Integrand size = 24, antiderivative size = 112 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {(d e-c f) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d) (2+m)}+\frac {(b d e+b c f (1+m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^2 (1+m) (2+m)} \] Output:
(-c*f+d*e)*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/d/(-a*d+b*c)/(2+m)+(b*d*e+b*c*f*(1 +m)-a*d*f*(2+m))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/d/(-a*d+b*c)^2/(1+m)/(2+m)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.73 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-2-m} (b (c e (2+m)+d e x+c f (1+m) x)-a (c f+d e (1+m)+d f (2+m) x))}{(b c-a d)^2 (1+m) (2+m)} \] Input:
Integrate[(a + b*x)^m*(c + d*x)^(-3 - m)*(e + f*x),x]
Output:
((a + b*x)^(1 + m)*(c + d*x)^(-2 - m)*(b*(c*e*(2 + m) + d*e*x + c*f*(1 + m )*x) - a*(c*f + d*e*(1 + m) + d*f*(2 + m)*x)))/((b*c - a*d)^2*(1 + m)*(2 + m))
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {88, 48}
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.
| \(\displaystyle \int (e+f x) (a+b x)^m (c+d x)^{-m-3} \, dx\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {(-a d f (m+2)+b c f (m+1)+b d e) \int (a+b x)^m (c+d x)^{-m-2}dx}{d (m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+2)+b c f (m+1)+b d e)}{d (m+1) (m+2) (b c-a d)^2}\) |
Input:
Int[(a + b*x)^m*(c + d*x)^(-3 - m)*(e + f*x),x]
Output:
((d*e - c*f)*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d*(b*c - a*d)*(2 + m)) + ((b*d*e + b*c*f*(1 + m) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(- 1 - m))/(d*(b*c - a*d)^2*(1 + m)*(2 + m))
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Time = 0.76 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.41
| method | result | size |
| gosper | \(-\frac {\left (b x +a \right )^{1+m} \left (x d +c \right )^{-2-m} \left (a d f m x -b c f m x +a d e m +2 a d f x -b c e m -b c f x -b d e x +a c f +a d e -2 b c e \right )}{a^{2} d^{2} m^{2}-2 a b c d \,m^{2}+b^{2} c^{2} m^{2}+3 a^{2} d^{2} m -6 a b c d m +3 b^{2} c^{2} m +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(158\) |
| orering | \(-\frac {\left (b x +a \right ) \left (x d +c \right ) \left (a d f m x -b c f m x +a d e m +2 a d f x -b c e m -b c f x -b d e x +a c f +a d e -2 b c e \right ) \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m}}{a^{2} d^{2} m^{2}-2 a b c d \,m^{2}+b^{2} c^{2} m^{2}+3 a^{2} d^{2} m -6 a b c d m +3 b^{2} c^{2} m +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(166\) |
| parallelrisch | \(-\frac {-x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c^{2} d f m -x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c \,d^{2} e m +2 x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} c \,d^{2} f +x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b \,d^{3} e m +x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b c \,d^{2} f m -x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c^{2} d e m +3 x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b c \,d^{2} f -2 x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} c \,d^{2} e +\left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b c \,d^{2} e m -\left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} c^{2} d e m +x^{3} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} d^{3} f m -x^{3} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c \,d^{2} f m +x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b \,d^{3} f m +x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} d^{3} e m -x^{3} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} d^{3} e -x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} c^{2} d f m +2 x^{3} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} d^{3} f -x^{3} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c \,d^{2} f +2 x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b \,d^{3} f -x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c^{2} d f -3 x^{2} \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c \,d^{2} e +x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b \,d^{3} e -2 x \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} b^{3} c^{2} d e +\left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b \,c^{2} d f +\left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a^{2} b c \,d^{2} e -2 \left (b x +a \right )^{m} \left (x d +c \right )^{-3-m} a \,b^{2} c^{2} d e}{\left (a^{2} d^{2} m^{2}-2 a b c d \,m^{2}+b^{2} c^{2} m^{2}+3 a^{2} d^{2} m -6 a b c d m +3 b^{2} c^{2} m +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) d b}\) | \(876\) |
Input:
int((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x,method=_RETURNVERBOSE)
Output:
-(b*x+a)^(1+m)*(d*x+c)^(-2-m)/(a^2*d^2*m^2-2*a*b*c*d*m^2+b^2*c^2*m^2+3*a^2 *d^2*m-6*a*b*c*d*m+3*b^2*c^2*m+2*a^2*d^2-4*a*b*c*d+2*b^2*c^2)*(a*d*f*m*x-b *c*f*m*x+a*d*e*m+2*a*d*f*x-b*c*e*m-b*c*f*x-b*d*e*x+a*c*f+a*d*e-2*b*c*e)
Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (114) = 228\).
