3.439 \(\int x^m (a x^j+b x^n)^p \, dx\)

Optimal. Leaf size=89 \[ \frac{x^{m+1} \left (a+b x^{n-j}\right ) \left (a x^j+b x^n\right )^p \, _2F_1\left (1,p+\frac{m+j p+1}{n-j}+1;\frac{m+j p+1}{n-j}+1;-\frac{b x^{n-j}}{a}\right )}{a (j p+m+1)} \]

[Out]

(x^(1 + m)*(a*x^j + b*x^n)^p*(a + b*x^(-j + n))*Hypergeometric2F1[1, 1 + p + (1 + m + j*p)/(-j + n), 1 + (1 +
m + j*p)/(-j + n), -((b*x^(-j + n))/a)])/(a*(1 + m + j*p))

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Rubi [A]  time = 0.0683331, antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2032, 365, 364} \[ \frac{x^{m+1} \left (\frac{a x^{j-n}}{b}+1\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac{m+n p+1}{j-n};\frac{m+n p+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{m+n p+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a*x^j + b*x^n)^p,x]

[Out]

(x^(1 + m)*(a*x^j + b*x^n)^p*Hypergeometric2F1[-p, (1 + m + n*p)/(j - n), 1 + (1 + m + n*p)/(j - n), -((a*x^(j
- n))/b)])/((1 + m + n*p)*(1 + (a*x^(j - n))/b)^p)

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[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 x^m \left (a x^j+b x^n\right )^p \, dx &=\left (x^{-n p} \left (b+a x^{j-n}\right )^{-p} \left (a x^j+b x^n\right )^p\right ) \int x^{m+n p} \left (b+a x^{j-n}\right )^p \, dx\\ &=\left (x^{-n p} \left (1+\frac{a x^{j-n}}{b}\right )^{-p} \left (a x^j+b x^n\right )^p\right ) \int x^{m+n p} \left (1+\frac{a x^{j-n}}{b}\right )^p \, dx\\ &=\frac{x^{1+m} \left (1+\frac{a x^{j-n}}{b}\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac{1+m+n p}{j-n};1+\frac{1+m+n p}{j-n};-\frac{a x^{j-n}}{b}\right )}{1+m+n p}\\ \end{align*}

Mathematica [A]  time = 0.103634, size = 92, normalized size = 1.03 \[ \frac{x^{m+1} \left (\frac{a x^{j-n}}{b}+1\right )^{-p} \left (a x^j+b x^n\right )^p \, _2F_1\left (-p,\frac{m+n p+1}{j-n};\frac{m+n p+1}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{m+n p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a*x^j + b*x^n)^p,x]

[Out]

(x^(1 + m)*(a*x^j + b*x^n)^p*Hypergeometric2F1[-p, (1 + m + n*p)/(j - n), 1 + (1 + m + n*p)/(j - n), -((a*x^(j
- n))/b)])/((1 + m + n*p)*(1 + (a*x^(j - n))/b)^p)

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Maple [F]  time = 0.582, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a{x}^{j}+b{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a*x^j+b*x^n)^p,x)

[Out]

int(x^m*(a*x^j+b*x^n)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a*x^j+b*x^n)^p,x, algorithm="maxima")

[Out]

integrate((a*x^j + b*x^n)^p*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a*x^j+b*x^n)^p,x, algorithm="fricas")

[Out]

integral((a*x^j + b*x^n)^p*x^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a*x**j+b*x**n)**p,x)

[Out]

Integral(x**m*(a*x**j + b*x**n)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a*x^j+b*x^n)^p,x, algorithm="giac")

[Out]

integrate((a*x^j + b*x^n)^p*x^m, x)