Integrand size = 22, antiderivative size = 77 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\frac {b c-a d}{9 a b \left (a+b x^6\right )^{3/2}}+\frac {c}{3 a^2 \sqrt {a+b x^6}}-\frac {c \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{3 a^{5/2}} \] Output:
1/9*(-a*d+b*c)/a/b/(b*x^6+a)^(3/2)+1/3*c/a^2/(b*x^6+a)^(1/2)-1/3*c*arctanh ((b*x^6+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\frac {4 a b c-a^2 d+3 b^2 c x^6}{9 a^2 b \left (a+b x^6\right )^{3/2}}-\frac {c \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{3 a^{5/2}} \] Input:
Integrate[(c + d*x^6)/(x*(a + b*x^6)^(5/2)),x]
Output:
(4*a*b*c - a^2*d + 3*b^2*c*x^6)/(9*a^2*b*(a + b*x^6)^(3/2)) - (c*ArcTanh[S qrt[a + b*x^6]/Sqrt[a]])/(3*a^(5/2))
Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 87, 61, 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.
\(\displaystyle \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{6} \int \frac {d x^6+c}{x^6 \left (b x^6+a\right )^{5/2}}dx^6\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{6} \left (\frac {c \int \frac {1}{x^6 \left (b x^6+a\right )^{3/2}}dx^6}{a}+\frac {2 (b c-a d)}{3 a b \left (a+b x^6\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{6} \left (\frac {c \left (\frac {\int \frac {1}{x^6 \sqrt {b x^6+a}}dx^6}{a}+\frac {2}{a \sqrt {a+b x^6}}\right )}{a}+\frac {2 (b c-a d)}{3 a b \left (a+b x^6\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} \left (\frac {c \left (\frac {2 \int \frac {1}{\frac {x^{12}}{b}-\frac {a}{b}}d\sqrt {b x^6+a}}{a b}+\frac {2}{a \sqrt {a+b x^6}}\right )}{a}+\frac {2 (b c-a d)}{3 a b \left (a+b x^6\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{6} \left (\frac {c \left (\frac {2}{a \sqrt {a+b x^6}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{a}+\frac {2 (b c-a d)}{3 a b \left (a+b x^6\right )^{3/2}}\right )\) |
Input:
Int[(c + d*x^6)/(x*(a + b*x^6)^(5/2)),x]
Output:
((2*(b*c - a*d))/(3*a*b*(a + b*x^6)^(3/2)) + (c*(2/(a*Sqrt[a + b*x^6]) - ( 2*ArcTanh[Sqrt[a + b*x^6]/Sqrt[a]])/a^(3/2)))/a)/6
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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 = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {3 b c \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{6}+a}}{\sqrt {a}}\right ) a^{2} \left (b \,x^{6}+a \right )^{\frac {3}{2}}+a^{\frac {5}{2}} \left (-3 b^{2} c \,x^{6}+a^{2} d -4 a b c \right )}{9 \left (b \,x^{6}+a \right )^{\frac {3}{2}} a^{\frac {9}{2}} b}\) | \(73\) |
Input:
int((d*x^6+c)/x/(b*x^6+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/9*(3*b*c*arctanh((b*x^6+a)^(1/2)/a^(1/2))*a^2*(b*x^6+a)^(3/2)+a^(5/2)*( -3*b^2*c*x^6+a^2*d-4*a*b*c))/(b*x^6+a)^(3/2)/a^(9/2)/b
Time = 0.09 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.12 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{3} c x^{12} + 2 \, a b^{2} c x^{6} + a^{2} b c\right )} \sqrt {a} \log \left (\frac {b x^{6} - 2 \, \sqrt {b x^{6} + a} \sqrt {a} + 2 \, a}{x^{6}}\right ) + 2 \, {\left (3 \, a b^{2} c x^{6} + 4 \, a^{2} b c - a^{3} d\right )} \sqrt {b x^{6} + a}}{18 \, {\left (a^{3} b^{3} x^{12} + 2 \, a^{4} b^{2} x^{6} + a^{5} b\right )}}, \frac {3 \, {\left (b^{3} c x^{12} + 2 \, a b^{2} c x^{6} + a^{2} b c\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{6} + a}}\right ) + {\left (3 \, a b^{2} c x^{6} + 4 \, a^{2} b c - a^{3} d\right )} \sqrt {b x^{6} + a}}{9 \, {\left (a^{3} b^{3} x^{12} + 2 \, a^{4} b^{2} x^{6} + a^{5} b\right )}}\right ] \] Input:
integrate((d*x^6+c)/x/(b*x^6+a)^(5/2),x, algorithm="fricas")
Output:
[1/18*(3*(b^3*c*x^12 + 2*a*b^2*c*x^6 + a^2*b*c)*sqrt(a)*log((b*x^6 - 2*sqr t(b*x^6 + a)*sqrt(a) + 2*a)/x^6) + 2*(3*a*b^2*c*x^6 + 4*a^2*b*c - a^3*d)*s qrt(b*x^6 + a))/(a^3*b^3*x^12 + 2*a^4*b^2*x^6 + a^5*b), 1/9*(3*(b^3*c*x^12 + 2*a*b^2*c*x^6 + a^2*b*c)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^6 + a)) + (3 *a*b^2*c*x^6 + 4*a^2*b*c - a^3*d)*sqrt(b*x^6 + a))/(a^3*b^3*x^12 + 2*a^4*b ^2*x^6 + a^5*b)]
