\(\int \frac {x^7 (c+d x^6)}{(a+b x^6)^{5/2}} \, dx\) [38]

Optimal result

Integrand size = 22, antiderivative size = 297 \[ \int \frac {x^7 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {(b c-a d) x^2}{9 b^2 \left (a+b x^6\right )^{3/2}}+\frac {(2 b c-11 a d) x^2}{27 a b^2 \sqrt {a+b x^6}}+\frac {2 \sqrt {2+\sqrt {3}} (b c+8 a d) \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} a b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}} \] Output:

-1/9*(-a*d+b*c)*x^2/b^2/(b*x^6+a)^(3/2)+1/27*(-11*a*d+2*b*c)*x^2/a/b^2/(b* 
x^6+a)^(1/2)+2/81*(1/2*6^(1/2)+1/2*2^(1/2))*(8*a*d+b*c)*(a^(1/3)+b^(1/3)*x 
^2)*((a^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/((1+3^(1/2))*a^(1/3)+b^(1/3 
)*x^2)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x^2)/((1+3^(1/2))*a 
^(1/3)+b^(1/3)*x^2),I*3^(1/2)+2*I)*3^(3/4)/a/b^(7/3)/(a^(1/3)*(a^(1/3)+b^( 
1/3)*x^2)/((1+3^(1/2))*a^(1/3)+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 = 10.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.34 \[ \int \frac {x^7 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {x^2 \left (-8 a^2 d+2 b^2 c x^6-a b \left (c+11 d x^6\right )+(b c+8 a d) \left (a+b x^6\right ) \sqrt {1+\frac {b x^6}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^6}{a}\right )\right )}{27 a b^2 \left (a+b x^6\right )^{3/2}} \] Input:

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

Output:

(x^2*(-8*a^2*d + 2*b^2*c*x^6 - a*b*(c + 11*d*x^6) + (b*c + 8*a*d)*(a + b*x 
^6)*Sqrt[1 + (b*x^6)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^6)/a)]))/( 
27*a*b^2*(a + b*x^6)^(3/2))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {957, 807, 817, 759}

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.

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

\(\Big \downarrow \) 957

\(\displaystyle \frac {(8 a d+b c) \int \frac {x^7}{\left (b x^6+a\right )^{3/2}}dx}{9 a b}+\frac {x^8 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {(8 a d+b c) \int \frac {x^6}{\left (b x^6+a\right )^{3/2}}dx^2}{18 a b}+\frac {x^8 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {(8 a d+b c) \left (\frac {2 \int \frac {1}{\sqrt {b x^6+a}}dx^2}{3 b}-\frac {2 x^2}{3 b \sqrt {a+b x^6}}\right )}{18 a b}+\frac {x^8 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {(8 a d+b c) \left (\frac {4 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x^2+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x^2+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x^2\right )^2}} \sqrt {a+b x^6}}-\frac {2 x^2}{3 b \sqrt {a+b x^6}}\right )}{18 a b}+\frac {x^8 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\)

Input:

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

Output:

((b*c - a*d)*x^8)/(9*a*b*(a + b*x^6)^(3/2)) + ((b*c + 8*a*d)*((-2*x^2)/(3* 
b*Sqrt[a + b*x^6]) + (4*Sqrt[2 + Sqrt[3]]*(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)/((1 + Sqrt[3])*a^(1/3) + b^(1/3 
)*x^2)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x^2)/((1 + Sqr 
t[3])*a^(1/3) + b^(1/3)*x^2)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*b^(4/3)*Sqrt[(a 
^(1/3)*(a^(1/3) + b^(1/3)*x^2))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x^2)^2]*S 
qrt[a + b*x^6])))/(18*a*b)
 

Defintions of rubi rules used

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

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 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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.49 \[ \int \frac {x^7 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (b^{3} c + 8 \, a b^{2} d\right )} x^{12} + 2 \, {\left (a b^{2} c + 8 \, a^{2} b d\right )} x^{6} + a^{2} b c + 8 \, a^{3} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x^{2}\right ) + {\left ({\left (2 \, b^{3} c - 11 \, a b^{2} d\right )} x^{8} - {\left (a b^{2} c + 8 \, a^{2} b d\right )} x^{2}\right )} \sqrt {b x^{6} + a}}{27 \, {\left (a b^{5} x^{12} + 2 \, a^{2} b^{4} x^{6} + a^{3} b^{3}\right )}} \] Input:

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

Output:

1/27*(2*((b^3*c + 8*a*b^2*d)*x^12 + 2*(a*b^2*c + 8*a^2*b*d)*x^6 + a^2*b*c 
+ 8*a^3*d)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x^2) + ((2*b^3*c - 11*a* 
b^2*d)*x^8 - (a*b^2*c + 8*a^2*b*d)*x^2)*sqrt(b*x^6 + a))/(a*b^5*x^12 + 2*a 
^2*b^4*x^6 + a^3*b^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^7 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-8 \sqrt {b \,x^{6}+a}\, a d \,x^{2}-\sqrt {b \,x^{6}+a}\, b c \,x^{2}-7 \sqrt {b \,x^{6}+a}\, b d \,x^{8}+16 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{4} d +2 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b c +32 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{3} b d \,x^{6}+4 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{2} c \,x^{6}+16 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a^{2} b^{2} d \,x^{12}+2 \left (\int \frac {\sqrt {b \,x^{6}+a}\, x}{b^{3} x^{18}+3 a \,b^{2} x^{12}+3 a^{2} b \,x^{6}+a^{3}}d x \right ) a \,b^{3} c \,x^{12}}{7 b^{2} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:

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

Output:

( - 8*sqrt(a + b*x**6)*a*d*x**2 - sqrt(a + b*x**6)*b*c*x**2 - 7*sqrt(a + b 
*x**6)*b*d*x**8 + 16*int((sqrt(a + b*x**6)*x)/(a**3 + 3*a**2*b*x**6 + 3*a* 
b**2*x**12 + b**3*x**18),x)*a**4*d + 2*int((sqrt(a + b*x**6)*x)/(a**3 + 3* 
a**2*b*x**6 + 3*a*b**2*x**12 + b**3*x**18),x)*a**3*b*c + 32*int((sqrt(a + 
b*x**6)*x)/(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((sqrt(a + b*x**6)*x)/(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 + 16*int((sqrt(a + b*x**6)*x)/(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)*x)/(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)/(7*b**2*(a**2 + 2*a*b*x**6 + b**2*x**12))