Integrand size = 30, antiderivative size = 220 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {(b C d-b c D-2 a d D) (c+d x)^{1+n}}{b^3 d^2 (1+n)}-\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) (c+d x)^{1+n}}{(b c-a d) (a+b x)}+\frac {D (c+d x)^{2+n}}{b^2 d^2 (2+n)}+\frac {\left (a^3 d D (3+n)-b^3 (B c+A d n)+a b^2 (2 c C+B d (1+n))-a^2 b (3 c D+C d (2+n))\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 (1+n)} \]
(C*b*d-2*D*a*d-D*b*c)*(d*x+c)^(1+n)/b^3/d^2/(1+n)-(A-a*(B*b^2-C*a*b+D*a^2) /b^3)*(d*x+c)^(1+n)/(-a*d+b*c)/(b*x+a)+D*(d*x+c)^(2+n)/b^2/d^2/(2+n)+(a^3* d*D*(3+n)-b^3*(A*d*n+B*c)+a*b^2*(2*C*c+B*d*(1+n))-a^2*b*(3*D*c+C*d*(2+n))) *(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)^2/(1+n)
Time = 0.69 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.82 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\frac {(c+d x)^{1+n} \left (\frac {b C d-b c D-2 a d D}{d^2 (1+n)}+\frac {b D (c+d x)}{d^2 (2+n)}-\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) (1+n)}+\frac {d \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^2 (1+n)}\right )}{b^3} \]
((c + d*x)^(1 + n)*((b*C*d - b*c*D - 2*a*d*D)/(d^2*(1 + n)) + (b*D*(c + d* x))/(d^2*(2 + n)) - ((b^2*B - 2*a*b*C + 3*a^2*D)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)*(1 + n)) + (d*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[2, 1 + n, 2 + n, (b*(c + d*x) )/(b*c - a*d)])/((b*c - a*d)^2*(1 + n))))/b^3
Time = 0.64 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2124, 1195, 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.
\(\displaystyle \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx\) |
\(\Big \downarrow \) 2124 |
\(\displaystyle -\frac {\int \frac {(c+d x)^n \left (-\left (\left (c-\frac {a d}{b}\right ) D x^2\right )-\frac {(b c-a d) (b C-a D) x}{b^2}+\frac {d D (n+1) a^3-b (c D+C d (n+1)) a^2+b^2 (c C+B d (n+1)) a-b^3 (B c+A d n)}{b^3}\right )}{a+b x}dx}{b c-a d}-\frac {(c+d x)^{n+1} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle -\frac {\int \left (-\frac {(b c-a d) (b C d-2 a D d-b c D) (c+d x)^n}{b^3 d}+\frac {\left (d D (n+3) a^3-b (3 c D+C d (n+2)) a^2+b^2 (2 c C+B d (n+1)) a-b^3 (B c+A d n)\right ) (c+d x)^n}{b^3 (a+b x)}-\frac {(b c-a d) D (c+d x)^{n+1}}{b^2 d}\right )dx}{b c-a d}-\frac {(c+d x)^{n+1} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(c+d x)^{n+1} \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}-\frac {-\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right ) \left (a^3 d D (n+3)-a^2 b (3 c D+C d (n+2))+a b^2 (B d (n+1)+2 c C)-b^3 (A d n+B c)\right )}{b^3 (n+1) (b c-a d)}-\frac {(b c-a d) (c+d x)^{n+1} (-2 a d D-b c D+b C d)}{b^3 d^2 (n+1)}-\frac {D (b c-a d) (c+d x)^{n+2}}{b^2 d^2 (n+2)}}{b c-a d}\) |
-(((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*(c + d*x)^(1 + n))/((b*c - a*d)*( a + b*x))) - (-(((b*c - a*d)*(b*C*d - b*c*D - 2*a*d*D)*(c + d*x)^(1 + n))/ (b^3*d^2*(1 + n))) - ((b*c - a*d)*D*(c + d*x)^(2 + n))/(b^2*d^2*(2 + n)) - ((a^3*d*D*(3 + n) - b^3*(B*c + A*d*n) + a*b^2*(2*c*C + B*d*(1 + n)) - a^2 *b*(3*c*D + C*d*(2 + n)))*(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)))/(b*c - a*d)
3.1.30.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : > With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px , a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - a*d))), x] + Simp[1/((m + 1)*(b*c - a*d)) Int[(a + b*x)^(m + 1)*(c + d*x )^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] || ! ILtQ[n, -1])
\[\int \frac {\left (d x +c \right )^{n} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{2}}d x\]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\int { \frac {{\left (\mathit {capitalD} x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}} \,d x } \]
\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^2} \, dx=\int \frac {{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^2} \,d x \]