\(\int \frac {x^{14} (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [32]

Optimal result

Integrand size = 22, antiderivative size = 122 \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {a (b c-a d) x^3}{9 b^3 \left (a+b x^6\right )^{3/2}}-\frac {(4 b c-7 a d) x^3}{9 b^3 \sqrt {a+b x^6}}+\frac {d x^3 \sqrt {a+b x^6}}{6 b^3}+\frac {(2 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} x^3}{\sqrt {a+b x^6}}\right )}{6 b^{7/2}} \] Output:

1/9*a*(-a*d+b*c)*x^3/b^3/(b*x^6+a)^(3/2)-1/9*(-7*a*d+4*b*c)*x^3/b^3/(b*x^6 
+a)^(1/2)+1/6*d*x^3*(b*x^6+a)^(1/2)/b^3+1/6*(-5*a*d+2*b*c)*arctanh(b^(1/2) 
*x^3/(b*x^6+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82 \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^3 \left (-6 a b c+15 a^2 d-8 b^2 c x^6+20 a b d x^6+3 b^2 d x^{12}\right )}{18 b^3 \left (a+b x^6\right )^{3/2}}+\frac {(2 b c-5 a d) \log \left (\sqrt {b} x^3+\sqrt {a+b x^6}\right )}{6 b^{7/2}} \] Input:

Integrate[(x^14*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(x^3*(-6*a*b*c + 15*a^2*d - 8*b^2*c*x^6 + 20*a*b*d*x^6 + 3*b^2*d*x^12))/(1 
8*b^3*(a + b*x^6)^(3/2)) + ((2*b*c - 5*a*d)*Log[Sqrt[b]*x^3 + Sqrt[a + b*x 
^6]])/(6*b^(7/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 807, 252, 252, 224, 219}

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.

\(\displaystyle \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(2 b c-5 a d) \int \frac {x^{14}}{\left (b x^6+a\right )^{5/2}}dx}{2 b}+\frac {d x^{15}}{6 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(2 b c-5 a d) \int \frac {x^{12}}{\left (b x^6+a\right )^{5/2}}dx^3}{6 b}+\frac {d x^{15}}{6 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {(2 b c-5 a d) \left (\frac {\int \frac {x^6}{\left (b x^6+a\right )^{3/2}}dx^3}{b}-\frac {x^9}{3 b \left (a+b x^6\right )^{3/2}}\right )}{6 b}+\frac {d x^{15}}{6 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {(2 b c-5 a d) \left (\frac {\frac {\int \frac {1}{\sqrt {b x^6+a}}dx^3}{b}-\frac {x^3}{b \sqrt {a+b x^6}}}{b}-\frac {x^9}{3 b \left (a+b x^6\right )^{3/2}}\right )}{6 b}+\frac {d x^{15}}{6 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {(2 b c-5 a d) \left (\frac {\frac {\int \frac {1}{1-b x^6}d\frac {x^3}{\sqrt {b x^6+a}}}{b}-\frac {x^3}{b \sqrt {a+b x^6}}}{b}-\frac {x^9}{3 b \left (a+b x^6\right )^{3/2}}\right )}{6 b}+\frac {d x^{15}}{6 b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x^3}{\sqrt {a+b x^6}}\right )}{b^{3/2}}-\frac {x^3}{b \sqrt {a+b x^6}}}{b}-\frac {x^9}{3 b \left (a+b x^6\right )^{3/2}}\right ) (2 b c-5 a d)}{6 b}+\frac {d x^{15}}{6 b \left (a+b x^6\right )^{3/2}}\)

Input:

Int[(x^14*(c + d*x^6))/(a + b*x^6)^(5/2),x]
 

Output:

(d*x^15)/(6*b*(a + b*x^6)^(3/2)) + ((2*b*c - 5*a*d)*(-1/3*x^9/(b*(a + b*x^ 
6)^(3/2)) + (-(x^3/(b*Sqrt[a + b*x^6])) + ArcTanh[(Sqrt[b]*x^3)/Sqrt[a + b 
*x^6]]/b^(3/2))/b))/(6*b)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

Maple [A] (verified)

Time = 9.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\frac {x^{3} a \left (-10 d \,x^{6}+3 c \right ) b^{\frac {3}{2}}+\frac {x^{9} \left (-3 d \,x^{6}+8 c \right ) b^{\frac {5}{2}}}{2}-\frac {15 \sqrt {b}\, a^{2} d \,x^{3}}{2}+\frac {3 \left (5 a d -2 c b \right ) \left (b \,x^{6}+a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{6}+a}}{x^{3} \sqrt {b}}\right )}{2}}{9 \left (b \,x^{6}+a \right )^{\frac {3}{2}} b^{\frac {7}{2}}}\)	\(101\)

