Integrand size = 24, antiderivative size = 188 \[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=-\frac {(a (6 c C-3 B d)+b C (2-p)) (c+d x)^2 \left (a+\frac {b}{c+d x}\right )^{1+p}}{6 a^2 d^3}+\frac {C (c+d x)^3 \left (a+\frac {b}{c+d x}\right )^{1+p}}{3 a d^3}-\frac {b \left (6 a^2 \left (c^2 C-B c d+A d^2\right )+b (a (6 c C-3 B d)+b C (2-p)) (1-p)\right ) \left (a+\frac {b}{c+d x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {b}{a (c+d x)}\right )}{6 a^4 d^3 (1+p)} \] Output:
-1/6*(a*(-3*B*d+6*C*c)+b*C*(2-p))*(d*x+c)^2*(a+b/(d*x+c))^(p+1)/a^2/d^3+1/ 3*C*(d*x+c)^3*(a+b/(d*x+c))^(p+1)/a/d^3-1/6*b*(6*a^2*(A*d^2-B*c*d+C*c^2)+b *(a*(-3*B*d+6*C*c)+b*C*(2-p))*(1-p))*(a+b/(d*x+c))^(p+1)*hypergeom([2, p+1 ],[2+p],1+b/a/(d*x+c))/a^4/d^3/(p+1)
\[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx \] Input:
Integrate[(A + B*x + C*x^2)*(a + b/(c + d*x))^p,x]
Output:
Integrate[(A + B*x + C*x^2)*(a + b/(c + d*x))^p, x]
Time = 1.33 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.65, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7293, 2009}
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.
\(\displaystyle \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (A \left (a+\frac {b}{c+d x}\right )^p+B x \left (a+\frac {b}{c+d x}\right )^p+C x^2 \left (a+\frac {b}{c+d x}\right )^p\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b B (2 a c+b (-p)+b) \left (a+\frac {b}{c+d x}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {b}{a (c+d x)}+1\right )}{2 a^3 d^2 (p+1)}-\frac {A b \left (a+\frac {b}{c+d x}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {b}{a (c+d x)}+1\right )}{a^2 d (p+1)}-\frac {C (c+d x)^2 (6 a c+b (2-p)) \left (a+\frac {b}{c+d x}\right )^{p+1}}{6 a^2 d^3}-\frac {b C \left (6 a^2 c^2+b (1-p) (6 a c+b (2-p))\right ) \left (a+\frac {b}{c+d x}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {b}{a (c+d x)}+1\right )}{6 a^4 d^3 (p+1)}+\frac {B (c+d x)^2 \left (a+\frac {b}{c+d x}\right )^{p+1}}{2 a d^2}+\frac {C (c+d x)^3 \left (a+\frac {b}{c+d x}\right )^{p+1}}{3 a d^3}\) |
Input:
Int[(A + B*x + C*x^2)*(a + b/(c + d*x))^p,x]
Output:
(B*(c + d*x)^2*(a + b/(c + d*x))^(1 + p))/(2*a*d^2) - (C*(6*a*c + b*(2 - p ))*(c + d*x)^2*(a + b/(c + d*x))^(1 + p))/(6*a^2*d^3) + (C*(c + d*x)^3*(a + b/(c + d*x))^(1 + p))/(3*a*d^3) - (A*b*(a + b/(c + d*x))^(1 + p)*Hyperge ometric2F1[2, 1 + p, 2 + p, 1 + b/(a*(c + d*x))])/(a^2*d*(1 + p)) - (b*C*( 6*a^2*c^2 + b*(6*a*c + b*(2 - p))*(1 - p))*(a + b/(c + d*x))^(1 + p)*Hyper geometric2F1[2, 1 + p, 2 + p, 1 + b/(a*(c + d*x))])/(6*a^4*d^3*(1 + p)) + (b*B*(b + 2*a*c - b*p)*(a + b/(c + d*x))^(1 + p)*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + b/(a*(c + d*x))])/(2*a^3*d^2*(1 + p))
\[\int \left (C \,x^{2}+B x +A \right ) \left (a +\frac {b}{d x +c}\right )^{p}d x\]
Input:
int((C*x^2+B*x+A)*(a+b/(d*x+c))^p,x)
Output:
int((C*x^2+B*x+A)*(a+b/(d*x+c))^p,x)
\[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (a + \frac {b}{d x + c}\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a+b/(d*x+c))^p,x, algorithm="fricas")
Output:
integral((C*x^2 + B*x + A)*((a*d*x + a*c + b)/(d*x + c))^p, x)
\[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\int \left (\frac {a c + a d x + b}{c + d x}\right )^{p} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((C*x**2+B*x+A)*(a+b/(d*x+c))**p,x)
Output:
Integral(((a*c + a*d*x + b)/(c + d*x))**p*(A + B*x + C*x**2), x)
\[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (a + \frac {b}{d x + c}\right )}^{p} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a+b/(d*x+c))^p,x, algorithm="maxima")
Output:
integrate((C*x^2 + B*x + A)*(a + b/(d*x + c))^p, x)
Exception generated. \[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((C*x^2+B*x+A)*(a+b/(d*x+c))^p,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{-1,[ 0,1,1,0]%%%} / %%%{1,[0,0,0,1]%%%} Error: Bad Argument Value
Timed out. \[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\int {\left (a+\frac {b}{c+d\,x}\right )}^p\,\left (C\,x^2+B\,x+A\right ) \,d x \] Input:
int((a + b/(c + d*x))^p*(A + B*x + C*x^2),x)
Output:
int((a + b/(c + d*x))^p*(A + B*x + C*x^2), x)
\[ \int \left (A+B x+C x^2\right ) \left (a+\frac {b}{c+d x}\right )^p \, dx=\left (\int \frac {\left (a d x +a c +b \right )^{p}}{\left (d x +c \right )^{p}}d x \right ) a +\left (\int \frac {\left (a d x +a c +b \right )^{p} x^{2}}{\left (d x +c \right )^{p}}d x \right ) c +\left (\int \frac {\left (a d x +a c +b \right )^{p} x}{\left (d x +c \right )^{p}}d x \right ) b \] Input:
int((C*x^2+B*x+A)*(a+b/(d*x+c))^p,x)
Output:
int((a*c + a*d*x + b)**p/(c + d*x)**p,x)*a + int(((a*c + a*d*x + b)**p*x** 2)/(c + d*x)**p,x)*c + int(((a*c + a*d*x + b)**p*x)/(c + d*x)**p,x)*b