\(\int \frac {c+d x^4}{x^8 (a+b x^4)^{13/4}} \, dx\) [170]

Optimal result

Integrand size = 22, antiderivative size = 143 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=-\frac {c}{7 a x^7 \left (a+b x^4\right )^{9/4}}-\frac {16 b c-7 a d}{63 a^2 x^3 \left (a+b x^4\right )^{9/4}}-\frac {4 (16 b c-7 a d)}{105 a^3 x^3 \left (a+b x^4\right )^{5/4}}-\frac {32 (16 b c-7 a d)}{105 a^4 x^3 \sqrt [4]{a+b x^4}}+\frac {128 (16 b c-7 a d) \left (a+b x^4\right )^{3/4}}{315 a^5 x^3} \] Output:

-1/7*c/a/x^7/(b*x^4+a)^(9/4)-1/63*(-7*a*d+16*b*c)/a^2/x^3/(b*x^4+a)^(9/4)- 
4/105*(-7*a*d+16*b*c)/a^3/x^3/(b*x^4+a)^(5/4)-32/105*(-7*a*d+16*b*c)/a^4/x 
^3/(b*x^4+a)^(1/4)+128/315*(-7*a*d+16*b*c)*(b*x^4+a)^(3/4)/a^5/x^3
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.73 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\frac {2048 b^4 c x^{16}+60 a^3 b x^4 \left (4 c-21 d x^4\right )+288 a^2 b^2 x^8 \left (10 c-7 d x^4\right )+128 a b^3 x^{12} \left (36 c-7 d x^4\right )-15 a^4 \left (3 c+7 d x^4\right )}{315 a^5 x^7 \left (a+b x^4\right )^{9/4}} \] Input:

Integrate[(c + d*x^4)/(x^8*(a + b*x^4)^(13/4)),x]
 

Output:

(2048*b^4*c*x^16 + 60*a^3*b*x^4*(4*c - 21*d*x^4) + 288*a^2*b^2*x^8*(10*c - 
 7*d*x^4) + 128*a*b^3*x^12*(36*c - 7*d*x^4) - 15*a^4*(3*c + 7*d*x^4))/(315 
*a^5*x^7*(a + b*x^4)^(9/4))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {955, 803, 749, 749, 746}

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.

\(\displaystyle \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(16 b c-7 a d) \int \frac {1}{x^4 \left (b x^4+a\right )^{13/4}}dx}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{9/4}}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {(16 b c-7 a d) \left (-\frac {4 b \int \frac {1}{\left (b x^4+a\right )^{13/4}}dx}{a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{9/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{9/4}}\)

\(\Big \downarrow \) 749

\(\displaystyle -\frac {(16 b c-7 a d) \left (-\frac {4 b \left (\frac {8 \int \frac {1}{\left (b x^4+a\right )^{9/4}}dx}{9 a}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}}\right )}{a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{9/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{9/4}}\)

\(\Big \downarrow \) 749

\(\displaystyle -\frac {(16 b c-7 a d) \left (-\frac {4 b \left (\frac {8 \left (\frac {4 \int \frac {1}{\left (b x^4+a\right )^{5/4}}dx}{5 a}+\frac {x}{5 a \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}}\right )}{a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{9/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{9/4}}\)

\(\Big \downarrow \) 746

\(\displaystyle -\frac {\left (-\frac {4 b \left (\frac {8 \left (\frac {4 x}{5 a^2 \sqrt [4]{a+b x^4}}+\frac {x}{5 a \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}}\right )}{a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{9/4}}\right ) (16 b c-7 a d)}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{9/4}}\)

Input:

Int[(c + d*x^4)/(x^8*(a + b*x^4)^(13/4)),x]
 

Output:

-1/7*c/(a*x^7*(a + b*x^4)^(9/4)) - ((16*b*c - 7*a*d)*(-1/3*1/(a*x^3*(a + b 
*x^4)^(9/4)) - (4*b*(x/(9*a*(a + b*x^4)^(9/4)) + (8*(x/(5*a*(a + b*x^4)^(5 
/4)) + (4*x)/(5*a^2*(a + b*x^4)^(1/4))))/(9*a)))/a))/(7*a)
 

Defintions of rubi rules used

rule 746

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) 
/a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
 

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

rule 803

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( 
a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 
)))   Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I 
LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
 

rule 955

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {\left (-105 d \,x^{4}-45 c \right ) a^{4}+240 \left (-\frac {21 d \,x^{4}}{4}+c \right ) b \,x^{4} a^{3}+2880 \left (-\frac {7 d \,x^{4}}{10}+c \right ) b^{2} x^{8} a^{2}+4608 \left (-\frac {7 d \,x^{4}}{36}+c \right ) b^{3} x^{12} a +2048 b^{4} c \,x^{16}}{315 \left (b \,x^{4}+a \right )^{\frac {9}{4}} x^{7} a^{5}}\)	\(95\)
gosper	\(-\frac {896 a \,b^{3} d \,x^{16}-2048 b^{4} c \,x^{16}+2016 a^{2} b^{2} d \,x^{12}-4608 a \,b^{3} c \,x^{12}+1260 a^{3} b d \,x^{8}-2880 a^{2} b^{2} c \,x^{8}+105 a^{4} d \,x^{4}-240 a^{3} b c \,x^{4}+45 c \,a^{4}}{315 x^{7} \left (b \,x^{4}+a \right )^{\frac {9}{4}} a^{5}}\)	\(107\)
trager	\(-\frac {896 a \,b^{3} d \,x^{16}-2048 b^{4} c \,x^{16}+2016 a^{2} b^{2} d \,x^{12}-4608 a \,b^{3} c \,x^{12}+1260 a^{3} b d \,x^{8}-2880 a^{2} b^{2} c \,x^{8}+105 a^{4} d \,x^{4}-240 a^{3} b c \,x^{4}+45 c \,a^{4}}{315 x^{7} \left (b \,x^{4}+a \right )^{\frac {9}{4}} a^{5}}\)	\(107\)
orering	\(-\frac {896 a \,b^{3} d \,x^{16}-2048 b^{4} c \,x^{16}+2016 a^{2} b^{2} d \,x^{12}-4608 a \,b^{3} c \,x^{12}+1260 a^{3} b d \,x^{8}-2880 a^{2} b^{2} c \,x^{8}+105 a^{4} d \,x^{4}-240 a^{3} b c \,x^{4}+45 c \,a^{4}}{315 x^{7} \left (b \,x^{4}+a \right )^{\frac {9}{4}} a^{5}}\)	\(107\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (7 a d \,x^{4}-25 b c \,x^{4}+3 a c \right )}{21 a^{5} x^{7}}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} x \left (113 a \,b^{2} d \,x^{8}-239 c \,b^{3} x^{8}+243 a^{2} b d \,x^{4}-504 a \,b^{2} c \,x^{4}+135 a^{3} d -270 a^{2} b c \right ) b}{45 a^{5} \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )}\)	\(138\)

Input:

int((d*x^4+c)/x^8/(b*x^4+a)^(13/4),x,method=_RETURNVERBOSE)
 

Output:

1/315*((-105*d*x^4-45*c)*a^4+240*(-21/4*d*x^4+c)*b*x^4*a^3+2880*(-7/10*d*x 
^4+c)*b^2*x^8*a^2+4608*(-7/36*d*x^4+c)*b^3*x^12*a+2048*b^4*c*x^16)/(b*x^4+ 
a)^(9/4)/x^7/a^5
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\frac {{\left (128 \, {\left (16 \, b^{4} c - 7 \, a b^{3} d\right )} x^{16} + 288 \, {\left (16 \, a b^{3} c - 7 \, a^{2} b^{2} d\right )} x^{12} + 180 \, {\left (16 \, a^{2} b^{2} c - 7 \, a^{3} b d\right )} x^{8} - 45 \, a^{4} c + 15 \, {\left (16 \, a^{3} b c - 7 \, a^{4} d\right )} x^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{315 \, {\left (a^{5} b^{3} x^{19} + 3 \, a^{6} b^{2} x^{15} + 3 \, a^{7} b x^{11} + a^{8} x^{7}\right )}} \] Input:

integrate((d*x^4+c)/x^8/(b*x^4+a)^(13/4),x, algorithm="fricas")
 

Output:

1/315*(128*(16*b^4*c - 7*a*b^3*d)*x^16 + 288*(16*a*b^3*c - 7*a^2*b^2*d)*x^ 
12 + 180*(16*a^2*b^2*c - 7*a^3*b*d)*x^8 - 45*a^4*c + 15*(16*a^3*b*c - 7*a^ 
4*d)*x^4)*(b*x^4 + a)^(3/4)/(a^5*b^3*x^19 + 3*a^6*b^2*x^15 + 3*a^7*b*x^11 
+ a^8*x^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\text {Timed out} \] Input:

