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

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)

_______________________________________________________________________________________

Rubi [A]  time = 0.413251, 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 \[ \frac{d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)}+\frac{e x^{n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+\frac{f x^{2 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}+\frac{g x^{3 n+1} (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+3 n+1}{n},-p;\frac{m+4 n+1}{n};-\frac{b x^n}{a}\right )}{m+3 n+1} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.4421, size = 272, normalized size = 0.92 \[ \frac{d \left (c x\right )^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (m + 1\right )} + \frac{e x^{n} \left (c x\right )^{- n} \left (c x\right )^{m + n + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (m + n + 1\right )} + \frac{f x^{2 n} \left (c x\right )^{- 2 n} \left (c x\right )^{m + 2 n + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 2 n + 1}{n} \\ \frac{m + 3 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (m + 2 n + 1\right )} + \frac{g x^{3 n} \left (c x\right )^{- 3 n} \left (c x\right )^{m + 3 n + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 3 n + 1}{n} \\ \frac{m + 4 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{c \left (m + 3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Mathematica [A]  time = 1.55533, 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

_______________________________________________________________________________________

Maple [F]  time = 0.114, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( d+e{x}^{n}+f{x}^{2\,n}+g{x}^{3\,n} \right ) \, dx \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError