Integrand size = 51, antiderivative size = 97 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}+\frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c} \] Output:
-(d*x+c)^(a*d/(-a*d+b*c))/b/c/((b*x+a)^(b*c/(-a*d+b*c)))+(d*x+c)^(a*d/(-a* d+b*c))/a/b/c/((b*x+a)^(a*d/(-a*d+b*c)))
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.47 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {x (a+b x)^{\frac {b c}{-b c+a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a c} \] Input:
Integrate[(a + b*x)^((-2*b*c + a*d)/(b*c - a*d))*(c + d*x)^((b*c - 2*a*d)/ (-(b*c) + a*d)),x]
Output:
(x*(a + b*x)^((b*c)/(-(b*c) + a*d))*(c + d*x)^((a*d)/(b*c - a*d)))/(a*c)
Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {55, 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.
\(\displaystyle \int (a+b x)^{\frac {a d-2 b c}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{a d-b c}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {d \int (a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{-\frac {b c-2 a d}{b c-a d}}dx}{b c}-\frac {(c+d x)^{\frac {a d}{b c-a d}} (a+b x)^{-\frac {b c}{b c-a d}}}{b c}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c}-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}\) |
Input:
Int[(a + b*x)^((-2*b*c + a*d)/(b*c - a*d))*(c + d*x)^((b*c - 2*a*d)/(-(b*c ) + a*d)),x]
Output:
-((c + d*x)^((a*d)/(b*c - a*d))/(b*c*(a + b*x)^((b*c)/(b*c - a*d)))) + (c + d*x)^((a*d)/(b*c - a*d))/(a*b*c*(a + b*x)^((a*d)/(b*c - a*d)))
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_))^(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] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Time = 0.45 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(\frac {x \left (b x +a \right )^{1-\frac {a d -2 b c}{a d -b c}} \left (x d +c \right )^{1-\frac {2 a d -b c}{a d -b c}}}{a c}\) | \(66\) |
orering | \(\frac {\left (b x +a \right ) \left (x d +c \right ) x \left (b x +a \right )^{\frac {a d -2 b c}{-a d +b c}} \left (x d +c \right )^{\frac {-2 a d +b c}{a d -b c}}}{a c}\) | \(69\) |
parallelrisch | \(\frac {x^{3} \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (x d +c \right )^{-\frac {2 a d -b c}{a d -b c}} b^{2} d^{2}+x^{2} \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (x d +c \right )^{-\frac {2 a d -b c}{a d -b c}} a b \,d^{2}+x^{2} \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (x d +c \right )^{-\frac {2 a d -b c}{a d -b c}} b^{2} c d +x \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (x d +c \right )^{-\frac {2 a d -b c}{a d -b c}} a b c d}{a b c d}\) | \(261\) |
Input:
int((b*x+a)^((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)^((-2*a*d+b*c)/(a*d-b*c)),x,me thod=_RETURNVERBOSE)
Output:
x/a/c*(b*x+a)^(1-(a*d-2*b*c)/(a*d-b*c))*(d*x+c)^(1-(2*a*d-b*c)/(a*d-b*c))
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {b d x^{3} + a c x + {\left (b c + a d\right )} x^{2}}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}} a c} \] Input:
integrate((b*x+a)^((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)^((-2*a*d+b*c)/(a*d-b*c) ),x, algorithm="fricas")
Output:
(b*d*x^3 + a*c*x + (b*c + a*d)*x^2)/((b*x + a)^((2*b*c - a*d)/(b*c - a*d)) *(d*x + c)^((b*c - 2*a*d)/(b*c - a*d))*a*c)
Timed out. \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)**((-2*a*d+b*c)/(a*d-b* c)),x)
Output:
Timed out
\[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}}} \,d x } \] Input:
integrate((b*x+a)^((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)^((-2*a*d+b*c)/(a*d-b*c) ),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^((2*b*c - a*d)/(b*c - a*d))*(d*x + c)^((b*c - 2*a*d )/(b*c - a*d))), x)
\[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}}} \,d x } \] Input:
integrate((b*x+a)^((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)^((-2*a*d+b*c)/(a*d-b*c) ),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^((2*b*c - a*d)/(b*c - a*d))*(d*x + c)^((b*c - 2*a*d )/(b*c - a*d))), x)
Time = 0.48 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {\frac {x}{{\left (a+b\,x\right )}^{\frac {a\,d-2\,b\,c}{a\,d-b\,c}}}+\frac {x^2\,\left (a\,d+b\,c\right )}{a\,c\,{\left (a+b\,x\right )}^{\frac {a\,d-2\,b\,c}{a\,d-b\,c}}}+\frac {b\,d\,x^3}{a\,c\,{\left (a+b\,x\right )}^{\frac {a\,d-2\,b\,c}{a\,d-b\,c}}}}{{\left (c+d\,x\right )}^{\frac {2\,a\,d-b\,c}{a\,d-b\,c}}} \] Input:
int(1/((a + b*x)^((a*d - 2*b*c)/(a*d - b*c))*(c + d*x)^((2*a*d - b*c)/(a*d - b*c))),x)
Output:
(x/(a + b*x)^((a*d - 2*b*c)/(a*d - b*c)) + (x^2*(a*d + b*c))/(a*c*(a + b*x )^((a*d - 2*b*c)/(a*d - b*c))) + (b*d*x^3)/(a*c*(a + b*x)^((a*d - 2*b*c)/( a*d - b*c))))/(c + d*x)^((2*a*d - b*c)/(a*d - b*c))
\[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\int \frac {\left (b x +a \right )^{\frac {b c}{a d -b c}}}{\left (d x +c \right )^{\frac {b c}{a d -b c}} a \,c^{2}+2 \left (d x +c \right )^{\frac {b c}{a d -b c}} a c d x +\left (d x +c \right )^{\frac {b c}{a d -b c}} a \,d^{2} x^{2}+\left (d x +c \right )^{\frac {b c}{a d -b c}} b \,c^{2} x +2 \left (d x +c \right )^{\frac {b c}{a d -b c}} b c d \,x^{2}+\left (d x +c \right )^{\frac {b c}{a d -b c}} b \,d^{2} x^{3}}d x \] Input:
int((b*x+a)^((a*d-2*b*c)/(-a*d+b*c))*(d*x+c)^((-2*a*d+b*c)/(a*d-b*c)),x)
Output:
int((a + b*x)**((b*c)/(a*d - b*c))/((c + d*x)**((b*c)/(a*d - b*c))*a*c**2 + 2*(c + d*x)**((b*c)/(a*d - b*c))*a*c*d*x + (c + d*x)**((b*c)/(a*d - b*c) )*a*d**2*x**2 + (c + d*x)**((b*c)/(a*d - b*c))*b*c**2*x + 2*(c + d*x)**((b *c)/(a*d - b*c))*b*c*d*x**2 + (c + d*x)**((b*c)/(a*d - b*c))*b*d**2*x**3), x)