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

[Out]

d*(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([-p, (1+m)/n],[(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*hypergeom([-p, (2+m)/n],[(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*
hypergeom([-p, (3+m)/n],[(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*hypergeom([-
p, (4+m)/n],[(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 = 0.19, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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 (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*}

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Mathematica [A]  time = 0.32, 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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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{3} + f x^{2} + e x + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m}, x\right ) \]

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

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)

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maple [F]  time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (g \,x^{3}+f \,x^{2}+e x +d \right ) \left (c x \right )^{m} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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