\(\int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx\) [3153]

Optimal result

Integrand size = 77, antiderivative size = 101 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {b (a+b x)^{1+m} (c+d x)^{-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac {(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)} \]

[Out]

b*(b*x+a)^(1+m)*(f*x+e)^((-a*d+b*c)*f*(1+m)/b/(-c*f+d*e))/(-a*d+b*c)/(-a*f+b*e)/(1+m)/((d*x+c)^(d*(-a*f+b*e)*(
1+m)/b/(-c*f+d*e)))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {97} \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {b (a+b x)^{m+1} (c+d x)^{-\frac {d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac {f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]

[In]

Int[(a + b*x)^m*(c + d*x)^(-1 - (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)))*(e + f*x)^(-1 + ((b*c - a*d)*f*(1 + m
))/(b*(d*e - c*f))),x]

[Out]

(b*(a + b*x)^(1 + m)*(e + f*x)^(((b*c - a*d)*f*(1 + m))/(b*(d*e - c*f))))/((b*c - a*d)*(b*e - a*f)*(1 + m)*(c
+ d*x)^((d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f))))

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {b (a+b x)^{1+m} (c+d x)^{-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac {(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {b (a+b x)^{1+m} (c+d x)^{-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{\frac {(b c-a d) f (1+m)}{b (d e-c f)}}}{(b c-a d) (b e-a f) (1+m)} \]

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-1 - (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)))*(e + f*x)^(-1 + ((b*c - a*d)*f*
(1 + m))/(b*(d*e - c*f))),x]

[Out]

(b*(a + b*x)^(1 + m)*(e + f*x)^(((b*c - a*d)*f*(1 + m))/(b*(d*e - c*f))))/((b*c - a*d)*(b*e - a*f)*(1 + m)*(c
+ d*x)^((d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f))))

Maple [A] (verified)

Time = 39.86 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.60

method	result	size
gosper	\(\frac {b \left (b x +a \right )^{1+m} \left (d x +c \right )^{1-\frac {a d f m -b d e m +a d f +b c f -2 b d e}{b \left (c f -d e \right )}} \left (f x +e \right )^{1+\frac {a d f m -b c f m +a d f -2 b c f +b d e}{b \left (c f -d e \right )}}}{a^{2} d f m -a b c f m -a b d e m +b^{2} c e m +a^{2} d f -a c f b -a b d e +b^{2} c e}\)	\(162\)
parallelrisch	\(\frac {x^{3} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} d^{2} f^{2}+x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} d^{2} f^{2}+x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} c d \,f^{2}+x^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} d^{2} e f +x \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} c d \,f^{2}+x \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} d^{2} e f +x \left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} b^{3} c d e f +\left (b x +a \right )^{m} \left (d x +c \right )^{\frac {-b \left (c f -d e \right )+d \left (-a f +b e \right ) \left (1+m \right )}{b \left (c f -d e \right )}} \left (f x +e \right )^{\frac {-b \left (c f -d e \right )+\left (a d -b c \right ) f \left (1+m \right )}{b \left (c f -d e \right )}} a \,b^{2} c d e f}{b d f \left (a^{2} d f m -a b c f m -a b d e m +b^{2} c e m +a^{2} d f -a c f b -a b d e +b^{2} c e \right )}\)	\(934\)

[In]

int((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),x,meth
od=_RETURNVERBOSE)

[Out]

b*(b*x+a)^(1+m)*(d*x+c)^(1-(a*d*f*m-b*d*e*m+a*d*f+b*c*f-2*b*d*e)/b/(c*f-d*e))*(f*x+e)^(1+(a*d*f*m-b*c*f*m+a*d*
f-2*b*c*f+b*d*e)/b/(c*f-d*e))/(a^2*d*f*m-a*b*c*f*m-a*b*d*e*m+b^2*c*e*m+a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (102) = 204\).

Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.23 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {{\left (b^{2} d f x^{3} + a b c e + {\left (b^{2} d e + {\left (b^{2} c + a b d\right )} f\right )} x^{2} + {\left (a b c f + {\left (b^{2} c + a b d\right )} e\right )} x\right )} {\left (b x + a\right )}^{m}}{{\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f + {\left ({\left (b^{2} c - a b d\right )} e - {\left (a b c - a^{2} d\right )} f\right )} m\right )} {\left (d x + c\right )}^{\frac {2 \, b d e - {\left (b c + a d\right )} f + {\left (b d e - a d f\right )} m}{b d e - b c f}} {\left (f x + e\right )}^{\frac {b d e - {\left (b c - a d\right )} f m - {\left (2 \, b c - a d\right )} f}{b d e - b c f}}} \]

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),
x, algorithm="fricas")

[Out]

(b^2*d*f*x^3 + a*b*c*e + (b^2*d*e + (b^2*c + a*b*d)*f)*x^2 + (a*b*c*f + (b^2*c + a*b*d)*e)*x)*(b*x + a)^m/(((b
^2*c - a*b*d)*e - (a*b*c - a^2*d)*f + ((b^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*m)*(d*x + c)^((2*b*d*e - (b*c +
a*d)*f + (b*d*e - a*d*f)*m)/(b*d*e - b*c*f))*(f*x + e)^((b*d*e - (b*c - a*d)*f*m - (2*b*c - a*d)*f)/(b*d*e - b
*c*f)))

Sympy [F]

\[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\int \left (a + b x\right )^{m} \left (c + d x\right )^{-1 - \frac {d \left (m + 1\right ) \left (- a f + b e\right )}{b \left (- c f + d e\right )}} \left (e + f x\right )^{-1 + \frac {f \left (m + 1\right ) \left (- a d + b c\right )}{b \left (- c f + d e\right )}}\, dx \]

[In]

integrate((b*x+a)**m*(d*x+c)**(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)**(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e
)),x)

[Out]

Integral((a + b*x)**m*(c + d*x)**(-1 - d*(m + 1)*(-a*f + b*e)/(b*(-c*f + d*e)))*(e + f*x)**(-1 + f*(m + 1)*(-a
*d + b*c)/(b*(-c*f + d*e))), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (102) = 204\).

