\(\int \frac {1}{(a+b x^3)^{2/3} (c+d x^3)^3} \, dx\) [119]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (a+b x^3\right )^{2/3}} \]

[Out]

x*(1+b*x^3/a)^(2/3)*AppellF1(1/3,2/3,3,4/3,-b*x^3/a,-d*x^3/c)/c^3/(b*x^3+a)^(2/3)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {441, 440} \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (a+b x^3\right )^{2/3}} \]

[In]

Int[1/((a + b*x^3)^(2/3)*(c + d*x^3)^3),x]

[Out]

(x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 2/3, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c^3*(a + b*x^3)^(2/3))

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right )^3} \, dx}{\left (a+b x^3\right )^{2/3}} \\ & = \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (a+b x^3\right )^{2/3}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(59)=118\).

Time = 10.68 (sec) , antiderivative size = 418, normalized size of antiderivative = 7.08 \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\frac {x \left (5 b d (-2 b c+a d) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 c \left (-4 a c \left (3 a^2 d^2 \left (6 c+5 d x^3\right )+b^2 c \left (18 c^2+5 c d x^3-10 d^2 x^6\right )+a b d \left (-36 c^2-25 c d x^3+5 d^2 x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+d x^3 \left (a+b x^3\right ) \left (a d \left (8 c+5 d x^3\right )-b c \left (13 c+10 d x^3\right )\right ) \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right )^2 \left (-4 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}\right )}{72 c^3 (b c-a d)^2 \left (a+b x^3\right )^{2/3}} \]

[In]

Integrate[1/((a + b*x^3)^(2/3)*(c + d*x^3)^3),x]

[Out]

(x*(5*b*d*(-2*b*c + a*d)*x^3*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + (4
*c*(-4*a*c*(3*a^2*d^2*(6*c + 5*d*x^3) + b^2*c*(18*c^2 + 5*c*d*x^3 - 10*d^2*x^6) + a*b*d*(-36*c^2 - 25*c*d*x^3
+ 5*d^2*x^6))*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + d*x^3*(a + b*x^3)*(a*d*(8*c + 5*d*x^3)
- b*c*(13*c + 10*d*x^3))*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5
/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))/((c + d*x^3)^2*(-4*a*c*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((
d*x^3)/c)] + x^3*(3*a*d*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*b*c*AppellF1[4/3, 5/3, 1, 7
/3, -((b*x^3)/a), -((d*x^3)/c)])))))/(72*c^3*(b*c - a*d)^2*(a + b*x^3)^(2/3))

Maple [F]

\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )^{3}}d x\]

[In]

int(1/(b*x^3+a)^(2/3)/(d*x^3+c)^3,x)

[Out]

int(1/(b*x^3+a)^(2/3)/(d*x^3+c)^3,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(b*x^3+a)^(2/3)/(d*x^3+c)^3,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(1/(b*x**3+a)**(2/3)/(d*x**3+c)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]

[In]

integrate(1/(b*x^3+a)^(2/3)/(d*x^3+c)^3,x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)^3), x)

Giac [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]

[In]

integrate(1/(b*x^3+a)^(2/3)/(d*x^3+c)^3,x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{2/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]

[In]

int(1/((a + b*x^3)^(2/3)*(c + d*x^3)^3),x)

[Out]

int(1/((a + b*x^3)^(2/3)*(c + d*x^3)^3), x)