Integrand size = 30, antiderivative size = 203 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {\left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) (c+d x)^{1+n}}{b^3 d^3 (1+n)}+\frac {(b C d-2 b c D-a d D) (c+d x)^{2+n}}{b^2 d^3 (2+n)}+\frac {D (c+d x)^{3+n}}{b d^3 (3+n)}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d) (1+n)} \]
(a^2*d^2*D-a*b*d*(C*d-D*c)-b^2*(-B*d^2+C*c*d-D*c^2))*(d*x+c)^(1+n)/b^3/d^3 /(1+n)+(C*b*d-D*a*d-2*D*b*c)*(d*x+c)^(2+n)/b^2/d^3/(2+n)+D*(d*x+c)^(3+n)/b /d^3/(3+n)-(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(d*x+c)^(1+n)*hypergeom([1, 1+n], [2+n],b*(d*x+c)/(-a*d+b*c))/b^3/(-a*d+b*c)/(1+n)
Time = 0.36 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.89 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\frac {(c+d x)^{1+n} \left (\frac {a^2 d^2 D+a b d (-C d+c D)+b^2 \left (-c C d+B d^2+c^2 D\right )}{d^3 (1+n)}+\frac {b (b C d-2 b c D-a d D) (c+d x)}{d^3 (2+n)}+\frac {b^2 D (c+d x)^2}{d^3 (3+n)}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) (1+n)}\right )}{b^3} \]
((c + d*x)^(1 + n)*((a^2*d^2*D + a*b*d*(-(C*d) + c*D) + b^2*(-(c*C*d) + B* d^2 + c^2*D))/(d^3*(1 + n)) + (b*(b*C*d - 2*b*c*D - a*d*D)*(c + d*x))/(d^3 *(2 + n)) + (b^2*D*(c + d*x)^2)/(d^3*(3 + n)) - ((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/ ((b*c - a*d)*(1 + n))))/b^3
Time = 0.42 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2123, 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.
\(\displaystyle \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \int \left (\frac {(c+d x)^n \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^3 (a+b x)}+\frac {(c+d x)^n \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^2}+\frac {(c+d x)^{n+1} (-a d D-2 b c D+b C d)}{b^2 d^2}+\frac {D (c+d x)^{n+2}}{b d^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right )}{b^3 (n+1) (b c-a d)}+\frac {(c+d x)^{n+1} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3 (n+1)}+\frac {(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac {D (c+d x)^{n+3}}{b d^3 (n+3)}\) |
((a^2*d^2*D - a*b*d*(C*d - c*D) - b^2*(c*C*d - B*d^2 - c^2*D))*(c + d*x)^( 1 + n))/(b^3*d^3*(1 + n)) + ((b*C*d - 2*b*c*D - a*d*D)*(c + d*x)^(2 + n))/ (b^2*d^3*(2 + n)) + (D*(c + d*x)^(3 + n))/(b*d^3*(3 + n)) - ((A*b^3 - a*(b ^2*B - a*b*C + a^2*D))*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n , (b*(c + d*x))/(b*c - a*d)])/(b^3*(b*c - a*d)*(1 + n))
3.1.29.3.1 Defintions of rubi rules used
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
\[\int \frac {\left (d x +c \right )^{n} \left (D x^{3}+C \,x^{2}+B x +A \right )}{b x +a}d x\]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int { \frac {{\left (\mathit {capitalD} x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a} \,d x } \]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int \frac {\left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )}{a + b x}\, dx \]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a} \,d x } \]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx=\int \frac {{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{a+b\,x} \,d x \]