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

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*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*Hyp
ergeometric2F1[(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 = 0.376937, 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 \[ \frac{g (c x)^{m+4} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+4}{n},-p;\frac{m+n+4}{n};-\frac{b x^n}{a}\right )}{c^4 (m+4)}+\frac{f (c x)^{m+3} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3}{n},-p;\frac{m+n+3}{n};-\frac{b x^n}{a}\right )}{c^3 (m+3)}+\frac{e (c x)^{m+2} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2}{n},-p;\frac{m+n+2}{n};-\frac{b x^n}{a}\right )}{c^2 (m+2)}+\frac{d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)} \]

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*Hyp
ergeometric2F1[(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 in Sympy [A]  time = 52.1569, size = 214, normalized size = 0.78 \[ \frac{d \left (c x\right )^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (m + 1\right )} + \frac{e \left (c x\right )^{m + 2} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 2}{n} \\ \frac{m + n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c^{2} \left (m + 2\right )} + \frac{f \left (c x\right )^{m + 3} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 3}{n} \\ \frac{m + n + 3}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c^{3} \left (m + 3\right )} + \frac{g \left (c x\right )^{m + 4} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 4}{n} \\ \frac{m + n + 4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c^{4} \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.337604, 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*Hypergeometric2F1[(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 = 0.108, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m} \left ( g{x}^{3}+f{x}^{2}+ex+d \right ) \left ( a+b{x}^{n} \right ) ^{p}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m,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. \[{\rm integral}\left ({\left (g x^{3} + f x^{2} + e x + d\right )}{\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m,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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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