Integrand size = 24, antiderivative size = 185 \[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {b (2 b c-a d) \sqrt {c+d x^6}}{6 a^2 c (b c-a d) \left (a+b x^6\right )}-\frac {\sqrt {c+d x^6}}{6 a c x^6 \left (a+b x^6\right )}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{6 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {b c-a d}}\right )}{6 a^3 (b c-a d)^{3/2}} \]
1/6*(a*d+4*b*c)*arctanh((d*x^6+c)^(1/2)/c^(1/2))/a^3/c^(3/2)-1/6*b^(3/2)*( -5*a*d+4*b*c)*arctanh(b^(1/2)*(d*x^6+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+ b*c)^(3/2)-1/6*b*(-a*d+2*b*c)*(d*x^6+c)^(1/2)/a^2/c/(-a*d+b*c)/(b*x^6+a)-1 /6*(d*x^6+c)^(1/2)/a/c/x^6/(b*x^6+a)
Time = 1.01 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {\frac {a \sqrt {c+d x^6} \left (-a^2 d+2 b^2 c x^6+a b \left (c-d x^6\right )\right )}{c (-b c+a d) x^6 \left (a+b x^6\right )}-\frac {b^{3/2} (4 b c-5 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{c^{3/2}}}{6 a^3} \]
((a*Sqrt[c + d*x^6]*(-(a^2*d) + 2*b^2*c*x^6 + a*b*(c - d*x^6)))/(c*(-(b*c) + a*d)*x^6*(a + b*x^6)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2) + ((4*b*c + a*d)*ArcT anh[Sqrt[c + d*x^6]/Sqrt[c]])/c^(3/2))/(6*a^3)
Time = 0.35 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {948, 114, 27, 168, 174, 73, 221}
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.
| \(\displaystyle \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{6} \int \frac {1}{x^{12} \left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^6\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {3 b d x^6+4 b c+a d}{2 x^6 \left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^6}{a c}-\frac {\sqrt {c+d x^6}}{a c x^6 \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\int \frac {3 b d x^6+4 b c+a d}{x^6 \left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^6}{2 a c}-\frac {\sqrt {c+d x^6}}{a c x^6 \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\int \frac {b d (2 b c-a d) x^6+(b c-a d) (4 b c+a d)}{x^6 \left (b x^6+a\right ) \sqrt {d x^6+c}}dx^6}{a (b c-a d)}+\frac {2 b \sqrt {c+d x^6} (2 b c-a d)}{a \left (a+b x^6\right ) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x^6}}{a c x^6 \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\frac {(b c-a d) (a d+4 b c) \int \frac {1}{x^6 \sqrt {d x^6+c}}dx^6}{a}-\frac {b^2 c (4 b c-5 a d) \int \frac {1}{\left (b x^6+a\right ) \sqrt {d x^6+c}}dx^6}{a}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x^6} (2 b c-a d)}{a \left (a+b x^6\right ) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x^6}}{a c x^6 \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\frac {2 (b c-a d) (a d+4 b c) \int \frac {1}{\frac {x^{12}}{d}-\frac {c}{d}}d\sqrt {d x^6+c}}{a d}-\frac {2 b^2 c (4 b c-5 a d) \int \frac {1}{\frac {b x^{12}}{d}+a-\frac {b c}{d}}d\sqrt {d x^6+c}}{a d}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x^6} (2 b c-a d)}{a \left (a+b x^6\right ) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x^6}}{a c x^6 \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{6} \left (-\frac {\frac {\frac {2 b^{3/2} c (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^6}}{\sqrt {b c-a d}}\right )}{a \sqrt {b c-a d}}-\frac {2 (b c-a d) (a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^6}}{\sqrt {c}}\right )}{a \sqrt {c}}}{a (b c-a d)}+\frac {2 b \sqrt {c+d x^6} (2 b c-a d)}{a \left (a+b x^6\right ) (b c-a d)}}{2 a c}-\frac {\sqrt {c+d x^6}}{a c x^6 \left (a+b x^6\right )}\right )\) |
