Integrand size = 22, antiderivative size = 85 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {b c-a d}{3 a^2 \sqrt {a+b x^6}}-\frac {c \sqrt {a+b x^6}}{6 a^2 x^6}+\frac {(3 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{6 a^{5/2}} \] Output:
-1/3*(-a*d+b*c)/a^2/(b*x^6+a)^(1/2)-1/6*c*(b*x^6+a)^(1/2)/a^2/x^6+1/6*(-2* a*d+3*b*c)*arctanh((b*x^6+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-a c-3 b c x^6+2 a d x^6}{6 a^2 x^6 \sqrt {a+b x^6}}+\frac {(3 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{6 a^{5/2}} \] Input:
Integrate[(c + d*x^6)/(x^7*(a + b*x^6)^(3/2)),x]
Output:
(-(a*c) - 3*b*c*x^6 + 2*a*d*x^6)/(6*a^2*x^6*Sqrt[a + b*x^6]) + ((3*b*c - 2 *a*d)*ArcTanh[Sqrt[a + b*x^6]/Sqrt[a]])/(6*a^(5/2))
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98, 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.
\(\displaystyle \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{6} \int \frac {d x^6+c}{x^{12} \left (b x^6+a\right )^{3/2}}dx^6\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{6} \left (-\frac {(3 b c-2 a d) \int \frac {1}{x^6 \left (b x^6+a\right )^{3/2}}dx^6}{2 a}-\frac {c}{a x^6 \sqrt {a+b x^6}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{6} \left (-\frac {(3 b c-2 a d) \left (\frac {\int \frac {1}{x^6 \sqrt {b x^6+a}}dx^6}{a}+\frac {2}{a \sqrt {a+b x^6}}\right )}{2 a}-\frac {c}{a x^6 \sqrt {a+b x^6}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} \left (-\frac {(3 b c-2 a d) \left (\frac {2 \int \frac {1}{\frac {x^{12}}{b}-\frac {a}{b}}d\sqrt {b x^6+a}}{a b}+\frac {2}{a \sqrt {a+b x^6}}\right )}{2 a}-\frac {c}{a x^6 \sqrt {a+b x^6}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{6} \left (-\frac {\left (\frac {2}{a \sqrt {a+b x^6}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{a^{3/2}}\right ) (3 b c-2 a d)}{2 a}-\frac {c}{a x^6 \sqrt {a+b x^6}}\right )\) |
Input:
Int[(c + d*x^6)/(x^7*(a + b*x^6)^(3/2)),x]
Output:
(-(c/(a*x^6*Sqrt[a + b*x^6])) - ((3*b*c - 2*a*d)*(2/(a*Sqrt[a + b*x^6]) - (2*ArcTanh[Sqrt[a + b*x^6]/Sqrt[a]])/a^(3/2)))/(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 = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(-\frac {2 \sqrt {b \,x^{6}+a}\, x^{6} \left (a d -\frac {3 c b}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{6}+a}}{\sqrt {a}}\right )+\left (\left (-2 d \,x^{6}+c \right ) a +3 b c \,x^{6}\right ) \sqrt {a}}{6 \sqrt {b \,x^{6}+a}\, a^{\frac {5}{2}} x^{6}}\) | \(77\) |
Input:
int((d*x^6+c)/x^7/(b*x^6+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/6/(b*x^6+a)^(1/2)*(2*(b*x^6+a)^(1/2)*x^6*(a*d-3/2*c*b)*arctanh((b*x^6+a )^(1/2)/a^(1/2))+((-2*d*x^6+c)*a+3*b*c*x^6)*a^(1/2))/a^(5/2)/x^6
Time = 0.12 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.68 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{12} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x^{6}\right )} \sqrt {a} \log \left (\frac {b x^{6} - 2 \, \sqrt {b x^{6} + a} \sqrt {a} + 2 \, a}{x^{6}}\right ) + 2 \, {\left ({\left (3 \, a b c - 2 \, a^{2} d\right )} x^{6} + a^{2} c\right )} \sqrt {b x^{6} + a}}{12 \, {\left (a^{3} b x^{12} + a^{4} x^{6}\right )}}, -\frac {{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{12} + {\left (3 \, a b c - 2 \, a^{2} d\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{6} + a}}\right ) + {\left ({\left (3 \, a b c - 2 \, a^{2} d\right )} x^{6} + a^{2} c\right )} \sqrt {b x^{6} + a}}{6 \, {\left (a^{3} b x^{12} + a^{4} x^{6}\right )}}\right ] \] Input:
integrate((d*x^6+c)/x^7/(b*x^6+a)^(3/2),x, algorithm="fricas")
Output:
[-1/12*(((3*b^2*c - 2*a*b*d)*x^12 + (3*a*b*c - 2*a^2*d)*x^6)*sqrt(a)*log(( b*x^6 - 2*sqrt(b*x^6 + a)*sqrt(a) + 2*a)/x^6) + 2*((3*a*b*c - 2*a^2*d)*x^6 + a^2*c)*sqrt(b*x^6 + a))/(a^3*b*x^12 + a^4*x^6), -1/6*(((3*b^2*c - 2*a*b *d)*x^12 + (3*a*b*c - 2*a^2*d)*x^6)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^6 + a)) + ((3*a*b*c - 2*a^2*d)*x^6 + a^2*c)*sqrt(b*x^6 + a))/(a^3*b*x^12 + a^4 *x^6)]
Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (75) = 150\).
