Integrand size = 24, antiderivative size = 80 \[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {c+d x^6}}{3 a c x^3}-\frac {b \arctan \left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{3 a^{3/2} \sqrt {b c-a d}} \] Output:
-1/3*(d*x^6+c)^(1/2)/a/c/x^3-1/3*b*arctan((-a*d+b*c)^(1/2)*x^3/a^(1/2)/(d* x^6+c)^(1/2))/a^(3/2)/(-a*d+b*c)^(1/2)
Time = 0.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25 \[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {c+d x^6}}{3 a c x^3}-\frac {b \arctan \left (\frac {a \sqrt {d}+b x^3 \left (\sqrt {d} x^3+\sqrt {c+d x^6}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{3 a^{3/2} \sqrt {b c-a d}} \] Input:
Integrate[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]
Output:
-1/3*Sqrt[c + d*x^6]/(a*c*x^3) - (b*ArcTan[(a*Sqrt[d] + b*x^3*(Sqrt[d]*x^3 + Sqrt[c + d*x^6]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(3*a^(3/2)*Sqrt[b*c - a*d ])
Time = 0.38 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {965, 382, 25, 27, 291, 218}
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.
\(\displaystyle \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx\) |
\(\Big \downarrow \) 965 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^6 \left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3\) |
\(\Big \downarrow \) 382 |
\(\displaystyle \frac {1}{3} \left (\frac {\int -\frac {b c}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{a c}-\frac {\sqrt {c+d x^6}}{a c x^3}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {b c}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{a c}-\frac {\sqrt {c+d x^6}}{a c x^3}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^3}{a}-\frac {\sqrt {c+d x^6}}{a c x^3}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \int \frac {1}{a-(a d-b c) x^6}d\frac {x^3}{\sqrt {d x^6+c}}}{a}-\frac {\sqrt {c+d x^6}}{a c x^3}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {1}{3} \left (-\frac {b \arctan \left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{a^{3/2} \sqrt {b c-a d}}-\frac {\sqrt {c+d x^6}}{a c x^3}\right )\) |
Input:
Int[1/(x^4*(a + b*x^6)*Sqrt[c + d*x^6]),x]
Output:
(-(Sqrt[c + d*x^6]/(a*c*x^3)) - (b*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*S qrt[c + d*x^6])])/(a^(3/2)*Sqrt[b*c - a*d]))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ (a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b* x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m + 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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]
Time = 3.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{6}+c}}{x^{3}}-\frac {b c \,\operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{6}+c}}{x^{3} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}}{3 a c}\) | \(67\) |
Input:
int(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/a*(-(d*x^6+c)^(1/2)/x^3-b*c/(a*(a*d-b*c))^(1/2)*arctanh(a*(d*x^6+c)^(1 /2)/x^3/(a*(a*d-b*c))^(1/2)))/c
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (64) = 128\).
Time = 0.13 (sec) , antiderivative size = 332, normalized size of antiderivative = 4.15 \[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\left [-\frac {\sqrt {-a b c + a^{2} d} b c x^{3} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 4 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )}}{12 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}}, -\frac {\sqrt {a b c - a^{2} d} b c x^{3} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right ) + 2 \, \sqrt {d x^{6} + c} {\left (a b c - a^{2} d\right )}}{6 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} x^{3}}\right ] \] Input:
integrate(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")
Output:
[-1/12*(sqrt(-a*b*c + a^2*d)*b*c*x^3*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2 )*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 + 4*((b*c - 2*a*d)*x^9 - a*c*x^3)*sqrt(d*x^6 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6 + a^2 )) + 4*sqrt(d*x^6 + c)*(a*b*c - a^2*d))/((a^2*b*c^2 - a^3*c*d)*x^3), -1/6* (sqrt(a*b*c - a^2*d)*b*c*x^3*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x ^6 + c)*sqrt(a*b*c - a^2*d)/((a*b*c*d - a^2*d^2)*x^9 + (a*b*c^2 - a^2*c*d) *x^3)) + 2*sqrt(d*x^6 + c)*(a*b*c - a^2*d))/((a^2*b*c^2 - a^3*c*d)*x^3)]
\[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \] Input:
integrate(1/x**4/(b*x**6+a)/(d*x**6+c)**(1/2),x)
Output:
Integral(1/(x**4*(a + b*x**6)*sqrt(c + d*x**6)), x)
\[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{4}} \,d x } \] Input:
integrate(1/x^4/(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^4), x)
Timed out. \[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\text {Timed out} \] Input:
integrate(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^4\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \] Input:
int(1/(x^4*(a + b*x^6)*(c + d*x^6)^(1/2)),x)
Output:
int(1/(x^4*(a + b*x^6)*(c + d*x^6)^(1/2)), x)
\[ \int \frac {1}{x^4 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {\sqrt {d \,x^{6}+c}}{b d \,x^{16}+a d \,x^{10}+b c \,x^{10}+a c \,x^{4}}d x \] Input:
int(1/x^4/(b*x^6+a)/(d*x^6+c)^(1/2),x)
Output:
int(sqrt(c + d*x**6)/(a*c*x**4 + a*d*x**10 + b*c*x**10 + b*d*x**16),x)