Time = 0.09 (sec) , antiderivative size = 336, normalized size of antiderivative = 3.00 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=-\frac {{\left (a^{2} c^{2} f - {\left (b^{2} d^{2} e + {\left (b^{2} c d - a b d^{2}\right )} f m + {\left (b^{2} c d - 2 \, a b d^{2}\right )} f\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} e m - {\left (3 \, b^{2} c d e + {\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2}\right )} f + {\left ({\left (b^{2} c d - a b d^{2}\right )} e + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f\right )} m\right )} x^{2} - {\left (2 \, a b c^{2} - a^{2} c d\right )} e + {\left (3 \, a^{2} c d f - {\left (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2}\right )} e - {\left ({\left (b^{2} c^{2} - a^{2} d^{2}\right )} e + {\left (a b c^{2} - a^{2} c d\right )} f\right )} m\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m^{2} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m} \] Input:
integrate((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x, algorithm="fricas")
Output:
-(a^2*c^2*f - (b^2*d^2*e + (b^2*c*d - a*b*d^2)*f*m + (b^2*c*d - 2*a*b*d^2) *f)*x^3 - (a*b*c^2 - a^2*c*d)*e*m - (3*b^2*c*d*e + (b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2)*f + ((b^2*c*d - a*b*d^2)*e + (b^2*c^2 - a^2*d^2)*f)*m)*x^2 - (2 *a*b*c^2 - a^2*c*d)*e + (3*a^2*c*d*f - (2*b^2*c^2 + 2*a*b*c*d - a^2*d^2)*e - ((b^2*c^2 - a^2*d^2)*e + (a*b*c^2 - a^2*c*d)*f)*m)*x)*(b*x + a)^m*(d*x + c)^(-m - 3)/(2*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*m^2 + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*m)
Exception generated. \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)**m*(d*x+c)**(-3-m)*(f*x+e),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\int { {\left (f x + e\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x, algorithm="maxima")
Output:
integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (114) = 228\).
Time = 0.16 (sec) , antiderivative size = 1023, normalized size of antiderivative = 9.13 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x, algorithm="giac")
Output:
((b*x + a)^m*b^2*c*d*f*m*x^3*e^(-m*log(d*x + c) - 3*log(d*x + c)) - (b*x + a)^m*a*b*d^2*f*m*x^3*e^(-m*log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^m*b ^2*c*d*e*m*x^2*e^(-m*log(d*x + c) - 3*log(d*x + c)) - (b*x + a)^m*a*b*d^2* e*m*x^2*e^(-m*log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^m*b^2*c^2*f*m*x^2 *e^(-m*log(d*x + c) - 3*log(d*x + c)) - (b*x + a)^m*a^2*d^2*f*m*x^2*e^(-m* log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^m*b^2*d^2*e*x^3*e^(-m*log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^m*b^2*c*d*f*x^3*e^(-m*log(d*x + c) - 3*l og(d*x + c)) - 2*(b*x + a)^m*a*b*d^2*f*x^3*e^(-m*log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^m*b^2*c^2*e*m*x*e^(-m*log(d*x + c) - 3*log(d*x + c)) - ( b*x + a)^m*a^2*d^2*e*m*x*e^(-m*log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^ m*a*b*c^2*f*m*x*e^(-m*log(d*x + c) - 3*log(d*x + c)) - (b*x + a)^m*a^2*c*d *f*m*x*e^(-m*log(d*x + c) - 3*log(d*x + c)) + 3*(b*x + a)^m*b^2*c*d*e*x^2* e^(-m*log(d*x + c) - 3*log(d*x + c)) + (b*x + a)^m*b^2*c^2*f*x^2*e^(-m*log (d*x + c) - 3*log(d*x + c)) - 2*(b*x + a)^m*a*b*c*d*f*x^2*e^(-m*log(d*x + c) - 3*log(d*x + c)) - 2*(b*x + a)^m*a^2*d^2*f*x^2*e^(-m*log(d*x + c) - 3* log(d*x + c)) + (b*x + a)^m*a*b*c^2*e*m*e^(-m*log(d*x + c) - 3*log(d*x + c )) - (b*x + a)^m*a^2*c*d*e*m*e^(-m*log(d*x + c) - 3*log(d*x + c)) + 2*(b*x + a)^m*b^2*c^2*e*x*e^(-m*log(d*x + c) - 3*log(d*x + c)) + 2*(b*x + a)^m*a *b*c*d*e*x*e^(-m*log(d*x + c) - 3*log(d*x + c)) - (b*x + a)^m*a^2*d^2*e*x* e^(-m*log(d*x + c) - 3*log(d*x + c)) - 3*(b*x + a)^m*a^2*c*d*f*x*e^(-m*...
