\(\int (c x)^m (d x^2)^n (a+b x)^p \, dx\) [481]

Optimal result

Integrand size = 20, antiderivative size = 68 \[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\frac {(c x)^{1+m} \left (d x^2\right )^n (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1+m+2 n,-p,2+m+2 n,-\frac {b x}{a}\right )}{c (1+m+2 n)} \] Output:

(c*x)^(1+m)*(d*x^2)^n*(b*x+a)^p*hypergeom([-p, 1+m+2*n],[2+m+2*n],-b*x/a)/ 
c/(1+m+2*n)/((1+b*x/a)^p)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.94 \[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\frac {x (c x)^m \left (d x^2\right )^n (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1+m+2 n,-p,2+m+2 n,-\frac {b x}{a}\right )}{1+m+2 n} \] Input:

Integrate[(c*x)^m*(d*x^2)^n*(a + b*x)^p,x]
 

Output:

(x*(c*x)^m*(d*x^2)^n*(a + b*x)^p*Hypergeometric2F1[1 + m + 2*n, -p, 2 + m 
+ 2*n, -((b*x)/a)])/((1 + m + 2*n)*(1 + (b*x)/a)^p)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {30, 76, 74}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx\)

\(\Big \downarrow \) 30

\(\displaystyle (c x)^{-2 n} \left (d x^2\right )^n \int (c x)^{m+2 n} (a+b x)^pdx\)

\(\Big \downarrow \) 76

\(\displaystyle (c x)^{-2 n} \left (d x^2\right )^n (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \int (c x)^{m+2 n} \left (\frac {b x}{a}+1\right )^pdx\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {(c x)^{m+1} \left (d x^2\right )^n (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (m+2 n+1,-p,m+2 n+2,-\frac {b x}{a}\right )}{c (m+2 n+1)}\)

Input:

Int[(c*x)^m*(d*x^2)^n*(a + b*x)^p,x]
 

Output:

((c*x)^(1 + m)*(d*x^2)^n*(a + b*x)^p*Hypergeometric2F1[1 + m + 2*n, -p, 2 
+ m + 2*n, -((b*x)/a)])/(c*(1 + m + 2*n)*(1 + (b*x)/a)^p)
 

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

Maple [F]

\[\int \left (c x \right )^{m} \left (d \,x^{2}\right )^{n} \left (b x +a \right )^{p}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\int { \left (d x^{2}\right )^{n} {\left (b x + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(d*x^2)^n*(b*x+a)^p,x, algorithm="fricas")
 

Output:

integral((d*x^2)^n*(b*x + a)^p*(c*x)^m, x)
 

Sympy [F]

\[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\int \left (c x\right )^{m} \left (d x^{2}\right )^{n} \left (a + b x\right )^{p}\, dx \] Input:

integrate((c*x)**m*(d*x**2)**n*(b*x+a)**p,x)
 

Output:

Integral((c*x)**m*(d*x**2)**n*(a + b*x)**p, x)
 

Maxima [F]

\[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\int { \left (d x^{2}\right )^{n} {\left (b x + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(d*x^2)^n*(b*x+a)^p,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\int { \left (d x^{2}\right )^{n} {\left (b x + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(d*x^2)^n*(b*x+a)^p,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\int {\left (c\,x\right )}^m\,{\left (d\,x^2\right )}^n\,{\left (a+b\,x\right )}^p \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (c x)^m \left (d x^2\right )^n (a+b x)^p \, dx=\text {too large to display} \] Input:

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

Output:

(d**n*c**m*(x**(m + 2*n)*(a + b*x)**p*a*p + x**(m + 2*n)*(a + b*x)**p*b*m* 
x + 2*x**(m + 2*n)*(a + b*x)**p*b*n*x + x**(m + 2*n)*(a + b*x)**p*b*p*x - 
int((x**(m + 2*n)*(a + b*x)**p)/(a*m**2*x + 4*a*m*n*x + 2*a*m*p*x + a*m*x 
+ 4*a*n**2*x + 4*a*n*p*x + 2*a*n*x + a*p**2*x + a*p*x + b*m**2*x**2 + 4*b* 
m*n*x**2 + 2*b*m*p*x**2 + b*m*x**2 + 4*b*n**2*x**2 + 4*b*n*p*x**2 + 2*b*n* 
x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**3*p - 6*int((x**(m + 2*n)*(a + b 
*x)**p)/(a*m**2*x + 4*a*m*n*x + 2*a*m*p*x + a*m*x + 4*a*n**2*x + 4*a*n*p*x 
 + 2*a*n*x + a*p**2*x + a*p*x + b*m**2*x**2 + 4*b*m*n*x**2 + 2*b*m*p*x**2 
+ b*m*x**2 + 4*b*n**2*x**2 + 4*b*n*p*x**2 + 2*b*n*x**2 + b*p**2*x**2 + b*p 
*x**2),x)*a**2*m**2*n*p - 2*int((x**(m + 2*n)*(a + b*x)**p)/(a*m**2*x + 4* 
a*m*n*x + 2*a*m*p*x + a*m*x + 4*a*n**2*x + 4*a*n*p*x + 2*a*n*x + a*p**2*x 
+ a*p*x + b*m**2*x**2 + 4*b*m*n*x**2 + 2*b*m*p*x**2 + b*m*x**2 + 4*b*n**2* 
x**2 + 4*b*n*p*x**2 + 2*b*n*x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**2*p* 
*2 - int((x**(m + 2*n)*(a + b*x)**p)/(a*m**2*x + 4*a*m*n*x + 2*a*m*p*x + a 
*m*x + 4*a*n**2*x + 4*a*n*p*x + 2*a*n*x + a*p**2*x + a*p*x + b*m**2*x**2 + 
 4*b*m*n*x**2 + 2*b*m*p*x**2 + b*m*x**2 + 4*b*n**2*x**2 + 4*b*n*p*x**2 + 2 
*b*n*x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**2*p - 12*int((x**(m + 2*n)* 
(a + b*x)**p)/(a*m**2*x + 4*a*m*n*x + 2*a*m*p*x + a*m*x + 4*a*n**2*x + 4*a 
*n*p*x + 2*a*n*x + a*p**2*x + a*p*x + b*m**2*x**2 + 4*b*m*n*x**2 + 2*b*m*p 
*x**2 + b*m*x**2 + 4*b*n**2*x**2 + 4*b*n*p*x**2 + 2*b*n*x**2 + b*p**2*x...