]

Integral((b*x + 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} + b x\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x^3 + b*x)^n, x)