Time = 2.12 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.21 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\frac {b\,d\,x^3\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,e-2\,b^2\,c^2\,e+3\,a^2\,c\,d\,f+a^2\,d^2\,e\,m-b^2\,c^2\,e\,m-2\,a\,b\,c\,d\,e-a\,b\,c^2\,f\,m+a^2\,c\,d\,f\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (a\,c\,f+a\,d\,e-2\,b\,c\,e+a\,d\,e\,m-b\,c\,e\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (2\,a^2\,d^2\,f-b^2\,c^2\,f-3\,b^2\,c\,d\,e+a^2\,d^2\,f\,m-b^2\,c^2\,f\,m+2\,a\,b\,c\,d\,f+a\,b\,d^2\,e\,m-b^2\,c\,d\,e\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )} \] Input:
int(((e + f*x)*(a + b*x)^m)/(c + d*x)^(m + 3),x)
Output:
(b*d*x^3*(a + b*x)^m*(b*c*f - 2*a*d*f + b*d*e - a*d*f*m + b*c*f*m))/((a*d - b*c)^2*(c + d*x)^(m + 3)*(3*m + m^2 + 2)) - (x*(a + b*x)^m*(a^2*d^2*e - 2*b^2*c^2*e + 3*a^2*c*d*f + a^2*d^2*e*m - b^2*c^2*e*m - 2*a*b*c*d*e - a*b* c^2*f*m + a^2*c*d*f*m))/((a*d - b*c)^2*(c + d*x)^(m + 3)*(3*m + m^2 + 2)) - (a*c*(a + b*x)^m*(a*c*f + a*d*e - 2*b*c*e + a*d*e*m - b*c*e*m))/((a*d - b*c)^2*(c + d*x)^(m + 3)*(3*m + m^2 + 2)) - (x^2*(a + b*x)^m*(2*a^2*d^2*f - b^2*c^2*f - 3*b^2*c*d*e + a^2*d^2*f*m - b^2*c^2*f*m + 2*a*b*c*d*f + a*b* d^2*e*m - b^2*c*d*e*m))/((a*d - b*c)^2*(c + d*x)^(m + 3)*(3*m + m^2 + 2))
\[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx=\left (\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{m} c^{3}+3 \left (d x +c \right )^{m} c^{2} d x +3 \left (d x +c \right )^{m} c \,d^{2} x^{2}+\left (d x +c \right )^{m} d^{3} x^{3}}d x \right ) e +\left (\int \frac {\left (b x +a \right )^{m} x}{\left (d x +c \right )^{m} c^{3}+3 \left (d x +c \right )^{m} c^{2} d x +3 \left (d x +c \right )^{m} c \,d^{2} x^{2}+\left (d x +c \right )^{m} d^{3} x^{3}}d x \right ) f \] Input:
int((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x)
Output:
int((a + b*x)**m/((c + d*x)**m*c**3 + 3*(c + d*x)**m*c**2*d*x + 3*(c + d*x )**m*c*d**2*x**2 + (c + d*x)**m*d**3*x**3),x)*e + int(((a + b*x)**m*x)/((c + d*x)**m*c**3 + 3*(c + d*x)**m*c**2*d*x + 3*(c + d*x)**m*c*d**2*x**2 + ( c + d*x)**m*d**3*x**3),x)*f