\(\int \frac {x^6 (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [43]

Optimal result

Integrand size = 22, antiderivative size = 284 \[ \int \frac {x^6 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {(b c-a d) x}{9 b^2 \left (a+b x^6\right )^{3/2}}+\frac {(b c-10 a d) x}{27 a b^2 \sqrt {a+b x^6}}+\frac {(2 b c+7 a d) x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{54 \sqrt [4]{3} a^{4/3} b^2 \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/9*(-a*d+b*c)*x/b^2/(b*x^6+a)^(3/2)+1/27*(-10*a*d+b*c)*x/a/b^2/(b*x^6+a) 
^(1/2)+1/162*(7*a*d+2*b*c)*x*(a^(1/3)+b^(1/3)*x^2)*((a^(2/3)-a^(1/3)*b^(1/ 
3)*x^2+b^(2/3)*x^4)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2)*InverseJaco 
biAM(arccos((a^(1/3)+(1-3^(1/2))*b^(1/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3) 
*x^2)),1/4*6^(1/2)+1/4*2^(1/2))*3^(3/4)/a^(4/3)/b^2/(b^(1/3)*x^2*(a^(1/3)+ 
b^(1/3)*x^2)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x^2)^2)^(1/2)/(b*x^6+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.36 \[ \int \frac {x^6 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-7 a^2 d x+b^2 c x^7-2 a b x \left (c+5 d x^6\right )+(2 b c+7 a d) x \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^6}{a}\right )}{27 a b^2 \left (a+b x^6\right )^{3/2}} \] Input:

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

Output:

(-7*a^2*d*x + b^2*c*x^7 - 2*a*b*x*(c + 5*d*x^6) + (2*b*c + 7*a*d)*x*(a + b 
*x^6)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[1/6, 1/2, 7/6, -((b*x^6)/a)])/ 
(27*a*b^2*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {957, 817, 766}

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.

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

\(\Big \downarrow \) 957

\(\displaystyle \frac {(7 a d+2 b c) \int \frac {x^6}{\left (b x^6+a\right )^{3/2}}dx}{9 a b}+\frac {x^7 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(7 a d+2 b c) \left (\frac {\int \frac {1}{\sqrt {b x^6+a}}dx}{3 b}-\frac {x}{3 b \sqrt {a+b x^6}}\right )}{9 a b}+\frac {x^7 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {(7 a d+2 b c) \left (\frac {x \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2+\sqrt [3]{a}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{6 \sqrt [4]{3} \sqrt [3]{a} b \sqrt {\frac {\sqrt [3]{b} x^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {x}{3 b \sqrt {a+b x^6}}\right )}{9 a b}+\frac {x^7 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

Input:

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

Output:

((b*c - a*d)*x^7)/(9*a*b*(a + b*x^6)^(3/2)) + ((2*b*c + 7*a*d)*(-1/3*x/(b* 
Sqrt[a + b*x^6]) + (x*(a^(1/3) + b^(1/3)*x^2)*Sqrt[(a^(2/3) - a^(1/3)*b^(1 
/3)*x^2 + b^(2/3)*x^4)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*EllipticF[ 
ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1 
/3)*x^2)], (2 + Sqrt[3])/4])/(6*3^(1/4)*a^(1/3)*b*Sqrt[(b^(1/3)*x^2*(a^(1/ 
3) + b^(1/3)*x^2))/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x^2)^2]*Sqrt[a + b*x^6 
])))/(9*a*b)
 

Defintions of rubi rules used

rule 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 817

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

rule 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

Maple [F]

\[\int \frac {x^{6} \left (d \,x^{6}+c \right )}{\left (b \,x^{6}+a \right )^{\frac {5}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x^12 + c*x^6)*sqrt(b*x^6 + a)/(b^3*x^18 + 3*a*b^2*x^12 + 3*a^2 
*b*x^6 + a^3), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^6 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-7 \sqrt {b \,x^{6}+a}\, a d x -2 \sqrt {b \,x^{6}+a}\, b c x -8 \sqrt {b \,x^{6}+a}\, b d \,x^{7}+7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} d +2 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b c +14 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b d \,x^{6}+4 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{2} c \,x^{6}+7 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{2} d \,x^{12}+2 \left (\int \frac {\sqrt {b \,x^{6}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{3} c \,x^{12}}{16 b^{2} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

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

Output:

( - 7*sqrt(a + b*x**6)*a*d*x - 2*sqrt(a + b*x**6)*b*c*x - 8*sqrt(a + b*x** 
6)*b*d*x**7 + 7*int(sqrt(a + b*x**6)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**1 
2 + b**3*x**18),x)*a**4*d + 2*int(sqrt(a + b*x**6)/(a**3 + 3*a**2*b*x**6 + 
 3*a*b**2*x**12 + b**3*x**18),x)*a**3*b*c + 14*int(sqrt(a + b*x**6)/(a**3 
+ 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**3*b*d*x**6 + 4*int(sq 
rt(a + b*x**6)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a** 
2*b**2*c*x**6 + 7*int(sqrt(a + b*x**6)/(a**3 + 3*a**2*b*x**6 + 3*a*b**2*x* 
*12 + b**3*x**18),x)*a**2*b**2*d*x**12 + 2*int(sqrt(a + b*x**6)/(a**3 + 3* 
a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a*b**3*c*x**12)/(16*b**2*(a* 
*2 + 2*a*b*x**6 + b**2*x**12))