Integrand size = 24, antiderivative size = 64 \[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {x^4 \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {2}{3},2,\frac {1}{2},\frac {5}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{4 a^2 \sqrt {c+d x^6}} \] Output:
1/4*x^4*(1+d*x^6/c)^(1/2)*AppellF1(2/3,2,1/2,5/3,-b*x^6/a,-d*x^6/c)/a^2/(d *x^6+c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(64)=128\).
Time = 10.14 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.62 \[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {x^4 \left (-10 a b \left (c+d x^6\right )-5 (b c-3 a d) \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+b d x^6 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )\right )}{60 a^2 (b c-a d) \left (a+b x^6\right ) \sqrt {c+d x^6}} \] Input:
Integrate[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
Output:
-1/60*(x^4*(-10*a*b*(c + d*x^6) - 5*(b*c - 3*a*d)*(a + b*x^6)*Sqrt[1 + (d* x^6)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)] + b*d*x^6*( a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), -( (b*x^6)/a)]))/(a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])
Time = 0.37 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {965, 1013, 1012}
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.
\(\displaystyle \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{\left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^2\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {\frac {d x^6}{c}+1} \int \frac {x^2}{\left (b x^6+a\right )^2 \sqrt {\frac {d x^6}{c}+1}}dx^2}{2 \sqrt {c+d x^6}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {x^4 \sqrt {\frac {d x^6}{c}+1} \operatorname {AppellF1}\left (\frac {2}{3},2,\frac {1}{2},\frac {5}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{4 a^2 \sqrt {c+d x^6}}\) |
Input:
Int[x^3/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
Output:
(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 2, 1/2, 5/3, -((b*x^6)/a), -((d*x^6 )/c)])/(4*a^2*Sqrt[c + d*x^6])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {x^{3}}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}d x\]
Input:
int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
Output:
int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
Timed out. \[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\text {Timed out} \] Input:
integrate(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^{3}}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \] Input:
integrate(x**3/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
Output:
Integral(x**3/((a + b*x**6)**2*sqrt(c + d*x**6)), x)
\[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \] Input:
integrate(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")
Output:
integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)
\[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \] Input:
integrate(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")
Output:
integrate(x^3/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)
Timed out. \[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x^3}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \] Input:
int(x^3/((a + b*x^6)^2*(c + d*x^6)^(1/2)),x)
Output:
int(x^3/((a + b*x^6)^2*(c + d*x^6)^(1/2)), x)
\[ \int \frac {x^3}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {\sqrt {d \,x^{6}+c}\, x^{3}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \] Input:
int(x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
Output:
int((sqrt(c + d*x**6)*x**3)/(a**2*c + a**2*d*x**6 + 2*a*b*c*x**6 + 2*a*b*d *x**12 + b**2*c*x**12 + b**2*d*x**18),x)