\(\int \frac {x^8 (c+d x^8)}{(a+b x^8)^{5/4}} \, dx\) [136]

Optimal result

Integrand size = 22, antiderivative size = 89 \[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\frac {d x^9}{7 b \sqrt [4]{a+b x^8}}+\frac {(7 b c-9 a d) x^9 \sqrt [4]{1+\frac {b x^8}{a}} \operatorname {Hypergeometric2F1}\left (\frac {9}{8},\frac {5}{4},\frac {17}{8},-\frac {b x^8}{a}\right )}{63 a b \sqrt [4]{a+b x^8}} \] Output:

1/7*d*x^9/b/(b*x^8+a)^(1/4)+1/63*(-9*a*d+7*b*c)*x^9*(1+b*x^8/a)^(1/4)*hype 
rgeom([9/8, 5/4],[17/8],-b*x^8/a)/a/b/(b*x^8+a)^(1/4)
 

Mathematica [A] (verified)

Time = 10.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\frac {x^9 \sqrt [4]{1+\frac {b x^8}{a}} \left (17 c \operatorname {Hypergeometric2F1}\left (\frac {9}{8},\frac {5}{4},\frac {17}{8},-\frac {b x^8}{a}\right )+9 d x^8 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {17}{8},\frac {25}{8},-\frac {b x^8}{a}\right )\right )}{153 a \sqrt [4]{a+b x^8}} \] Input:

Integrate[(x^8*(c + d*x^8))/(a + b*x^8)^(5/4),x]
 

Output:

(x^9*(1 + (b*x^8)/a)^(1/4)*(17*c*Hypergeometric2F1[9/8, 5/4, 17/8, -((b*x^ 
8)/a)] + 9*d*x^8*Hypergeometric2F1[5/4, 17/8, 25/8, -((b*x^8)/a)]))/(153*a 
*(a + b*x^8)^(1/4))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 889, 888}

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.

\(\displaystyle \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {(7 b c-9 a d) \int \frac {x^8}{\left (b x^8+a\right )^{5/4}}dx}{7 b}+\frac {d x^9}{7 b \sqrt [4]{a+b x^8}}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {\sqrt [4]{\frac {b x^8}{a}+1} (7 b c-9 a d) \int \frac {x^8}{\left (\frac {b x^8}{a}+1\right )^{5/4}}dx}{7 a b \sqrt [4]{a+b x^8}}+\frac {d x^9}{7 b \sqrt [4]{a+b x^8}}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^9 \sqrt [4]{\frac {b x^8}{a}+1} (7 b c-9 a d) \operatorname {Hypergeometric2F1}\left (\frac {9}{8},\frac {5}{4},\frac {17}{8},-\frac {b x^8}{a}\right )}{63 a b \sqrt [4]{a+b x^8}}+\frac {d x^9}{7 b \sqrt [4]{a+b x^8}}\)

Input:

Int[(x^8*(c + d*x^8))/(a + b*x^8)^(5/4),x]
 

Output:

(d*x^9)/(7*b*(a + b*x^8)^(1/4)) + ((7*b*c - 9*a*d)*x^9*(1 + (b*x^8)/a)^(1/ 
4)*Hypergeometric2F1[9/8, 5/4, 17/8, -((b*x^8)/a)])/(63*a*b*(a + b*x^8)^(1 
/4))
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

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

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 [F]

\[\int \frac {x^{8} \left (d \,x^{8}+c \right )}{\left (b \,x^{8}+a \right )^{\frac {5}{4}}}d x\]

Input:

int(x^8*(d*x^8+c)/(b*x^8+a)^(5/4),x)
 

Output:

int(x^8*(d*x^8+c)/(b*x^8+a)^(5/4),x)
 

Fricas [F]

\[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{8} + c\right )} x^{8}}{{\left (b x^{8} + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate(x^8*(d*x^8+c)/(b*x^8+a)^(5/4),x, algorithm="fricas")
 

Output:

integral((d*x^16 + c*x^8)*(b*x^8 + a)^(3/4)/(b^2*x^16 + 2*a*b*x^8 + a^2), 
x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 51.88 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\frac {c x^{9} \Gamma \left (\frac {9}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{8}, \frac {5}{4} \\ \frac {17}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 a^{\frac {5}{4}} \Gamma \left (\frac {17}{8}\right )} + \frac {d x^{17} \Gamma \left (\frac {17}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {17}{8} \\ \frac {25}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 a^{\frac {5}{4}} \Gamma \left (\frac {25}{8}\right )} \] Input:

integrate(x**8*(d*x**8+c)/(b*x**8+a)**(5/4),x)
 

Output:

c*x**9*gamma(9/8)*hyper((9/8, 5/4), (17/8,), b*x**8*exp_polar(I*pi)/a)/(8* 
a**(5/4)*gamma(17/8)) + d*x**17*gamma(17/8)*hyper((5/4, 17/8), (25/8,), b* 
x**8*exp_polar(I*pi)/a)/(8*a**(5/4)*gamma(25/8))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{8} + c\right )} x^{8}}{{\left (b x^{8} + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate(x^8*(d*x^8+c)/(b*x^8+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate((d*x^8 + c)*x^8/(b*x^8 + a)^(5/4), x)
 

Giac [F]

\[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\int { \frac {{\left (d x^{8} + c\right )} x^{8}}{{\left (b x^{8} + a\right )}^{\frac {5}{4}}} \,d x } \] Input:

integrate(x^8*(d*x^8+c)/(b*x^8+a)^(5/4),x, algorithm="giac")
 

Output:

integrate((d*x^8 + c)*x^8/(b*x^8 + a)^(5/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\int \frac {x^8\,\left (d\,x^8+c\right )}{{\left (b\,x^8+a\right )}^{5/4}} \,d x \] Input:

int((x^8*(c + d*x^8))/(a + b*x^8)^(5/4),x)
 

Output:

int((x^8*(c + d*x^8))/(a + b*x^8)^(5/4), x)
 

Reduce [F]

\[ \int \frac {x^8 \left (c+d x^8\right )}{\left (a+b x^8\right )^{5/4}} \, dx=\left (\int \frac {x^{16}}{\left (b \,x^{8}+a \right )^{\frac {1}{4}} a +\left (b \,x^{8}+a \right )^{\frac {1}{4}} b \,x^{8}}d x \right ) d +\left (\int \frac {x^{8}}{\left (b \,x^{8}+a \right )^{\frac {1}{4}} a +\left (b \,x^{8}+a \right )^{\frac {1}{4}} b \,x^{8}}d x \right ) c \] Input:

int(x^8*(d*x^8+c)/(b*x^8+a)^(5/4),x)
 

Output:

int(x**16/((a + b*x**8)**(1/4)*a + (a + b*x**8)**(1/4)*b*x**8),x)*d + int( 
x**8/((a + b*x**8)**(1/4)*a + (a + b*x**8)**(1/4)*b*x**8),x)*c