Integrand size = 18, antiderivative size = 63 \[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\frac {x^{1+m} \left (d x^2\right )^n (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1+m+2 n,-p,2+m+2 n,-\frac {b x}{a}\right )}{1+m+2 n} \] Output:
x^(1+m)*(d*x^2)^n*(b*x+a)^p*hypergeom([-p, 1+m+2*n],[2+m+2*n],-b*x/a)/(1+m +2*n)/((1+b*x/a)^p)
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\frac {x^{1+m} \left (d x^2\right )^n (a+b x)^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1+m+2 n,-p,2+m+2 n,-\frac {b x}{a}\right )}{1+m+2 n} \] Input:
Integrate[x^m*(d*x^2)^n*(a + b*x)^p,x]
Output:
(x^(1 + 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)
Time = 0.29 (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.
\(\displaystyle \int x^m \left (d x^2\right )^n (a+b x)^p \, dx\) |
\(\Big \downarrow \) 30 |
\(\displaystyle x^{-2 n} \left (d x^2\right )^n \int x^{m+2 n} (a+b x)^pdx\) |
\(\Big \downarrow \) 76 |
\(\displaystyle x^{-2 n} \left (d x^2\right )^n (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \int x^{m+2 n} \left (\frac {b x}{a}+1\right )^pdx\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {x^{m+1} \left (d x^2\right )^n (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (m+2 n+1,-p,m+2 n+2,-\frac {b x}{a}\right )}{m+2 n+1}\) |
Input:
Int[x^m*(d*x^2)^n*(a + b*x)^p,x]
Output:
(x^(1 + 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)
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])
\[\int x^{m} \left (d \,x^{2}\right )^{n} \left (b x +a \right )^{p}d x\]
Input:
int(x^m*(d*x^2)^n*(b*x+a)^p,x)
Output:
int(x^m*(d*x^2)^n*(b*x+a)^p,x)
\[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\int { \left (d x^{2}\right )^{n} {\left (b x + a\right )}^{p} x^{m} \,d x } \] Input:
integrate(x^m*(d*x^2)^n*(b*x+a)^p,x, algorithm="fricas")
Output:
integral((d*x^2)^n*(b*x + a)^p*x^m, x)
\[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\int x^{m} \left (d x^{2}\right )^{n} \left (a + b x\right )^{p}\, dx \] Input:
integrate(x**m*(d*x**2)**n*(b*x+a)**p,x)
Output:
Integral(x**m*(d*x**2)**n*(a + b*x)**p, x)
\[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\int { \left (d x^{2}\right )^{n} {\left (b x + a\right )}^{p} x^{m} \,d x } \] Input:
integrate(x^m*(d*x^2)^n*(b*x+a)^p,x, algorithm="maxima")
Output:
integrate((d*x^2)^n*(b*x + a)^p*x^m, x)
\[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\int { \left (d x^{2}\right )^{n} {\left (b x + a\right )}^{p} x^{m} \,d x } \] Input:
integrate(x^m*(d*x^2)^n*(b*x+a)^p,x, algorithm="giac")
Output:
integrate((d*x^2)^n*(b*x + a)^p*x^m, x)
Timed out. \[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\int x^m\,{\left (d\,x^2\right )}^n\,{\left (a+b\,x\right )}^p \,d x \] Input:
int(x^m*(d*x^2)^n*(a + b*x)^p,x)
Output:
int(x^m*(d*x^2)^n*(a + b*x)^p, x)
\[ \int x^m \left (d x^2\right )^n (a+b x)^p \, dx=\text {too large to display} \] Input:
int(x^m*(d*x^2)^n*(b*x+a)^p,x)
Output:
(d**n*(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**2 +...