integrate((d*x**4+c)/x**8/(b*x**4+a)**(13/4),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\frac {1}{315} \, {\left (\frac {7 \, {\left (5 \, b^{4} - \frac {36 \, {\left (b x^{4} + a\right )} b^{3}}{x^{4}} + \frac {270 \, {\left (b x^{4} + a\right )}^{2} b^{2}}{x^{8}}\right )} x^{9}}{{\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{5}} + \frac {15 \, {\left (\frac {28 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b}{x^{3}} - \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}}}{x^{7}}\right )}}{a^{5}}\right )} c - \frac {1}{45} \, {\left (\frac {{\left (5 \, b^{3} - \frac {27 \, {\left (b x^{4} + a\right )} b^{2}}{x^{4}} + \frac {135 \, {\left (b x^{4} + a\right )}^{2} b}{x^{8}}\right )} x^{9}}{{\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{4}} + \frac {15 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{a^{4} x^{3}}\right )} d \] Input:

integrate((d*x^4+c)/x^8/(b*x^4+a)^(13/4),x, algorithm="maxima")
 

Output:

1/315*(7*(5*b^4 - 36*(b*x^4 + a)*b^3/x^4 + 270*(b*x^4 + a)^2*b^2/x^8)*x^9/ 
((b*x^4 + a)^(9/4)*a^5) + 15*(28*(b*x^4 + a)^(3/4)*b/x^3 - 3*(b*x^4 + a)^( 
7/4)/x^7)/a^5)*c - 1/45*((5*b^3 - 27*(b*x^4 + a)*b^2/x^4 + 135*(b*x^4 + a) 
^2*b/x^8)*x^9/((b*x^4 + a)^(9/4)*a^4) + 15*(b*x^4 + a)^(3/4)/(a^4*x^3))*d
 

Giac [F]

\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {13}{4}} x^{8}} \,d x } \] Input:

integrate((d*x^4+c)/x^8/(b*x^4+a)^(13/4),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(13/4)*x^8), x)
 

Mupad [B] (verification not implemented)

Time = 4.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.21 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\frac {x\,\left (\frac {b\,\left (224\,a\,d-521\,b\,c\right )}{126\,a^4}+\frac {a\,\left (\frac {4\,b^3\,c}{35\,a^5}-\frac {b^2\,\left (224\,a\,d-521\,b\,c\right )}{90\,a^5}\right )}{b}\right )}{{\left (b\,x^4+a\right )}^{5/4}}-\frac {x^4\,\left (\frac {2\,b^2\,c}{7\,a^3}+\frac {4\,b\,\left (7\,a^4\,d-25\,a^3\,b\,c\right )}{63\,a^6}\right )+\frac {7\,a^4\,d-25\,a^3\,b\,c}{21\,a^5}}{x^3\,{\left (b\,x^4+a\right )}^{9/4}}-\frac {c\,{\left (b\,x^4+a\right )}^{3/4}}{7\,a^4\,x^7}+\frac {x\,\left (2048\,b^2\,c-896\,a\,b\,d\right )}{315\,a^5\,{\left (b\,x^4+a\right )}^{1/4}} \] Input:

int((c + d*x^4)/(x^8*(a + b*x^4)^(13/4)),x)
 

Output:

(x*((b*(224*a*d - 521*b*c))/(126*a^4) + (a*((4*b^3*c)/(35*a^5) - (b^2*(224 
*a*d - 521*b*c))/(90*a^5)))/b))/(a + b*x^4)^(5/4) - (x^4*((2*b^2*c)/(7*a^3 
) + (4*b*(7*a^4*d - 25*a^3*b*c))/(63*a^6)) + (7*a^4*d - 25*a^3*b*c)/(21*a^ 
5))/(x^3*(a + b*x^4)^(9/4)) - (c*(a + b*x^4)^(3/4))/(7*a^4*x^7) + (x*(2048 
*b^2*c - 896*a*b*d))/(315*a^5*(a + b*x^4)^(1/4))
 

Reduce [F]

\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{13/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} x^{8}+3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b \,x^{12}+3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{2} x^{16}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{3} x^{20}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} x^{4}+3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b \,x^{8}+3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{2} x^{12}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{3} x^{16}}d x \right ) d \] Input:

int((d*x^4+c)/x^8/(b*x^4+a)^(13/4),x)
 

Output:

int(1/((a + b*x**4)**(1/4)*a**3*x**8 + 3*(a + b*x**4)**(1/4)*a**2*b*x**12 
+ 3*(a + b*x**4)**(1/4)*a*b**2*x**16 + (a + b*x**4)**(1/4)*b**3*x**20),x)* 
c + int(1/((a + b*x**4)**(1/4)*a**3*x**4 + 3*(a + b*x**4)**(1/4)*a**2*b*x* 
*8 + 3*(a + b*x**4)**(1/4)*a*b**2*x**12 + (a + b*x**4)**(1/4)*b**3*x**16), 
x)*d