3.586 \(\int (c x)^m (d+e x+f x^2+g x^3) (a+b x^n)^p \, dx\)

Optimal. Leaf size=273 \[ \frac{e (c x)^{m+2} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{n},-p;\frac{m+n+2}{n};-\frac{b x^n}{a}\right )}{c^2 (m+2)}+\frac{f (c x)^{m+3} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3}{n},-p;\frac{m+n+3}{n};-\frac{b x^n}{a}\right )}{c^3 (m+3)}+\frac{g (c x)^{m+4} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+4}{n},-p;\frac{m+n+4}{n};-\frac{b x^n}{a}\right )}{c^4 (m+4)}+\frac{d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)} \]

[Out]

(d*(c*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(c*(1 + m)*(1 +
(b*x^n)/a)^p) + (e*(c*x)^(2 + m)*(a + b*x^n)^p*Hypergeometric2F1[(2 + m)/n, -p, (2 + m + n)/n, -((b*x^n)/a)])/
(c^2*(2 + m)*(1 + (b*x^n)/a)^p) + (f*(c*x)^(3 + m)*(a + b*x^n)^p*Hypergeometric2F1[(3 + m)/n, -p, (3 + m + n)/
n, -((b*x^n)/a)])/(c^3*(3 + m)*(1 + (b*x^n)/a)^p) + (g*(c*x)^(4 + m)*(a + b*x^n)^p*Hypergeometric2F1[(4 + m)/n
, -p, (4 + m + n)/n, -((b*x^n)/a)])/(c^4*(4 + m)*(1 + (b*x^n)/a)^p)

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Rubi [A]  time = 0.18508, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1844, 365, 364} \[ \frac{e (c x)^{m+2} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{n},-p;\frac{m+n+2}{n};-\frac{b x^n}{a}\right )}{c^2 (m+2)}+\frac{f (c x)^{m+3} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3}{n},-p;\frac{m+n+3}{n};-\frac{b x^n}{a}\right )}{c^3 (m+3)}+\frac{g (c x)^{m+4} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+4}{n},-p;\frac{m+n+4}{n};-\frac{b x^n}{a}\right )}{c^4 (m+4)}+\frac{d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(d + e*x + f*x^2 + g*x^3)*(a + b*x^n)^p,x]

[Out]

(d*(c*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(c*(1 + m)*(1 +
(b*x^n)/a)^p) + (e*(c*x)^(2 + m)*(a + b*x^n)^p*Hypergeometric2F1[(2 + m)/n, -p, (2 + m + n)/n, -((b*x^n)/a)])/
(c^2*(2 + m)*(1 + (b*x^n)/a)^p) + (f*(c*x)^(3 + m)*(a + b*x^n)^p*Hypergeometric2F1[(3 + m)/n, -p, (3 + m + n)/
n, -((b*x^n)/a)])/(c^3*(3 + m)*(1 + (b*x^n)/a)^p) + (g*(c*x)^(4 + m)*(a + b*x^n)^p*Hypergeometric2F1[(4 + m)/n
, -p, (4 + m + n)/n, -((b*x^n)/a)])/(c^4*(4 + m)*(1 + (b*x^n)/a)^p)

Rule 1844

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]

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 (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx &=\int \left (d (c x)^m \left (a+b x^n\right )^p+\frac{e (c x)^{1+m} \left (a+b x^n\right )^p}{c}+\frac{f (c x)^{2+m} \left (a+b x^n\right )^p}{c^2}+\frac{g (c x)^{3+m} \left (a+b x^n\right )^p}{c^3}\right ) \, dx\\ &=d \int (c x)^m \left (a+b x^n\right )^p \, dx+\frac{e \int (c x)^{1+m} \left (a+b x^n\right )^p \, dx}{c}+\frac{f \int (c x)^{2+m} \left (a+b x^n\right )^p \, dx}{c^2}+\frac{g \int (c x)^{3+m} \left (a+b x^n\right )^p \, dx}{c^3}\\ &=\left (d \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (c x)^m \left (1+\frac{b x^n}{a}\right )^p \, dx+\frac{\left (e \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (c x)^{1+m} \left (1+\frac{b x^n}{a}\right )^p \, dx}{c}+\frac{\left (f \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (c x)^{2+m} \left (1+\frac{b x^n}{a}\right )^p \, dx}{c^2}+\frac{\left (g \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (c x)^{3+m} \left (1+\frac{b x^n}{a}\right )^p \, dx}{c^3}\\ &=\frac{d (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{n},-p;\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{c (1+m)}+\frac{e (c x)^{2+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{2+m}{n},-p;\frac{2+m+n}{n};-\frac{b x^n}{a}\right )}{c^2 (2+m)}+\frac{f (c x)^{3+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{3+m}{n},-p;\frac{3+m+n}{n};-\frac{b x^n}{a}\right )}{c^3 (3+m)}+\frac{g (c x)^{4+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \, _2F_1\left (\frac{4+m}{n},-p;\frac{4+m+n}{n};-\frac{b x^n}{a}\right )}{c^4 (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.24311, size = 178, normalized size = 0.65 \[ x (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (\frac{d \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}+x \left (\frac{e \, _2F_1\left (\frac{m+2}{n},-p;\frac{m+n+2}{n};-\frac{b x^n}{a}\right )}{m+2}+x \left (\frac{f \, _2F_1\left (\frac{m+3}{n},-p;\frac{m+n+3}{n};-\frac{b x^n}{a}\right )}{m+3}+\frac{g x \, _2F_1\left (\frac{m+4}{n},-p;\frac{m+n+4}{n};-\frac{b x^n}{a}\right )}{m+4}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(d + e*x + f*x^2 + g*x^3)*(a + b*x^n)^p,x]

[Out]

(x*(c*x)^m*(a + b*x^n)^p*((d*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(1 + m) + x*((e*Hy
pergeometric2F1[(2 + m)/n, -p, (2 + m + n)/n, -((b*x^n)/a)])/(2 + m) + x*((f*Hypergeometric2F1[(3 + m)/n, -p,
(3 + m + n)/n, -((b*x^n)/a)])/(3 + m) + (g*x*Hypergeometric2F1[(4 + m)/n, -p, (4 + m + n)/n, -((b*x^n)/a)])/(4
+ m)))))/(1 + (b*x^n)/a)^p

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Maple [F]  time = 0.437, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) \left ( a+b{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x)

[Out]

int((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x, algorithm="maxima")

[Out]

integrate((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x, algorithm="fricas")

[Out]

integral((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**m*(g*x**3+f*x**2+e*x+d)*(a+b*x**n)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x, algorithm="giac")

[Out]

integrate((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m, x)