Input:

int(x^14*(d*x^6+c)/(b*x^6+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/9*(x^3*a*(-10*d*x^6+3*c)*b^(3/2)+1/2*x^9*(-3*d*x^6+8*c)*b^(5/2)-15/2*b^ 
(1/2)*a^2*d*x^3+3/2*(5*a*d-2*b*c)*(b*x^6+a)^(3/2)*arctanh((b*x^6+a)^(1/2)/ 
x^3/b^(1/2)))/(b*x^6+a)^(3/2)/b^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.80 \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{12} + 2 \, {\left (2 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{6} + 2 \, a^{2} b c - 5 \, a^{3} d\right )} \sqrt {b} \log \left (-2 \, b x^{6} + 2 \, \sqrt {b x^{6} + a} \sqrt {b} x^{3} - a\right ) - 2 \, {\left (3 \, b^{3} d x^{15} - 4 \, {\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{9} - 3 \, {\left (2 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{3}\right )} \sqrt {b x^{6} + a}}{36 \, {\left (b^{6} x^{12} + 2 \, a b^{5} x^{6} + a^{2} b^{4}\right )}}, -\frac {3 \, {\left ({\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{12} + 2 \, {\left (2 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{6} + 2 \, a^{2} b c - 5 \, a^{3} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{3}}{\sqrt {b x^{6} + a}}\right ) - {\left (3 \, b^{3} d x^{15} - 4 \, {\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} x^{9} - 3 \, {\left (2 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{3}\right )} \sqrt {b x^{6} + a}}{18 \, {\left (b^{6} x^{12} + 2 \, a b^{5} x^{6} + a^{2} b^{4}\right )}}\right ] \] Input:

integrate(x^14*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/36*(3*((2*b^3*c - 5*a*b^2*d)*x^12 + 2*(2*a*b^2*c - 5*a^2*b*d)*x^6 + 2* 
a^2*b*c - 5*a^3*d)*sqrt(b)*log(-2*b*x^6 + 2*sqrt(b*x^6 + a)*sqrt(b)*x^3 - 
a) - 2*(3*b^3*d*x^15 - 4*(2*b^3*c - 5*a*b^2*d)*x^9 - 3*(2*a*b^2*c - 5*a^2* 
b*d)*x^3)*sqrt(b*x^6 + a))/(b^6*x^12 + 2*a*b^5*x^6 + a^2*b^4), -1/18*(3*(( 
2*b^3*c - 5*a*b^2*d)*x^12 + 2*(2*a*b^2*c - 5*a^2*b*d)*x^6 + 2*a^2*b*c - 5* 
a^3*d)*sqrt(-b)*arctan(sqrt(-b)*x^3/sqrt(b*x^6 + a)) - (3*b^3*d*x^15 - 4*( 
2*b^3*c - 5*a*b^2*d)*x^9 - 3*(2*a*b^2*c - 5*a^2*b*d)*x^3)*sqrt(b*x^6 + a)) 
/(b^6*x^12 + 2*a*b^5*x^6 + a^2*b^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**14*(d*x**6+c)/(b*x**6+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (102) = 204\).

Time = 0.11 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.68 \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {1}{18} \, {\left (\frac {2 \, {\left (b + \frac {3 \, {\left (b x^{6} + a\right )}}{x^{6}}\right )} x^{9}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{6} + a}}{x^{3}}}{\sqrt {b} + \frac {\sqrt {b x^{6} + a}}{x^{3}}}\right )}{b^{\frac {5}{2}}}\right )} c + \frac {1}{36} \, d {\left (\frac {2 \, {\left (2 \, a b^{2} + \frac {10 \, {\left (b x^{6} + a\right )} a b}{x^{6}} - \frac {15 \, {\left (b x^{6} + a\right )}^{2} a}{x^{12}}\right )}}{\frac {{\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{4}}{x^{9}} - \frac {{\left (b x^{6} + a\right )}^{\frac {5}{2}} b^{3}}{x^{15}}} + \frac {15 \, a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{6} + a}}{x^{3}}}{\sqrt {b} + \frac {\sqrt {b x^{6} + a}}{x^{3}}}\right )}{b^{\frac {7}{2}}}\right )} \] Input:

integrate(x^14*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
 

Output:

