Integrand size = 22, antiderivative size = 173 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}-\frac {20 b c-7 a d}{91 a^2 x^3 \left (a+b x^4\right )^{13/4}}-\frac {16 (20 b c-7 a d)}{819 a^3 x^3 \left (a+b x^4\right )^{9/4}}-\frac {64 (20 b c-7 a d)}{1365 a^4 x^3 \left (a+b x^4\right )^{5/4}}-\frac {512 (20 b c-7 a d)}{1365 a^5 x^3 \sqrt [4]{a+b x^4}}+\frac {2048 (20 b c-7 a d) \left (a+b x^4\right )^{3/4}}{4095 a^6 x^3} \] Output:
-1/7*c/a/x^7/(b*x^4+a)^(13/4)-1/91*(-7*a*d+20*b*c)/a^2/x^3/(b*x^4+a)^(13/4 )-16/819*(-7*a*d+20*b*c)/a^3/x^3/(b*x^4+a)^(9/4)-64/1365*(-7*a*d+20*b*c)/a ^4/x^3/(b*x^4+a)^(5/4)-512/1365*(-7*a*d+20*b*c)/a^5/x^3/(b*x^4+a)^(1/4)+20 48/4095*(-7*a*d+20*b*c)*(b*x^4+a)^(3/4)/a^6/x^3
Time = 1.70 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.73 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=\frac {40960 b^5 c x^{20}+780 a^4 b x^4 \left (5 c-28 d x^4\right )+2496 a^3 b^2 x^8 \left (25 c-21 d x^4\right )+3328 a^2 b^3 x^{12} \left (45 c-14 d x^4\right )+2048 a b^4 x^{16} \left (65 c-7 d x^4\right )-195 a^5 \left (3 c+7 d x^4\right )}{4095 a^6 x^7 \left (a+b x^4\right )^{13/4}} \] Input:
Integrate[(c + d*x^4)/(x^8*(a + b*x^4)^(17/4)),x]
Output:
(40960*b^5*c*x^20 + 780*a^4*b*x^4*(5*c - 28*d*x^4) + 2496*a^3*b^2*x^8*(25* c - 21*d*x^4) + 3328*a^2*b^3*x^12*(45*c - 14*d*x^4) + 2048*a*b^4*x^16*(65* c - 7*d*x^4) - 195*a^5*(3*c + 7*d*x^4))/(4095*a^6*x^7*(a + b*x^4)^(13/4))
Time = 0.47 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 803, 749, 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.
| \(\displaystyle \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(20 b c-7 a d) \int \frac {1}{x^4 \left (b x^4+a\right )^{17/4}}dx}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}\) |
| \(\Big \downarrow \) 803 |
| \(\displaystyle -\frac {(20 b c-7 a d) \left (-\frac {16 b \int \frac {1}{\left (b x^4+a\right )^{17/4}}dx}{3 a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{13/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {(20 b c-7 a d) \left (-\frac {16 b \left (\frac {12 \int \frac {1}{\left (b x^4+a\right )^{13/4}}dx}{13 a}+\frac {x}{13 a \left (a+b x^4\right )^{13/4}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{13/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {(20 b c-7 a d) \left (-\frac {16 b \left (\frac {12 \left (\frac {8 \int \frac {1}{\left (b x^4+a\right )^{9/4}}dx}{9 a}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {x}{13 a \left (a+b x^4\right )^{13/4}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{13/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle -\frac {(20 b c-7 a d) \left (-\frac {16 b \left (\frac {12 \left (\frac {8 \left (\frac {4 \int \frac {1}{\left (b x^4+a\right )^{5/4}}dx}{5 a}+\frac {x}{5 a \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {x}{13 a \left (a+b x^4\right )^{13/4}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{13/4}}\right )}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}\) |
| \(\Big \downarrow \) 746 |
| \(\displaystyle -\frac {\left (-\frac {16 b \left (\frac {12 \left (\frac {8 \left (\frac {4 x}{5 a^2 \sqrt [4]{a+b x^4}}+\frac {x}{5 a \left (a+b x^4\right )^{5/4}}\right )}{9 a}+\frac {x}{9 a \left (a+b x^4\right )^{9/4}}\right )}{13 a}+\frac {x}{13 a \left (a+b x^4\right )^{13/4}}\right )}{3 a}-\frac {1}{3 a x^3 \left (a+b x^4\right )^{13/4}}\right ) (20 b c-7 a d)}{7 a}-\frac {c}{7 a x^7 \left (a+b x^4\right )^{13/4}}\) |
Input:
Int[(c + d*x^4)/(x^8*(a + b*x^4)^(17/4)),x]
Output:
-1/7*c/(a*x^7*(a + b*x^4)^(13/4)) - ((20*b*c - 7*a*d)*(-1/3*1/(a*x^3*(a + b*x^4)^(13/4)) - (16*b*(x/(13*a*(a + b*x^4)^(13/4)) + (12*(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)))/(13*a)))/(3*a)))/(7*a)
