Integrand size = 22, antiderivative size = 64 \[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {x^2 \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {1}{3},2,\frac {1}{2},\frac {4}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 \sqrt {c+d x^6}} \] Output:
1/2*x^2*(1+d*x^6/c)^(1/2)*AppellF1(1/3,2,1/2,4/3,-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. \(172\) vs. \(2(64)=128\).
Time = 10.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.69 \[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {8 a b x^2 \left (c+d x^6\right )+8 (2 b c-3 a d) x^2 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+b d x^8 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{48 a^2 (b c-a d) \left (a+b x^6\right ) \sqrt {c+d x^6}} \] Input:
Integrate[x/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
Output:
(8*a*b*x^2*(c + d*x^6) + 8*(2*b*c - 3*a*d)*x^2*(a + b*x^6)*Sqrt[1 + (d*x^6 )/c]*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^6)/c), -((b*x^6)/a)] + b*d*x^8*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b* x^6)/a)])/(48*a^2*(b*c - a*d)*(a + b*x^6)*Sqrt[c + d*x^6])
Time = 0.34 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {965, 937, 936}
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.
\(\displaystyle \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^2\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {d x^6}{c}+1} \int \frac {1}{\left (b x^6+a\right )^2 \sqrt {\frac {d x^6}{c}+1}}dx^2}{2 \sqrt {c+d x^6}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x^2 \sqrt {\frac {d x^6}{c}+1} \operatorname {AppellF1}\left (\frac {1}{3},2,\frac {1}{2},\frac {4}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 \sqrt {c+d x^6}}\) |
Input:
Int[x/((a + b*x^6)^2*Sqrt[c + d*x^6]),x]
Output:
(x^2*Sqrt[1 + (d*x^6)/c]*AppellF1[1/3, 2, 1/2, 4/3, -((b*x^6)/a), -((d*x^6 )/c)])/(2*a^2*Sqrt[c + d*x^6])
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])
Int[((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[(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])
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]
\[\int \frac {x}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}d x\]
Input:
int(x/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
Output:
int(x/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
Timed out. \[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\text {Timed out} \] Input:
integrate(x/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x}{\left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \] Input:
integrate(x/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
Output:
Integral(x/((a + b*x**6)**2*sqrt(c + d*x**6)), x)
\[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \] Input:
integrate(x/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")
Output:
integrate(x/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)
\[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {x}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c}} \,d x } \] Input:
integrate(x/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")
Output:
integrate(x/((b*x^6 + a)^2*sqrt(d*x^6 + c)), x)
Timed out. \[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {x}{{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \] Input:
int(x/((a + b*x^6)^2*(c + d*x^6)^(1/2)),x)
Output:
int(x/((a + b*x^6)^2*(c + d*x^6)^(1/2)), x)
\[ \int \frac {x}{\left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {\sqrt {d \,x^{6}+c}\, x}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \] Input:
int(x/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
Output:
int((sqrt(c + d*x**6)*x)/(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)