-1/18*(2*(b + 3*(b*x^6 + a)/x^6)*x^9/((b*x^6 + a)^(3/2)*b^2) + 3*log(-(sqr 
t(b) - sqrt(b*x^6 + a)/x^3)/(sqrt(b) + sqrt(b*x^6 + a)/x^3))/b^(5/2))*c + 
1/36*d*(2*(2*a*b^2 + 10*(b*x^6 + a)*a*b/x^6 - 15*(b*x^6 + a)^2*a/x^12)/((b 
*x^6 + a)^(3/2)*b^4/x^9 - (b*x^6 + a)^(5/2)*b^3/x^15) + 15*a*log(-(sqrt(b) 
 - sqrt(b*x^6 + a)/x^3)/(sqrt(b) + sqrt(b*x^6 + a)/x^3))/b^(7/2))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Exception raised: NotImplementedError} \] Input:

integrate(x^14*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: Recu 
rsive assumption sageVARa>=(-sageVARb/t_nostep^6) ignored2*((86093442*sage 
VARb^10*sageVARd*sageVARa^5*1/1033121304/sageVARb^11/sageVARa^5*sageVARx*s 
ageVARx*sageVARx*sa
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^{14}\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:

int((x^14*(c + d*x^6))/(a + b*x^6)^(5/2),x)
 

Output:

int((x^14*(c + d*x^6))/(a + b*x^6)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.61 \[ \int \frac {x^{14} \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {30 \sqrt {b \,x^{6}+a}\, a^{2} b d \,x^{3}-12 \sqrt {b \,x^{6}+a}\, a \,b^{2} c \,x^{3}+40 \sqrt {b \,x^{6}+a}\, a \,b^{2} d \,x^{9}-16 \sqrt {b \,x^{6}+a}\, b^{3} c \,x^{9}+6 \sqrt {b \,x^{6}+a}\, b^{3} d \,x^{15}+15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a^{3} d -6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a^{2} b c +30 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a^{2} b d \,x^{6}-12 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a \,b^{2} c \,x^{6}+15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a \,b^{2} d \,x^{12}-6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) b^{3} c \,x^{12}-15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a^{3} d +6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a^{2} b c -30 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a^{2} b d \,x^{6}+12 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a \,b^{2} c \,x^{6}-15 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a \,b^{2} d \,x^{12}+6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) b^{3} c \,x^{12}}{36 b^{4} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

int(x^14*(d*x^6+c)/(b*x^6+a)^(5/2),x)
 

Output:

(30*sqrt(a + b*x**6)*a**2*b*d*x**3 - 12*sqrt(a + b*x**6)*a*b**2*c*x**3 + 4 
0*sqrt(a + b*x**6)*a*b**2*d*x**9 - 16*sqrt(a + b*x**6)*b**3*c*x**9 + 6*sqr 
t(a + b*x**6)*b**3*d*x**15 + 15*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x** 
3)*a**3*d - 6*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x**3)*a**2*b*c + 30*s 
qrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x**3)*a**2*b*d*x**6 - 12*sqrt(b)*log 
(sqrt(a + b*x**6) - sqrt(b)*x**3)*a*b**2*c*x**6 + 15*sqrt(b)*log(sqrt(a + 
b*x**6) - sqrt(b)*x**3)*a*b**2*d*x**12 - 6*sqrt(b)*log(sqrt(a + b*x**6) - 
sqrt(b)*x**3)*b**3*c*x**12 - 15*sqrt(b)*log(sqrt(a + b*x**6) + sqrt(b)*x** 
3)*a**3*d + 6*sqrt(b)*log(sqrt(a + b*x**6) + sqrt(b)*x**3)*a**2*b*c - 30*s 
qrt(b)*log(sqrt(a + b*x**6) + sqrt(b)*x**3)*a**2*b*d*x**6 + 12*sqrt(b)*log 
(sqrt(a + b*x**6) + sqrt(b)*x**3)*a*b**2*c*x**6 - 15*sqrt(b)*log(sqrt(a + 
b*x**6) + sqrt(b)*x**3)*a*b**2*d*x**12 + 6*sqrt(b)*log(sqrt(a + b*x**6) + 
sqrt(b)*x**3)*b**3*c*x**12)/(36*b**4*(a**2 + 2*a*b*x**6 + b**2*x**12))