Time = 62.68 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.08 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=c \left (- \frac {1}{6 a \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{6}} + 1}} - \frac {\sqrt {b}}{2 a^{2} x^{3} \sqrt {\frac {a}{b x^{6}} + 1}} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{3}} \right )}}{2 a^{\frac {5}{2}}}\right ) + d \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{6}}{a}}}{6 a^{\frac {9}{2}} + 6 a^{\frac {7}{2}} b x^{6}} + \frac {a^{3} \log {\left (\frac {b x^{6}}{a} \right )}}{6 a^{\frac {9}{2}} + 6 a^{\frac {7}{2}} b x^{6}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{6}}{a}} + 1 \right )}}{6 a^{\frac {9}{2}} + 6 a^{\frac {7}{2}} b x^{6}} + \frac {a^{2} b x^{6} \log {\left (\frac {b x^{6}}{a} \right )}}{6 a^{\frac {9}{2}} + 6 a^{\frac {7}{2}} b x^{6}} - \frac {2 a^{2} b x^{6} \log {\left (\sqrt {1 + \frac {b x^{6}}{a}} + 1 \right )}}{6 a^{\frac {9}{2}} + 6 a^{\frac {7}{2}} b x^{6}}\right ) \] Input:
integrate((d*x**6+c)/x**7/(b*x**6+a)**(3/2),x)
Output:
c*(-1/(6*a*sqrt(b)*x**9*sqrt(a/(b*x**6) + 1)) - sqrt(b)/(2*a**2*x**3*sqrt( a/(b*x**6) + 1)) + b*asinh(sqrt(a)/(sqrt(b)*x**3))/(2*a**(5/2))) + d*(2*a* *3*sqrt(1 + b*x**6/a)/(6*a**(9/2) + 6*a**(7/2)*b*x**6) + a**3*log(b*x**6/a )/(6*a**(9/2) + 6*a**(7/2)*b*x**6) - 2*a**3*log(sqrt(1 + b*x**6/a) + 1)/(6 *a**(9/2) + 6*a**(7/2)*b*x**6) + a**2*b*x**6*log(b*x**6/a)/(6*a**(9/2) + 6 *a**(7/2)*b*x**6) - 2*a**2*b*x**6*log(sqrt(1 + b*x**6/a) + 1)/(6*a**(9/2) + 6*a**(7/2)*b*x**6))
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (69) = 138\).
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.69 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {1}{12} \, c {\left (\frac {2 \, {\left (3 \, {\left (b x^{6} + a\right )} b - 2 \, a b\right )}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {b x^{6} + a} a^{3}} + \frac {3 \, b \log \left (\frac {\sqrt {b x^{6} + a} - \sqrt {a}}{\sqrt {b x^{6} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )} + \frac {1}{6} \, d {\left (\frac {\log \left (\frac {\sqrt {b x^{6} + a} - \sqrt {a}}{\sqrt {b x^{6} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2}{\sqrt {b x^{6} + a} a}\right )} \] Input:
integrate((d*x^6+c)/x^7/(b*x^6+a)^(3/2),x, algorithm="maxima")
Output:
-1/12*c*(2*(3*(b*x^6 + a)*b - 2*a*b)/((b*x^6 + a)^(3/2)*a^2 - sqrt(b*x^6 + a)*a^3) + 3*b*log((sqrt(b*x^6 + a) - sqrt(a))/(sqrt(b*x^6 + a) + sqrt(a)) )/a^(5/2)) + 1/6*d*(log((sqrt(b*x^6 + a) - sqrt(a))/(sqrt(b*x^6 + a) + sqr t(a)))/a^(3/2) + 2/(sqrt(b*x^6 + a)*a))
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.16 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=-\frac {{\left (3 \, b c - 2 \, a d\right )} \arctan \left (\frac {\sqrt {b x^{6} + a}}{\sqrt {-a}}\right )}{6 \, \sqrt {-a} a^{2}} - \frac {3 \, {\left (b x^{6} + a\right )} b c - 2 \, a b c - 2 \, {\left (b x^{6} + a\right )} a d + 2 \, a^{2} d}{6 \, {\left ({\left (b x^{6} + a\right )}^{\frac {3}{2}} - \sqrt {b x^{6} + a} a\right )} a^{2}} \] Input:
integrate((d*x^6+c)/x^7/(b*x^6+a)^(3/2),x, algorithm="giac")
Output:
-1/6*(3*b*c - 2*a*d)*arctan(sqrt(b*x^6 + a)/sqrt(-a))/(sqrt(-a)*a^2) - 1/6 *(3*(b*x^6 + a)*b*c - 2*a*b*c - 2*(b*x^6 + a)*a*d + 2*a^2*d)/(((b*x^6 + a) ^(3/2) - sqrt(b*x^6 + a)*a)*a^2)
Time = 4.97 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=\frac {d}{3\,a\,\sqrt {b\,x^6+a}}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^6+a}}{\sqrt {a}}\right )}{3\,a^{3/2}}-\frac {c}{6\,a\,x^6\,\sqrt {b\,x^6+a}}+\frac {b\,c\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^6+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {b\,c}{2\,a^2\,\sqrt {b\,x^6+a}} \] Input:
int((c + d*x^6)/(x^7*(a + b*x^6)^(3/2)),x)
Output:
d/(3*a*(a + b*x^6)^(1/2)) - (d*atanh((a + b*x^6)^(1/2)/a^(1/2)))/(3*a^(3/2 )) - c/(6*a*x^6*(a + b*x^6)^(1/2)) + (b*c*atanh((a + b*x^6)^(1/2)/a^(1/2)) )/(2*a^(5/2)) - (b*c)/(2*a^2*(a + b*x^6)^(1/2))
Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.98 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{3/2}} \, dx=\frac {-2 \sqrt {b \,x^{6}+a}\, a^{2} c +4 \sqrt {b \,x^{6}+a}\, a^{2} d \,x^{6}-6 \sqrt {b \,x^{6}+a}\, a b c \,x^{6}+2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a^{2} d \,x^{6}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a b c \,x^{6}+2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a b d \,x^{12}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) b^{2} c \,x^{12}-2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a^{2} d \,x^{6}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a b c \,x^{6}-2 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a b d \,x^{12}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) b^{2} c \,x^{12}}{12 a^{3} x^{6} \left (b \,x^{6}+a \right )} \] Input:
int((d*x^6+c)/x^7/(b*x^6+a)^(3/2),x)
Output:
( - 2*sqrt(a + b*x**6)*a**2*c + 4*sqrt(a + b*x**6)*a**2*d*x**6 - 6*sqrt(a + b*x**6)*a*b*c*x**6 + 2*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a**2*d*x* *6 - 3*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a*b*c*x**6 + 2*sqrt(a)*log( sqrt(a + b*x**6) - sqrt(a))*a*b*d*x**12 - 3*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*b**2*c*x**12 - 2*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*a**2*d* x**6 + 3*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*a*b*c*x**6 - 2*sqrt(a)*lo g(sqrt(a + b*x**6) + sqrt(a))*a*b*d*x**12 + 3*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*b**2*c*x**12)/(12*a**3*x**6*(a + b*x**6))