Integrand size = 18, antiderivative size = 63 \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{1+m+2 p} \]
x^(1+m)*(c*x^2)^p*(b*x+a)^n*hypergeom([-n, 1+m+2*p],[2+m+2*p],-b*x/a)/(1+m +2*p)/((1+b*x/a)^n)
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\frac {x^{1+m} \left (c x^2\right )^p (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,1+m+2 p,2+m+2 p,-\frac {b x}{a}\right )}{1+m+2 p} \]
(x^(1 + m)*(c*x^2)^p*(a + b*x)^n*Hypergeometric2F1[-n, 1 + m + 2*p, 2 + m + 2*p, -((b*x)/a)])/((1 + m + 2*p)*(1 + (b*x)/a)^n)
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
\(\displaystyle \int x^m \left (c x^2\right )^p (a+b x)^n \, dx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle x^{-2 p} \left (c x^2\right )^p \int x^{m+2 p} (a+b x)^ndx\) |
\(\Big \downarrow \) 76 |
\(\displaystyle x^{-2 p} \left (c x^2\right )^p (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \int x^{m+2 p} \left (\frac {b x}{a}+1\right )^ndx\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {x^{m+1} \left (c x^2\right )^p (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,m+2 p+1,m+2 p+2,-\frac {b x}{a}\right )}{m+2 p+1}\) |
(x^(1 + m)*(c*x^2)^p*(a + b*x)^n*Hypergeometric2F1[-n, 1 + m + 2*p, 2 + m + 2*p, -((b*x)/a)])/((1 + m + 2*p)*(1 + (b*x)/a)^n)
3.10.97.3.1 Defintions of rubi rules used
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]
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])))
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])
\[\int x^{m} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{n}d x\]
\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} x^{m} \,d x } \]
\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int x^{m} \left (c x^{2}\right )^{p} \left (a + b x\right )^{n}\, dx \]
\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} x^{m} \,d x } \]
\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int { \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{n} x^{m} \,d x } \]
Timed out. \[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\int x^m\,{\left (c\,x^2\right )}^p\,{\left (a+b\,x\right )}^n \,d x \]
\[ \int x^m \left (c x^2\right )^p (a+b x)^n \, dx=\text {too large to display} \]
(c**p*(x**m*(a + b*x)**n*abs(x)**(2*p)*a*n + x**m*(a + b*x)**n*abs(x)**(2* p)*b*m*x + x**m*(a + b*x)**n*abs(x)**(2*p)*b*n*x + 2*x**m*(a + b*x)**n*abs (x)**(2*p)*b*p*x - int((x**m*(a + b*x)**n*abs(x)**(2*p))/(a*m**2*x + 2*a*m *n*x + 4*a*m*p*x + a*m*x + a*n**2*x + 4*a*n*p*x + a*n*x + 4*a*p**2*x + 2*a *p*x + b*m**2*x**2 + 2*b*m*n*x**2 + 4*b*m*p*x**2 + b*m*x**2 + b*n**2*x**2 + 4*b*n*p*x**2 + b*n*x**2 + 4*b*p**2*x**2 + 2*b*p*x**2),x)*a**2*m**3*n - 2 *int((x**m*(a + b*x)**n*abs(x)**(2*p))/(a*m**2*x + 2*a*m*n*x + 4*a*m*p*x + a*m*x + a*n**2*x + 4*a*n*p*x + a*n*x + 4*a*p**2*x + 2*a*p*x + b*m**2*x**2 + 2*b*m*n*x**2 + 4*b*m*p*x**2 + b*m*x**2 + b*n**2*x**2 + 4*b*n*p*x**2 + b *n*x**2 + 4*b*p**2*x**2 + 2*b*p*x**2),x)*a**2*m**2*n**2 - 6*int((x**m*(a + b*x)**n*abs(x)**(2*p))/(a*m**2*x + 2*a*m*n*x + 4*a*m*p*x + a*m*x + a*n**2 *x + 4*a*n*p*x + a*n*x + 4*a*p**2*x + 2*a*p*x + b*m**2*x**2 + 2*b*m*n*x**2 + 4*b*m*p*x**2 + b*m*x**2 + b*n**2*x**2 + 4*b*n*p*x**2 + b*n*x**2 + 4*b*p **2*x**2 + 2*b*p*x**2),x)*a**2*m**2*n*p - int((x**m*(a + b*x)**n*abs(x)**( 2*p))/(a*m**2*x + 2*a*m*n*x + 4*a*m*p*x + a*m*x + a*n**2*x + 4*a*n*p*x + a *n*x + 4*a*p**2*x + 2*a*p*x + b*m**2*x**2 + 2*b*m*n*x**2 + 4*b*m*p*x**2 + b*m*x**2 + b*n**2*x**2 + 4*b*n*p*x**2 + b*n*x**2 + 4*b*p**2*x**2 + 2*b*p*x **2),x)*a**2*m**2*n - int((x**m*(a + b*x)**n*abs(x)**(2*p))/(a*m**2*x + 2* a*m*n*x + 4*a*m*p*x + a*m*x + a*n**2*x + 4*a*n*p*x + a*n*x + 4*a*p**2*x + 2*a*p*x + b*m**2*x**2 + 2*b*m*n*x**2 + 4*b*m*p*x**2 + b*m*x**2 + b*n**2...