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

Optimal. Leaf size=297 \[ \frac{d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)}+\frac{e x^{n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+\frac{f x^{2 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}+\frac{g x^{3 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3 n+1}{n},-p;\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{m+3 n+1} \]

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

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Rubi [A]  time = 0.208185, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1844, 365, 364, 20} \[ \frac{d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)}+\frac{e x^{n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+\frac{f x^{2 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}+\frac{g x^{3 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3 n+1}{n},-p;\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{m+3 n+1} \]

Antiderivative was successfully verified.

[In]

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

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rubi steps

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

Mathematica [A]  time = 0.273287, size = 204, normalized size = 0.69 \[ x (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (\frac{d \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}+x^n \left (\frac{e \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+x^n \left (\frac{f \, _2F_1\left (\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}+\frac{g x^n \, _2F_1\left (\frac{m+3 n+1}{n},-p;\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{m+3 n+1}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 0.385, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( d+e{x}^{n}+f{x}^{2\,n}+g{x}^{3\,n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

[Out]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError