\(\int \frac {c+d x^6}{x^7 (a+b x^6)^{5/2}} \, dx\) [31]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=-\frac {b c-a d}{9 a^2 \left (a+b x^6\right )^{3/2}}-\frac {2 b c-a d}{3 a^3 \sqrt {a+b x^6}}-\frac {c \sqrt {a+b x^6}}{6 a^3 x^6}+\frac {(5 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{6 a^{7/2}} \] Output:

-1/9*(-a*d+b*c)/a^2/(b*x^6+a)^(3/2)-1/3*(-a*d+2*b*c)/a^3/(b*x^6+a)^(1/2)-1 
/6*c*(b*x^6+a)^(1/2)/a^3/x^6+1/6*(-2*a*d+5*b*c)*arctanh((b*x^6+a)^(1/2)/a^ 
(1/2))/a^(7/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=\frac {-3 a^2 c-20 a b c x^6+8 a^2 d x^6-15 b^2 c x^{12}+6 a b d x^{12}}{18 a^3 x^6 \left (a+b x^6\right )^{3/2}}+\frac {(5 b c-2 a d) \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{6 a^{7/2}} \] Input:

Integrate[(c + d*x^6)/(x^7*(a + b*x^6)^(5/2)),x]
 

Output:

(-3*a^2*c - 20*a*b*c*x^6 + 8*a^2*d*x^6 - 15*b^2*c*x^12 + 6*a*b*d*x^12)/(18 
*a^3*x^6*(a + b*x^6)^(3/2)) + ((5*b*c - 2*a*d)*ArcTanh[Sqrt[a + b*x^6]/Sqr 
t[a]])/(6*a^(7/2))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {948, 87, 61, 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.

\(\displaystyle \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 948

\(\displaystyle \frac {1}{6} \int \frac {d x^6+c}{x^{12} \left (b x^6+a\right )^{5/2}}dx^6\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{6} \left (-\frac {(5 b c-2 a d) \int \frac {1}{x^6 \left (b x^6+a\right )^{5/2}}dx^6}{2 a}-\frac {c}{a x^6 \left (a+b x^6\right )^{3/2}}\right )\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{6} \left (-\frac {(5 b c-2 a d) \left (\frac {\int \frac {1}{x^6 \left (b x^6+a\right )^{3/2}}dx^6}{a}+\frac {2}{3 a \left (a+b x^6\right )^{3/2}}\right )}{2 a}-\frac {c}{a x^6 \left (a+b x^6\right )^{3/2}}\right )\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {1}{6} \left (-\frac {(5 b c-2 a d) \left (\frac {\frac {\int \frac {1}{x^6 \sqrt {b x^6+a}}dx^6}{a}+\frac {2}{a \sqrt {a+b x^6}}}{a}+\frac {2}{3 a \left (a+b x^6\right )^{3/2}}\right )}{2 a}-\frac {c}{a x^6 \left (a+b x^6\right )^{3/2}}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} \left (-\frac {(5 b c-2 a d) \left (\frac {\frac {2 \int \frac {1}{\frac {x^{12}}{b}-\frac {a}{b}}d\sqrt {b x^6+a}}{a b}+\frac {2}{a \sqrt {a+b x^6}}}{a}+\frac {2}{3 a \left (a+b x^6\right )^{3/2}}\right )}{2 a}-\frac {c}{a x^6 \left (a+b x^6\right )^{3/2}}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{6} \left (-\frac {\left (\frac {\frac {2}{a \sqrt {a+b x^6}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x^6}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a \left (a+b x^6\right )^{3/2}}\right ) (5 b c-2 a d)}{2 a}-\frac {c}{a x^6 \left (a+b x^6\right )^{3/2}}\right )\)

Input:

Int[(c + d*x^6)/(x^7*(a + b*x^6)^(5/2)),x]
 

Output:

(-(c/(a*x^6*(a + b*x^6)^(3/2))) - ((5*b*c - 2*a*d)*(2/(3*a*(a + b*x^6)^(3/ 
2)) + (2/(a*Sqrt[a + b*x^6]) - (2*ArcTanh[Sqrt[a + b*x^6]/Sqrt[a]])/a^(3/2 
))/a))/(2*a))/6
 

Defintions of rubi rules used

rule 61

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]
 

rule 73

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]
 

rule 87

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]))))
 

rule 221

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]
 

rule 948

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]]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(-\frac {\left (b \,x^{6}+a \right )^{\frac {3}{2}} x^{6} \left (a d -\frac {5 c b}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{6}+a}}{\sqrt {a}}\right )+\frac {10 \left (-\frac {3 d \,x^{6}}{10}+c \right ) b \,x^{6} a^{\frac {3}{2}}}{3}+\left (-\frac {4 d \,x^{6}}{3}+\frac {c}{2}\right ) a^{\frac {5}{2}}+\frac {5 \sqrt {a}\, b^{2} c \,x^{12}}{2}}{3 a^{\frac {7}{2}} \left (b \,x^{6}+a \right )^{\frac {3}{2}} x^{6}}\)	\(97\)

Input:

int((d*x^6+c)/x^7/(b*x^6+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3/a^(7/2)/(b*x^6+a)^(3/2)*((b*x^6+a)^(3/2)*x^6*(a*d-5/2*c*b)*arctanh((b 
*x^6+a)^(1/2)/a^(1/2))+10/3*(-3/10*d*x^6+c)*b*x^6*a^(3/2)+(-4/3*d*x^6+1/2* 
c)*a^(5/2)+5/2*a^(1/2)*b^2*c*x^12)/x^6
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 348, normalized size of antiderivative = 3.11 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{3} c - 2 \, a b^{2} d\right )} x^{18} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{12} + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{6}\right )} \sqrt {a} \log \left (\frac {b x^{6} - 2 \, \sqrt {b x^{6} + a} \sqrt {a} + 2 \, a}{x^{6}}\right ) + 2 \, {\left (3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{12} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{6} + 3 \, a^{3} c\right )} \sqrt {b x^{6} + a}}{36 \, {\left (a^{4} b^{2} x^{18} + 2 \, a^{5} b x^{12} + a^{6} x^{6}\right )}}, -\frac {3 \, {\left ({\left (5 \, b^{3} c - 2 \, a b^{2} d\right )} x^{18} + 2 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{12} + {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{6} + a}}\right ) + {\left (3 \, {\left (5 \, a b^{2} c - 2 \, a^{2} b d\right )} x^{12} + 4 \, {\left (5 \, a^{2} b c - 2 \, a^{3} d\right )} x^{6} + 3 \, a^{3} c\right )} \sqrt {b x^{6} + a}}{18 \, {\left (a^{4} b^{2} x^{18} + 2 \, a^{5} b x^{12} + a^{6} x^{6}\right )}}\right ] \] Input:

integrate((d*x^6+c)/x^7/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/36*(3*((5*b^3*c - 2*a*b^2*d)*x^18 + 2*(5*a*b^2*c - 2*a^2*b*d)*x^12 + ( 
5*a^2*b*c - 2*a^3*d)*x^6)*sqrt(a)*log((b*x^6 - 2*sqrt(b*x^6 + a)*sqrt(a) + 
 2*a)/x^6) + 2*(3*(5*a*b^2*c - 2*a^2*b*d)*x^12 + 4*(5*a^2*b*c - 2*a^3*d)*x 
^6 + 3*a^3*c)*sqrt(b*x^6 + a))/(a^4*b^2*x^18 + 2*a^5*b*x^12 + a^6*x^6), -1 
/18*(3*((5*b^3*c - 2*a*b^2*d)*x^18 + 2*(5*a*b^2*c - 2*a^2*b*d)*x^12 + (5*a 
^2*b*c - 2*a^3*d)*x^6)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^6 + a)) + (3*(5*a 
*b^2*c - 2*a^2*b*d)*x^12 + 4*(5*a^2*b*c - 2*a^3*d)*x^6 + 3*a^3*c)*sqrt(b*x 
^6 + a))/(a^4*b^2*x^18 + 2*a^5*b*x^12 + a^6*x^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x**6+c)/x**7/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.52 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=-\frac {1}{36} \, c {\left (\frac {2 \, {\left (15 \, {\left (b x^{6} + a\right )}^{2} b - 10 \, {\left (b x^{6} + a\right )} a b - 2 \, a^{2} b\right )}}{{\left (b x^{6} + a\right )}^{\frac {5}{2}} a^{3} - {\left (b x^{6} + a\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b \log \left (\frac {\sqrt {b x^{6} + a} - \sqrt {a}}{\sqrt {b x^{6} + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}}}\right )} + \frac {1}{18} \, d {\left (\frac {3 \, \log \left (\frac {\sqrt {b x^{6} + a} - \sqrt {a}}{\sqrt {b x^{6} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x^{6} + 4 \, a\right )}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} a^{2}}\right )} \] Input:

integrate((d*x^6+c)/x^7/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

-1/36*c*(2*(15*(b*x^6 + a)^2*b - 10*(b*x^6 + a)*a*b - 2*a^2*b)/((b*x^6 + a 
)^(5/2)*a^3 - (b*x^6 + a)^(3/2)*a^4) + 15*b*log((sqrt(b*x^6 + a) - sqrt(a) 
)/(sqrt(b*x^6 + a) + sqrt(a)))/a^(7/2)) + 1/18*d*(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))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=-\frac {{\left (5 \, b c - 2 \, a d\right )} \arctan \left (\frac {\sqrt {b x^{6} + a}}{\sqrt {-a}}\right )}{6 \, \sqrt {-a} a^{3}} - \frac {6 \, {\left (b x^{6} + a\right )} b c + a b c - 3 \, {\left (b x^{6} + a\right )} a d - a^{2} d}{9 \, {\left (b x^{6} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {\sqrt {b x^{6} + a} c}{6 \, a^{3} x^{6}} \] Input:

integrate((d*x^6+c)/x^7/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

-1/6*(5*b*c - 2*a*d)*arctan(sqrt(b*x^6 + a)/sqrt(-a))/(sqrt(-a)*a^3) - 1/9 
*(6*(b*x^6 + a)*b*c + a*b*c - 3*(b*x^6 + a)*a*d - a^2*d)/((b*x^6 + a)^(3/2 
)*a^3) - 1/6*sqrt(b*x^6 + a)*c/(a^3*x^6)
 

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.13 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=\frac {\frac {d}{3\,a}+\frac {d\,\left (b\,x^6+a\right )}{a^2}}{3\,{\left (b\,x^6+a\right )}^{3/2}}-\frac {d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^6+a}}{\sqrt {a}}\right )}{3\,a^{5/2}}-\frac {c}{6\,a\,x^6\,{\left (b\,x^6+a\right )}^{3/2}}+\frac {5\,b\,c\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^6+a}}{\sqrt {a}}\right )}{6\,a^{7/2}}-\frac {10\,b\,c}{9\,a^2\,{\left (b\,x^6+a\right )}^{3/2}}-\frac {5\,b^2\,c\,x^6}{6\,a^3\,{\left (b\,x^6+a\right )}^{3/2}} \] Input:

int((c + d*x^6)/(x^7*(a + b*x^6)^(5/2)),x)
 

Output:

(d/(3*a) + (d*(a + b*x^6))/a^2)/(3*(a + b*x^6)^(3/2)) - (d*atanh((a + b*x^ 
6)^(1/2)/a^(1/2)))/(3*a^(5/2)) - c/(6*a*x^6*(a + b*x^6)^(3/2)) + (5*b*c*at 
anh((a + b*x^6)^(1/2)/a^(1/2)))/(6*a^(7/2)) - (10*b*c)/(9*a^2*(a + b*x^6)^ 
(3/2)) - (5*b^2*c*x^6)/(6*a^3*(a + b*x^6)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.66 \[ \int \frac {c+d x^6}{x^7 \left (a+b x^6\right )^{5/2}} \, dx=\frac {-6 \sqrt {b \,x^{6}+a}\, a^{3} c +16 \sqrt {b \,x^{6}+a}\, a^{3} d \,x^{6}-40 \sqrt {b \,x^{6}+a}\, a^{2} b c \,x^{6}+12 \sqrt {b \,x^{6}+a}\, a^{2} b d \,x^{12}-30 \sqrt {b \,x^{6}+a}\, a \,b^{2} c \,x^{12}+6 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a^{3} d \,x^{6}-15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a^{2} b c \,x^{6}+12 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a^{2} b d \,x^{12}-30 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a \,b^{2} c \,x^{12}+6 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) a \,b^{2} d \,x^{18}-15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {a}\right ) b^{3} c \,x^{18}-6 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a^{3} d \,x^{6}+15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a^{2} b c \,x^{6}-12 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a^{2} b d \,x^{12}+30 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a \,b^{2} c \,x^{12}-6 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) a \,b^{2} d \,x^{18}+15 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {a}\right ) b^{3} c \,x^{18}}{36 a^{4} x^{6} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int((d*x^6+c)/x^7/(b*x^6+a)^(5/2),x)
 

Output:

( - 6*sqrt(a + b*x**6)*a**3*c + 16*sqrt(a + b*x**6)*a**3*d*x**6 - 40*sqrt( 
a + b*x**6)*a**2*b*c*x**6 + 12*sqrt(a + b*x**6)*a**2*b*d*x**12 - 30*sqrt(a 
 + b*x**6)*a*b**2*c*x**12 + 6*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a**3 
*d*x**6 - 15*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a**2*b*c*x**6 + 12*sq 
rt(a)*log(sqrt(a + b*x**6) - sqrt(a))*a**2*b*d*x**12 - 30*sqrt(a)*log(sqrt 
(a + b*x**6) - sqrt(a))*a*b**2*c*x**12 + 6*sqrt(a)*log(sqrt(a + b*x**6) - 
sqrt(a))*a*b**2*d*x**18 - 15*sqrt(a)*log(sqrt(a + b*x**6) - sqrt(a))*b**3* 
c*x**18 - 6*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*a**3*d*x**6 + 15*sqrt( 
a)*log(sqrt(a + b*x**6) + sqrt(a))*a**2*b*c*x**6 - 12*sqrt(a)*log(sqrt(a + 
 b*x**6) + sqrt(a))*a**2*b*d*x**12 + 30*sqrt(a)*log(sqrt(a + b*x**6) + sqr 
t(a))*a*b**2*c*x**12 - 6*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*a*b**2*d* 
x**18 + 15*sqrt(a)*log(sqrt(a + b*x**6) + sqrt(a))*b**3*c*x**18)/(36*a**4* 
x**6*(a**2 + 2*a*b*x**6 + b**2*x**12))