3.6.86 \(\int (c x)^m (d+e x+f x^2+g x^3) (a+b x^n)^p \, dx\) [586]

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

[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 = 0.13, 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 = {1858, 372, 371} \begin {gather*} \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)} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 1858

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 = 0.24, size = 178, normalized size = 0.65 \begin {gather*} x (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (\frac {d \, _2F_1\left (\frac {1+m}{n},-p;\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{1+m}+x \left (\frac {e \, _2F_1\left (\frac {2+m}{n},-p;\frac {2+m+n}{n};-\frac {b x^n}{a}\right )}{2+m}+x \left (\frac {f \, _2F_1\left (\frac {3+m}{n},-p;\frac {3+m+n}{n};-\frac {b x^n}{a}\right )}{3+m}+\frac {g x \, _2F_1\left (\frac {4+m}{n},-p;\frac {4+m+n}{n};-\frac {b x^n}{a}\right )}{4+m}\right )\right )\right ) \end {gather*}

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 = 0.09, size = 0, normalized size = 0.00 \[\int \left (c x \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right ) \left (a +b \,x^{n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m*(g*x^3+f*x^2+e*x+d)*(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 + x*e + d)*(b*x^n + a)^p*(c*x)^m, x)

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Fricas [F]
time = 0.39, size = 32, normalized size = 0.12 \begin {gather*} {\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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 110.51, size = 248, normalized size = 0.91 \begin {gather*} \frac {a^{p} c^{m} d x x^{m} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {a^{p} c^{m} e x^{2} x^{m} \Gamma \left (\frac {m}{n} + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {2}{n} \\ \frac {m}{n} + 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {2}{n}\right )} + \frac {a^{p} c^{m} f x^{3} x^{m} \Gamma \left (\frac {m}{n} + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {3}{n} \\ \frac {m}{n} + 1 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {3}{n}\right )} + \frac {a^{p} c^{m} g x^{4} x^{m} \Gamma \left (\frac {m}{n} + \frac {4}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {4}{n} \\ \frac {m}{n} + 1 + \frac {4}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {4}{n}\right )} \end {gather*}

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]

a**p*c**m*d*x*x**m*gamma(m/n + 1/n)*hyper((-p, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamm
a(m/n + 1 + 1/n)) + a**p*c**m*e*x**2*x**m*gamma(m/n + 2/n)*hyper((-p, m/n + 2/n), (m/n + 1 + 2/n,), b*x**n*exp
_polar(I*pi)/a)/(n*gamma(m/n + 1 + 2/n)) + a**p*c**m*f*x**3*x**m*gamma(m/n + 3/n)*hyper((-p, m/n + 3/n), (m/n
+ 1 + 3/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 1 + 3/n)) + a**p*c**m*g*x**4*x**m*gamma(m/n + 4/n)*hyper
((-p, m/n + 4/n), (m/n + 1 + 4/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 1 + 4/n))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 + x*e + 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 \begin {gather*} \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 \end {gather*}

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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