(-(Sqrt[c + d*x^6]/(a*c*x^6*(a + b*x^6))) - ((2*b*(2*b*c - a*d)*Sqrt[c + d *x^6])/(a*(b*c - a*d)*(a + b*x^6)) + ((-2*(b*c - a*d)*(4*b*c + a*d)*ArcTan h[Sqrt[c + d*x^6]/Sqrt[c]])/(a*Sqrt[c]) + (2*b^(3/2)*c*(4*b*c - 5*a*d)*Arc Tanh[(Sqrt[b]*Sqrt[c + d*x^6])/Sqrt[b*c - a*d]])/(a*Sqrt[b*c - a*d]))/(a*( b*c - a*d)))/(2*a*c))/6
3.9.74.3.1 Defintions of rubi rules used
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 5.45 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-4 x^{6} \left (b \,x^{6}+a \right ) \left (b c -\frac {5 a d}{4}\right ) b^{2} c^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {d \,x^{6}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {\left (a d -b c \right ) b}\, \left (c \,x^{6} \left (b \,x^{6}+a \right ) \left (a d +4 b c \right ) \left (a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{6}+c}}{\sqrt {c}}\right )+\left (-a d \left (b \,x^{6}+a \right ) c^{\frac {3}{2}}+b \left (2 b \,x^{6}+a \right ) c^{\frac {5}{2}}\right ) \sqrt {d \,x^{6}+c}\, a \right )}{6 \sqrt {\left (a d -b c \right ) b}\, c^{\frac {5}{2}} a^{3} \left (a d -b c \right ) \left (b \,x^{6}+a \right ) x^{6}}\) | \(189\) |
1/6/((a*d-b*c)*b)^(1/2)/c^(5/2)*(-4*x^6*(b*x^6+a)*(b*c-5/4*a*d)*b^2*c^(5/2 )*arctan(b*(d*x^6+c)^(1/2)/((a*d-b*c)*b)^(1/2))+((a*d-b*c)*b)^(1/2)*(c*x^6 *(b*x^6+a)*(a*d+4*b*c)*(a*d-b*c)*arctanh((d*x^6+c)^(1/2)/c^(1/2))+(-a*d*(b *x^6+a)*c^(3/2)+b*(2*b*x^6+a)*c^(5/2))*(d*x^6+c)^(1/2)*a))/a^3/(a*d-b*c)/( b*x^6+a)/x^6
Time = 0.48 (sec) , antiderivative size = 1236, normalized size of antiderivative = 6.68 \[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\text {Too large to display} \]
[1/12*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^12 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x ^6)*sqrt(b/(b*c - a*d))*log((b*d*x^6 + 2*b*c - a*d - 2*sqrt(d*x^6 + c)*(b* c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^6 + a)) + ((4*b^3*c^2 - 3*a*b^2*c*d - a ^2*b*d^2)*x^12 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^6)*sqrt(c)*log((d *x^6 + 2*sqrt(d*x^6 + c)*sqrt(c) + 2*c)/x^6) - 2*((2*a*b^2*c^2 - a^2*b*c*d )*x^6 + a^2*b*c^2 - a^3*c*d)*sqrt(d*x^6 + c))/((a^3*b^2*c^3 - a^4*b*c^2*d) *x^12 + (a^4*b*c^3 - a^5*c^2*d)*x^6), -1/12*(2*((4*b^3*c^3 - 5*a*b^2*c^2*d )*x^12 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^6)*sqrt(-b/(b*c - a*d))*arctan(-s qrt(d*x^6 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^6 + b*c)) - ((4*b^3 *c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^12 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^ 2)*x^6)*sqrt(c)*log((d*x^6 + 2*sqrt(d*x^6 + c)*sqrt(c) + 2*c)/x^6) + 2*((2 *a*b^2*c^2 - a^2*b*c*d)*x^6 + a^2*b*c^2 - a^3*c*d)*sqrt(d*x^6 + c))/((a^3* b^2*c^3 - a^4*b*c^2*d)*x^12 + (a^4*b*c^3 - a^5*c^2*d)*x^6), -1/12*(2*((4*b ^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^12 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3* d^2)*x^6)*sqrt(-c)*arctan(sqrt(d*x^6 + c)*sqrt(-c)/c) - ((4*b^3*c^3 - 5*a* b^2*c^2*d)*x^12 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^6)*sqrt(b/(b*c - a*d))*l og((b*d*x^6 + 2*b*c - a*d - 2*sqrt(d*x^6 + c)*(b*c - a*d)*sqrt(b/(b*c - a* d)))/(b*x^6 + a)) + 2*((2*a*b^2*c^2 - a^2*b*c*d)*x^6 + a^2*b*c^2 - a^3*c*d )*sqrt(d*x^6 + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^12 + (a^4*b*c^3 - a^5*c^ 2*d)*x^6), -1/6*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^12 + (4*a*b^2*c^3 - 5*a...
