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*hypergeom([-p, (1+m)/n],[(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*hypergeom([-p, (1+m+n)/n],[(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*hypergeom([-p, (1+m+2*n)/n],[(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*hypergeom([-p, (1+m+3*n)/n],[(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 = 0.21, antiderivative size = 297, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.30, 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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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m}, x\right ) \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp
lification assuming x near 0Simplification assuming c near 0Simplification assuming x near 0Simplification ass
uming c near 0Unable to divide, perhaps due to rounding error%%%{1,[2,0,6,4,0,2,4,4,1,0,0]%%%}+%%%{4,[2,0,6,4,
0,2,3,4,1,0,0]%%%}+%%%{6,[2,0,6,4,0,2,2,4,1,0,0]%%%}+%%%{4,[2,0,6,4,0,2,1,4,1,0,0]%%%}+%%%{1,[2,0,6,4,0,2,0,4,
1,0,0]%%%}+%%%{1,[1,0,6,4,0,2,4,4,0,1,0]%%%}+%%%{4,[1,0,6,4,0,2,3,4,0,1,0]%%%}+%%%{6,[1,0,6,4,0,2,2,4,0,1,0]%%
%}+%%%{4,[1,0,6,4,0,2,1,4,0,1,0]%%%}+%%%{1,[1,0,6,4,0,2,0,4,0,1,0]%%%}+%%%{-1,[1,0,6,4,0,1,4,5,1,0,0]%%%}+%%%{
-4,[1,0,6,4,0,1,3,5,1,0,0]%%%}+%%%{-6,[1,0,6,4,0,1,2,5,1,0,0]%%%}+%%%{-4,[1,0,6,4,0,1,1,5,1,0,0]%%%}+%%%{-1,[1
,0,6,4,0,1,0,5,1,0,0]%%%}+%%%{1,[0,0,6,4,0,3,4,3,0,0,1]%%%}+%%%{4,[0,0,6,4,0,3,3,3,0,0,1]%%%}+%%%{6,[0,0,6,4,0
,3,2,3,0,0,1]%%%}+%%%{4,[0,0,6,4,0,3,1,3,0,0,1]%%%}+%%%{1,[0,0,6,4,0,3,0,3,0,0,1]%%%}+%%%{1,[0,0,6,3,1,3,3,3,0
,0,1]%%%}+%%%{3,[0,0,6,3,1,3,2,3,0,0,1]%%%}+%%%{3,[0,0,6,3,1,3,1,3,0,0,1]%%%}+%%%{1,[0,0,6,3,1,3,0,3,0,0,1]%%%
}+%%%{1,[0,0,6,3,1,1,3,5,0,1,0]%%%}+%%%{3,[0,0,6,3,1,1,2,5,0,1,0]%%%}+%%%{3,[0,0,6,3,1,1,1,5,0,1,0]%%%}+%%%{1,
[0,0,6,3,1,1,0,5,0,1,0]%%%}+%%%{-1,[0,0,6,3,1,0,3,6,1,0,0]%%%}+%%%{-3,[0,0,6,3,1,0,2,6,1,0,0]%%%}+%%%{-3,[0,0,
6,3,1,0,1,6,1,0,0]%%%}+%%%{-1,[0,0,6,3,1,0,0,6,1,0,0]%%%}+%%%{1,[0,0,6,3,0,3,3,3,0,0,1]%%%}+%%%{3,[0,0,6,3,0,3
,2,3,0,0,1]%%%}+%%%{3,[0,0,6,3,0,3,1,3,0,0,1]%%%}+%%%{1,[0,0,6,3,0,3,0,3,0,0,1]%%%}+%%%{1,[0,0,6,3,0,1,3,5,0,1
,0]%%%}+%%%{3,[0,0,6,3,0,1,2,5,0,1,0]%%%}+%%%{3,[0,0,6,3,0,1,1,5,0,1,0]%%%}+%%%{1,[0,0,6,3,0,1,0,5,0,1,0]%%%}+
%%%{-1,[0,0,6,3,0,0,3,6,1,0,0]%%%}+%%%{-3,[0,0,6,3,0,0,2,6,1,0,0]%%%}+%%%{-3,[0,0,6,3,0,0,1,6,1,0,0]%%%}+%%%{-
1,[0,0,6,3,0,0,0,6,1,0,0]%%%} / %%%{1,[0,0,7,4,0,3,4,4,0,0,0]%%%}+%%%{4,[0,0,7,4,0,3,3,4,0,0,0]%%%}+%%%{6,[0,0
,7,4,0,3,2,4,0,0,0]%%%}+%%%{4,[0,0,7,4,0,3,1,4,0,0,0]%%%}+%%%{1,[0,0,7,4,0,3,0,4,0,0,0]%%%} Error: Bad Argumen
t Value

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maple [F]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}+d \right ) \left (c x \right )^{m} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,x\right )}^m\,{\left (a+b\,x^n\right )}^p\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right ) \,d x \]

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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