Time = 0.34 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.28 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {{\left (b^{2} x + a b\right )} e^{\left (\frac {a d f m \log \left (d x + c\right )}{b d e - b c f} - \frac {a d f m \log \left (f x + e\right )}{b d e - b c f} + \frac {a d f \log \left (d x + c\right )}{b d e - b c f} - \frac {d e m \log \left (d x + c\right )}{d e - c f} - \frac {a d f \log \left (f x + e\right )}{b d e - b c f} + \frac {c f m \log \left (f x + e\right )}{d e - c f} + m \log \left (b x + a\right ) - \frac {d e \log \left (d x + c\right )}{d e - c f} + \frac {c f \log \left (f x + e\right )}{d e - c f}\right )}}{b^{2} c e {\left (m + 1\right )} + a^{2} d f {\left (m + 1\right )} - {\left (d e {\left (m + 1\right )} + c f {\left (m + 1\right )}\right )} a b} \]

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),
x, algorithm="maxima")

[Out]

(b^2*x + a*b)*e^(a*d*f*m*log(d*x + c)/(b*d*e - b*c*f) - a*d*f*m*log(f*x + e)/(b*d*e - b*c*f) + a*d*f*log(d*x +
c)/(b*d*e - b*c*f) - d*e*m*log(d*x + c)/(d*e - c*f) - a*d*f*log(f*x + e)/(b*d*e - b*c*f) + c*f*m*log(f*x + e)
/(d*e - c*f) + m*log(b*x + a) - d*e*log(d*x + c)/(d*e - c*f) + c*f*log(f*x + e)/(d*e - c*f))/(b^2*c*e*(m + 1)
+ a^2*d*f*(m + 1) - (d*e*(m + 1) + c*f*(m + 1))*a*b)

Giac [F]

\[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-\frac {{\left (b e - a f\right )} d {\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1} {\left (f x + e\right )}^{\frac {{\left (b c - a d\right )} f {\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1} \,d x } \]

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),
x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-(b*e - a*f)*d*(m + 1)/((d*e - c*f)*b) - 1)*(f*x + e)^((b*c - a*d)*f*(m + 1)/
((d*e - c*f)*b) - 1), x)

Mupad [B] (verification not implemented)

Time = 4.57 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.50 \[ \int (a+b x)^m (c+d x)^{-1-\frac {d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac {(b c-a d) f (1+m)}{b (d e-c f)}} \, dx=\frac {\frac {x^2\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m\,\left (b^2\,c\,f+b^2\,d\,e+a\,b\,d\,f\right )}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}+\frac {b\,x\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}+\frac {b^2\,d\,f\,x^3\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}+\frac {a\,b\,c\,e\,{\left (e+f\,x\right )}^{\frac {f\,\left (a\,d-b\,c\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}-1}\,{\left (a+b\,x\right )}^m}{\left (a\,d-b\,c\right )\,\left (a\,f-b\,e\right )\,\left (m+1\right )}}{{\left (c+d\,x\right )}^{\frac {d\,\left (a\,f-b\,e\right )\,\left (m+1\right )}{b\,\left (c\,f-d\,e\right )}+1}} \]

[In]

int(((e + f*x)^((f*(a*d - b*c)*(m + 1))/(b*(c*f - d*e)) - 1)*(a + b*x)^m)/(c + d*x)^((d*(a*f - b*e)*(m + 1))/(
b*(c*f - d*e)) + 1),x)

[Out]

((x^2*(e + f*x)^((f*(a*d - b*c)*(m + 1))/(b*(c*f - d*e)) - 1)*(a + b*x)^m*(b^2*c*f + b^2*d*e + a*b*d*f))/((a*d
- b*c)*(a*f - b*e)*(m + 1)) + (b*x*(e + f*x)^((f*(a*d - b*c)*(m + 1))/(b*(c*f - d*e)) - 1)*(a + b*x)^m*(a*c*f
+ a*d*e + b*c*e))/((a*d - b*c)*(a*f - b*e)*(m + 1)) + (b^2*d*f*x^3*(e + f*x)^((f*(a*d - b*c)*(m + 1))/(b*(c*f
- d*e)) - 1)*(a + b*x)^m)/((a*d - b*c)*(a*f - b*e)*(m + 1)) + (a*b*c*e*(e + f*x)^((f*(a*d - b*c)*(m + 1))/(b*
(c*f - d*e)) - 1)*(a + b*x)^m)/((a*d - b*c)*(a*f - b*e)*(m + 1)))/(c + d*x)^((d*(a*f - b*e)*(m + 1))/(b*(c*f -
d*e)) + 1)