\[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^{7} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \]
\[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{7}} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{6} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{6 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{6} + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x^{6} + c} b^{2} c^{2} d - {\left (d x^{6} + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x^{6} + c} a b c d^{2} - \sqrt {d x^{6} + c} a^{2} d^{3}}{6 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x^{6} + c\right )}^{2} b - 2 \, {\left (d x^{6} + c\right )} b c + b c^{2} + {\left (d x^{6} + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{6} + c}}{\sqrt {-c}}\right )}{6 \, a^{3} \sqrt {-c} c} \]
1/6*(4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^6 + c)*b/sqrt(-b^2*c + a*b*d))/( (a^3*b*c - a^4*d)*sqrt(-b^2*c + a*b*d)) - 1/6*(2*(d*x^6 + c)^(3/2)*b^2*c*d - 2*sqrt(d*x^6 + c)*b^2*c^2*d - (d*x^6 + c)^(3/2)*a*b*d^2 + 2*sqrt(d*x^6 + c)*a*b*c*d^2 - sqrt(d*x^6 + c)*a^2*d^3)/((a^2*b*c^2 - a^3*c*d)*((d*x^6 + c)^2*b - 2*(d*x^6 + c)*b*c + b*c^2 + (d*x^6 + c)*a*d - a*c*d)) - 1/6*(4*b *c + a*d)*arctan(sqrt(d*x^6 + c)/sqrt(-c))/(a^3*sqrt(-c)*c)
Time = 11.63 (sec) , antiderivative size = 3860, normalized size of antiderivative = 20.86 \[ \int \frac {1}{x^7 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\text {Too large to display} \]
(((c + d*x^6)^(1/2)*(a^2*d^3 + 2*b^2*c^2*d - 2*a*b*c*d^2))/(2*a^2*(b*c^2 - a*c*d)) + (b*(c + d*x^6)^(3/2)*(a*d^2 - 2*b*c*d))/(2*a^2*(b*c^2 - a*c*d)) )/((c + d*x^6)*(3*a*d - 6*b*c) + 3*b*(c + d*x^6)^2 + 3*b*c^2 - 3*a*c*d) + (atan((((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^6)^(1/2)*(a^ 4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b ^5*c^2*d^4))/(18*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) + ((-b^3*(a* d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*((144*a^9*b^2*c*d^6 + 288*a^6*b^5*c^4*d^ 3 - 576*a^7*b^4*c^3*d^4 + 144*a^8*b^3*c^2*d^5)/(216*(a^6*b^2*c^4 + a^8*c^2 *d^2 - 2*a^7*b*c^3*d)) - ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x^6)^(1/2)*(5* a*d - 4*b*c)*(288*a^6*b^5*c^5*d^2 - 720*a^7*b^4*c^4*d^3 + 576*a^8*b^3*c^3* d^4 - 144*a^9*b^2*c^2*d^5))/(216*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3* d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))/(12*(a^6*d ^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*1i)/(12*(a^6*d^3 - a ^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)) + ((-b^3*(a*d - b*c)^3)^(1/ 2)*(5*a*d - 4*b*c)*(((c + d*x^6)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64* a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(18*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) - ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c )*((144*a^9*b^2*c*d^6 + 288*a^6*b^5*c^4*d^3 - 576*a^7*b^4*c^3*d^4 + 144*a^ 8*b^3*c^2*d^5)/(216*(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d)) + ((-b^3* (a*d - b*c)^3)^(1/2)*(c + d*x^6)^(1/2)*(5*a*d - 4*b*c)*(288*a^6*b^5*c^5...