Integrand size = 24, antiderivative size = 64 \[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (-\frac {2}{3},1,\frac {1}{2},\frac {1}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{4 a x^4 \sqrt {c+d x^6}} \] Output:
-1/4*(1+d*x^6/c)^(1/2)*AppellF1(-2/3,1,1/2,1/3,-b*x^6/a,-d*x^6/c)/a/x^4/(d *x^6+c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(64)=128\).
Time = 10.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.20 \[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\frac {-8 a \left (c+d x^6\right )-4 (4 b c+a d) x^6 \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )-b d x^{12} \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{32 a^2 c x^4 \sqrt {c+d x^6}} \] Input:
Integrate[1/(x^5*(a + b*x^6)*Sqrt[c + d*x^6]),x]
Output:
(-8*a*(c + d*x^6) - 4*(4*b*c + a*d)*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^12*Sqrt[1 + (d*x^6)/c]*Ap pellF1[4/3, 1/2, 1, 7/3, -((d*x^6)/c), -((b*x^6)/a)])/(32*a^2*c*x^4*Sqrt[c + d*x^6])
Time = 0.40 (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.
\(\displaystyle \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (b x^6+a\right ) \sqrt {d x^6+c}}dx^2\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {\frac {d x^6}{c}+1} \int \frac {1}{x^6 \left (b x^6+a\right ) \sqrt {\frac {d x^6}{c}+1}}dx^2}{2 \sqrt {c+d x^6}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {\sqrt {\frac {d x^6}{c}+1} \operatorname {AppellF1}\left (-\frac {2}{3},1,\frac {1}{2},\frac {1}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{4 a x^4 \sqrt {c+d x^6}}\) |
Input:
Int[1/(x^5*(a + b*x^6)*Sqrt[c + d*x^6]),x]
Output:
-1/4*(Sqrt[1 + (d*x^6)/c]*AppellF1[-2/3, 1, 1/2, 1/3, -((b*x^6)/a), -((d*x ^6)/c)])/(a*x^4*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])
\[\int \frac {1}{x^{5} \left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}d x\]
Input:
int(1/x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x)
Output:
int(1/x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x)
Timed out. \[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\text {Timed out} \] Input:
integrate(1/x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^{5} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \] Input:
integrate(1/x**5/(b*x**6+a)/(d*x**6+c)**(1/2),x)
Output:
Integral(1/(x**5*(a + b*x**6)*sqrt(c + d*x**6)), x)
\[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{5}} \,d x } \] Input:
integrate(1/x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^5), x)
\[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{5}} \,d x } \] Input:
integrate(1/x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^5), x)
Timed out. \[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^5\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \] Input:
int(1/(x^5*(a + b*x^6)*(c + d*x^6)^(1/2)),x)
Output:
int(1/(x^5*(a + b*x^6)*(c + d*x^6)^(1/2)), x)
\[ \int \frac {1}{x^5 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {\sqrt {d \,x^{6}+c}}{b d \,x^{17}+a d \,x^{11}+b c \,x^{11}+a c \,x^{5}}d x \] Input:
int(1/x^5/(b*x^6+a)/(d*x^6+c)^(1/2),x)
Output:
int(sqrt(c + d*x**6)/(a*c*x**5 + a*d*x**11 + b*c*x**11 + b*d*x**17),x)