\(\int (a+b x^3)^{-1-\frac {b c}{3 b c-3 a d}} (c+d x^3)^{-1+\frac {a d}{3 b c-3 a d}} \, dx\) [145]

Optimal result

Integrand size = 50, antiderivative size = 53 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x \left (a+b x^3\right )^{-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}}}{a c} \]

[Out]

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)))

Rubi [A] (verified)

Time = 0.01 (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 = {389} \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x \left (a+b x^3\right )^{-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}}}{a c} \]

[In]

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]

[Out]

(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)))

Rule 389

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]

Rubi steps \begin{align*} \text {integral}& = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x \left (a+b x^3\right )^{\frac {b c}{-3 b c+3 a d}} \left (c+d x^3\right )^{\frac {a d}{3 b c-3 a d}}}{a c} \]

[In]

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]

[Out]

(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)

Maple [A] (verified)

Time = 7.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.34

method	result	size
gosper	\(\frac {x \left (b \,x^{3}+a \right )^{1-\frac {3 a d -4 b c}{3 \left (a d -b c \right )}} \left (d \,x^{3}+c \right )^{1-\frac {4 a d -3 b c}{3 \left (a d -b c \right )}}}{a c}\)	\(71\)

[In]

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)

[Out]

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))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.72 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {b d x^{7} + {\left (b c + a d\right )} x^{4} + a c x}{{\left (b x^{3} + a\right )}^{\frac {4 \, b c - 3 \, a d}{3 \, {\left (b c - a d\right )}}} {\left (d x^{3} + c\right )}^{\frac {3 \, b c - 4 \, a d}{3 \, {\left (b c - a d\right )}}} a c} \]

[In]

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")

[Out]

(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)

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\text {Timed out} \]

[In]

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)

[Out]

Timed out

Maxima [F]

\[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\int { {\left (b x^{3} + a\right )}^{-\frac {b c}{3 \, {\left (b c - a d\right )}} - 1} {\left (d x^{3} + c\right )}^{\frac {a d}{3 \, {\left (b c - a d\right )}} - 1} \,d x } \]

[In]

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")

[Out]

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)

Giac [F]

\[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\int { {\left (b x^{3} + a\right )}^{-\frac {b c}{3 \, {\left (b c - a d\right )}} - 1} {\left (d x^{3} + c\right )}^{\frac {a d}{3 \, {\left (b c - a d\right )}} - 1} \,d x } \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 5.92 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.47 \[ \int \left (a+b x^3\right )^{-1-\frac {b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac {a d}{3 b c-3 a d}} \, dx=\frac {x\,{\left (b\,x^3+a\right )}^{\frac {b\,c}{3\,a\,d-3\,b\,c}-1}+\frac {x^4\,{\left (b\,x^3+a\right )}^{\frac {b\,c}{3\,a\,d-3\,b\,c}-1}\,\left (a\,d+b\,c\right )}{a\,c}+\frac {b\,d\,x^7\,{\left (b\,x^3+a\right )}^{\frac {b\,c}{3\,a\,d-3\,b\,c}-1}}{a\,c}}{{\left (d\,x^3+c\right )}^{\frac {a\,d}{3\,a\,d-3\,b\,c}+1}} \]

[In]

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)

[Out]

(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)