Integrand size = 50, antiderivative size = 53 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x \left (a+b x^3\right )^{-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}}}{a c} \] Output:
x*(d*x^3+c)^(a*d/(-3*a*d+3*b*c))/a/c/((b*x^3+a)^(b*c/(-3*a*d+3*b*c)))
Time = 0.64 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x \left (a+b x^3\right )^{\frac {b c}{-3 b c+3 a d}} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}}}{a c} \] Input:
Integrate[(a + b*x^3)^(-1 - (b*c)/(3*b*c - 3*a*d))*(c + d*x^3)^(-1 + (a*d) /(3*b*c - 3*a*d)),x]
Output:
(x*(a + b*x^3)^((b*c)/(-3*b*c + 3*a*d))*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d) ))/(a*c)
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {906}
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.
\(\displaystyle \int \left (a+b x^3\right )^{-\frac {b c}{3 b c-3 a d}-1} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}-1} \, dx\) |
\(\Big \downarrow \) 906 |
\(\displaystyle \frac {x \left (a+b x^3\right )^{-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}}}{a c}\) |
Input:
Int[(a + b*x^3)^(-1 - (b*c)/(3*b*c - 3*a*d))*(c + d*x^3)^(-1 + (a*d)/(3*b* c - 3*a*d)),x]
Output:
(x*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c*(a + b*x^3)^((b*c)/(3*b*c - 3 *a*d)))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && EqQ[a*d*(p + 1) + b*c*(q + 1), 0]
Time = 4.70 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.34
method | result | size |
gosper | \(\frac {x \left (b \,x^{3}+a \right )^{1-\frac {3 a d -4 b c}{3 \left (a d -b c \right )}} \left (d \,x^{3}+c \right )^{1-\frac {4 a d -3 b c}{3 \left (a d -b c \right )}}}{a c}\) | \(71\) |
orering | \(\frac {\left (b \,x^{3}+a \right ) \left (d \,x^{3}+c \right ) x \left (b \,x^{3}+a \right )^{-1-\frac {b c}{-3 a d +3 b c}} \left (d \,x^{3}+c \right )^{-1+\frac {a d}{-3 a d +3 b c}}}{a c}\) | \(72\) |
Input:
int((b*x^3+a)^(-1-b*c/(-3*a*d+3*b*c))*(d*x^3+c)^(-1+a*d/(-3*a*d+3*b*c)),x, method=_RETURNVERBOSE)
Output:
x/a/c*(b*x^3+a)^(1-1/3*(3*a*d-4*b*c)/(a*d-b*c))*(d*x^3+c)^(1-1/3*(4*a*d-3* b*c)/(a*d-b*c))
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.72 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {b d x^{7} + {\left (b c + a d\right )} x^{4} + a c x}{{\left (b x^{3} + a\right )}^{\frac {4 \, b c - 3 \, a d}{3 \, {\left (b c - a d\right )}}} {\left (d x^{3} + c\right )}^{\frac {3 \, b c - 4 \, a d}{3 \, {\left (b c - a d\right )}}} a c} \] Input:
integrate((b*x^3+a)^(-1-b*c/(-3*a*d+3*b*c))*(d*x^3+c)^(-1+a*d/(-3*a*d+3*b* c)),x, algorithm="fricas")
Output:
(b*d*x^7 + (b*c + a*d)*x^4 + a*c*x)/((b*x^3 + a)^(1/3*(4*b*c - 3*a*d)/(b*c - a*d))*(d*x^3 + c)^(1/3*(3*b*c - 4*a*d)/(b*c - a*d))*a*c)
Timed out. \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\text {Timed out} \] Input:
integrate((b*x**3+a)**(-1-b*c/(-3*a*d+3*b*c))*(d*x**3+c)**(-1+a*d/(-3*a*d+ 3*b*c)),x)
Output:
Timed out
\[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\int { {\left (b x^{3} + a\right )}^{-\frac {b c}{3 \, {\left (b c - a d\right )}} - 1} {\left (d x^{3} + c\right )}^{\frac {a d}{3 \, {\left (b c - a d\right )}} - 1} \,d x } \] Input:
integrate((b*x^3+a)^(-1-b*c/(-3*a*d+3*b*c))*(d*x^3+c)^(-1+a*d/(-3*a*d+3*b* c)),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(-1/3*b*c/(b*c - a*d) - 1)*(d*x^3 + c)^(1/3*a*d/(b*c - a*d) - 1), x)
\[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\int { {\left (b x^{3} + a\right )}^{-\frac {b c}{3 \, {\left (b c - a d\right )}} - 1} {\left (d x^{3} + c\right )}^{\frac {a d}{3 \, {\left (b c - a d\right )}} - 1} \,d x } \] Input:
integrate((b*x^3+a)^(-1-b*c/(-3*a*d+3*b*c))*(d*x^3+c)^(-1+a*d/(-3*a*d+3*b* c)),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(-1/3*b*c/(b*c - a*d) - 1)*(d*x^3 + c)^(1/3*a*d/(b*c - a*d) - 1), x)
Time = 1.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.47 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x\,{\left (b\,x^3+a\right )}^{\frac {b\,c}{3\,a\,d-3\,b\,c}-1}+\frac {x^4\,{\left (b\,x^3+a\right )}^{\frac {b\,c}{3\,a\,d-3\,b\,c}-1}\,\left (a\,d+b\,c\right )}{a\,c}+\frac {b\,d\,x^7\,{\left (b\,x^3+a\right )}^{\frac {b\,c}{3\,a\,d-3\,b\,c}-1}}{a\,c}}{{\left (d\,x^3+c\right )}^{\frac {a\,d}{3\,a\,d-3\,b\,c}+1}} \] Input:
int((a + b*x^3)^((b*c)/(3*a*d - 3*b*c) - 1)/(c + d*x^3)^((a*d)/(3*a*d - 3* b*c) + 1),x)
Output:
(x*(a + b*x^3)^((b*c)/(3*a*d - 3*b*c) - 1) + (x^4*(a + b*x^3)^((b*c)/(3*a* d - 3*b*c) - 1)*(a*d + b*c))/(a*c) + (b*d*x^7*(a + b*x^3)^((b*c)/(3*a*d - 3*b*c) - 1))/(a*c))/(c + d*x^3)^((a*d)/(3*a*d - 3*b*c) + 1)
\[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\int \frac {\left (b \,x^{3}+a \right )^{\frac {b c}{3 a d -3 b c}}}{\left (d \,x^{3}+c \right )^{\frac {a d}{3 a d -3 b c}} a c +\left (d \,x^{3}+c \right )^{\frac {a d}{3 a d -3 b c}} a d \,x^{3}+\left (d \,x^{3}+c \right )^{\frac {a d}{3 a d -3 b c}} b c \,x^{3}+\left (d \,x^{3}+c \right )^{\frac {a d}{3 a d -3 b c}} b d \,x^{6}}d x \] Input:
int((b*x^3+a)^(-1-b*c/(-3*a*d+3*b*c))*(d*x^3+c)^(-1+a*d/(-3*a*d+3*b*c)),x)
Output:
int((a + b*x**3)**((b*c)/(3*a*d - 3*b*c))/((c + d*x**3)**((a*d)/(3*a*d - 3 *b*c))*a*c + (c + d*x**3)**((a*d)/(3*a*d - 3*b*c))*a*d*x**3 + (c + d*x**3) **((a*d)/(3*a*d - 3*b*c))*b*c*x**3 + (c + d*x**3)**((a*d)/(3*a*d - 3*b*c)) *b*d*x**6),x)