Time = 26.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.22 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {a d - b c}{18 a \left (a + b x^{6}\right )^{\frac {3}{2}}} + \frac {b c}{6 a^{2} \sqrt {a + b x^{6}}} + \frac {b c \operatorname {atan}{\left (\frac {\sqrt {a + b x^{6}}}{\sqrt {- a}} \right )}}{6 a^{2} \sqrt {- a}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {c \log {\left (d x^{6} \right )} + d x^{6}}{6 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((d*x**6+c)/x/(b*x**6+a)**(5/2),x)
Output:
Piecewise((2*(-(a*d - b*c)/(18*a*(a + b*x**6)**(3/2)) + b*c/(6*a**2*sqrt(a + b*x**6)) + b*c*atan(sqrt(a + b*x**6)/sqrt(-a))/(6*a**2*sqrt(-a)))/b, Ne (b, 0)), ((c*log(d*x**6) + d*x**6)/(6*a**(5/2)), True))
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\frac {1}{18} \, c {\left (\frac {3 \, \log \left (\frac {\sqrt {b x^{6} + a} - \sqrt {a}}{\sqrt {b x^{6} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x^{6} + 4 \, a\right )}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} a^{2}}\right )} - \frac {d}{9 \, {\left (b x^{6} + a\right )}^{\frac {3}{2}} b} \] Input:
integrate((d*x^6+c)/x/(b*x^6+a)^(5/2),x, algorithm="maxima")
Output:
1/18*c*(3*log((sqrt(b*x^6 + a) - sqrt(a))/(sqrt(b*x^6 + a) + sqrt(a)))/a^( 5/2) + 2*(3*b*x^6 + 4*a)/((b*x^6 + a)^(3/2)*a^2)) - 1/9*d/((b*x^6 + a)^(3/ 2)*b)
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.87 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\frac {c \arctan \left (\frac {\sqrt {b x^{6} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a^{2}} + \frac {3 \, {\left (b x^{6} + a\right )} b c + a b c - a^{2} d}{9 \, {\left (b x^{6} + a\right )}^{\frac {3}{2}} a^{2} b} \] Input:
integrate((d*x^6+c)/x/(b*x^6+a)^(5/2),x, algorithm="giac")
Output:
1/3*c*arctan(sqrt(b*x^6 + a)/sqrt(-a))/(sqrt(-a)*a^2) + 1/9*(3*(b*x^6 + a) *b*c + a*b*c - a^2*d)/((b*x^6 + a)^(3/2)*a^2*b)
Time = 4.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\frac {\frac {c}{3\,a}+\frac {c\,\left (b\,x^6+a\right )}{a^2}}{3\,{\left (b\,x^6+a\right )}^{3/2}}-\frac {d}{9\,b\,{\left (b\,x^6+a\right )}^{3/2}}-\frac {c\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^6+a}}{\sqrt {a}}\right )}{3\,a^{5/2}} \] Input:
int((c + d*x^6)/(x*(a + b*x^6)^(5/2)),x)
Output:
(c/(3*a) + (c*(a + b*x^6))/a^2)/(3*(a + b*x^6)^(3/2)) - d/(9*b*(a + b*x^6) ^(3/2)) - (c*atanh((a + b*x^6)^(1/2)/a^(1/2)))/(3*a^(5/2))
Time = 0.32 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.83 \[ \int \frac {c+d x^6}{x \left (a+b x^6\right )^{5/2}} \, dx=\frac {-2 \sqrt {b \,x^{6}+a}\, a^{3} d +8 \sqrt {b \,x^{6}+a}\, a^{2} b c +6 \sqrt {b \,x^{6}+a}\, a \,b^{2} c \,x^{6}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a^{2} b c +6 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a \,b^{2} c \,x^{6}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) b^{3} c \,x^{12}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a^{2} b c -6 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a \,b^{2} c \,x^{6}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) b^{3} c \,x^{12}}{18 a^{3} b \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:
int((d*x^6+c)/x/(b*x^6+a)^(5/2),x)
Output:
( - 2*sqrt(a + b*x**6)*a**3*d + 8*sqrt(a + b*x**6)*a**2*b*c + 6*sqrt(a + b *x**6)*a*b**2*c*x**6 + 3*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a**2*b*c + 6*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a*b**2*c*x**6 + 3*sqrt(a)*log( sqrt(a + b*x**6) - sqrt(a))*b**3*c*x**12 - 3*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*a**2*b*c - 6*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*a*b**2*c*x **6 - 3*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*b**3*c*x**12)/(18*a**3*b*( a**2 + 2*a*b*x**6 + b**2*x**12))