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]
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])
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]
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]
Time = 0.86 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {\left (-1365 d \,x^{4}-585 c \right ) a^{5}+3900 b \,x^{4} \left (-\frac {28 d \,x^{4}}{5}+c \right ) a^{4}+62400 \left (-\frac {21 d \,x^{4}}{25}+c \right ) b^{2} x^{8} a^{3}+149760 \left (-\frac {14 d \,x^{4}}{45}+c \right ) b^{3} x^{12} a^{2}+133120 \left (-\frac {7 d \,x^{4}}{65}+c \right ) b^{4} x^{16} a +40960 b^{5} c \,x^{20}}{4095 \left (b \,x^{4}+a \right )^{\frac {13}{4}} x^{7} a^{6}}\) | \(114\) |
| gosper | \(-\frac {14336 a \,b^{4} d \,x^{20}-40960 b^{5} c \,x^{20}+46592 a^{2} b^{3} d \,x^{16}-133120 a \,b^{4} c \,x^{16}+52416 a^{3} b^{2} d \,x^{12}-149760 a^{2} b^{3} c \,x^{12}+21840 a^{4} b d \,x^{8}-62400 a^{3} b^{2} c \,x^{8}+1365 a^{5} d \,x^{4}-3900 a^{4} b c \,x^{4}+585 c \,a^{5}}{4095 x^{7} \left (b \,x^{4}+a \right )^{\frac {13}{4}} a^{6}}\) | \(131\) |
| trager | \(-\frac {14336 a \,b^{4} d \,x^{20}-40960 b^{5} c \,x^{20}+46592 a^{2} b^{3} d \,x^{16}-133120 a \,b^{4} c \,x^{16}+52416 a^{3} b^{2} d \,x^{12}-149760 a^{2} b^{3} c \,x^{12}+21840 a^{4} b d \,x^{8}-62400 a^{3} b^{2} c \,x^{8}+1365 a^{5} d \,x^{4}-3900 a^{4} b c \,x^{4}+585 c \,a^{5}}{4095 x^{7} \left (b \,x^{4}+a \right )^{\frac {13}{4}} a^{6}}\) | \(131\) |
| orering | \(-\frac {14336 a \,b^{4} d \,x^{20}-40960 b^{5} c \,x^{20}+46592 a^{2} b^{3} d \,x^{16}-133120 a \,b^{4} c \,x^{16}+52416 a^{3} b^{2} d \,x^{12}-149760 a^{2} b^{3} c \,x^{12}+21840 a^{4} b d \,x^{8}-62400 a^{3} b^{2} c \,x^{8}+1365 a^{5} d \,x^{4}-3900 a^{4} b c \,x^{4}+585 c \,a^{5}}{4095 x^{7} \left (b \,x^{4}+a \right )^{\frac {13}{4}} a^{6}}\) | \(131\) |
| risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (7 a d \,x^{4}-32 b c \,x^{4}+3 a c \right )}{21 a^{6} x^{7}}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} x \left (1853 a \,b^{3} d \,x^{12}-4960 c \,b^{4} x^{12}+5876 a^{2} b^{2} d \,x^{8}-15535 a \,b^{3} c \,x^{8}+6318 a^{3} b d \,x^{4}-16380 a^{2} b^{2} c \,x^{4}+2340 a^{4} d -5850 a^{3} b c \right ) b}{585 a^{6} \left (b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} x^{8} b^{2}+4 x^{4} a^{3} b +a^{4}\right )}\) | \(173\) |
Input:
int((d*x^4+c)/x^8/(b*x^4+a)^(17/4),x,method=_RETURNVERBOSE)
Output:
1/4095*((-1365*d*x^4-585*c)*a^5+3900*b*x^4*(-28/5*d*x^4+c)*a^4+62400*(-21/ 25*d*x^4+c)*b^2*x^8*a^3+149760*(-14/45*d*x^4+c)*b^3*x^12*a^2+133120*(-7/65 *d*x^4+c)*b^4*x^16*a+40960*b^5*c*x^20)/(b*x^4+a)^(13/4)/x^7/a^6
Time = 0.12 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=\frac {{\left (2048 \, {\left (20 \, b^{5} c - 7 \, a b^{4} d\right )} x^{20} + 6656 \, {\left (20 \, a b^{4} c - 7 \, a^{2} b^{3} d\right )} x^{16} + 7488 \, {\left (20 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d\right )} x^{12} + 3120 \, {\left (20 \, a^{3} b^{2} c - 7 \, a^{4} b d\right )} x^{8} - 585 \, a^{5} c + 195 \, {\left (20 \, a^{4} b c - 7 \, a^{5} d\right )} x^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{4095 \, {\left (a^{6} b^{4} x^{23} + 4 \, a^{7} b^{3} x^{19} + 6 \, a^{8} b^{2} x^{15} + 4 \, a^{9} b x^{11} + a^{10} x^{7}\right )}} \] Input:
integrate((d*x^4+c)/x^8/(b*x^4+a)^(17/4),x, algorithm="fricas")
Output:
1/4095*(2048*(20*b^5*c - 7*a*b^4*d)*x^20 + 6656*(20*a*b^4*c - 7*a^2*b^3*d) *x^16 + 7488*(20*a^2*b^3*c - 7*a^3*b^2*d)*x^12 + 3120*(20*a^3*b^2*c - 7*a^ 4*b*d)*x^8 - 585*a^5*c + 195*(20*a^4*b*c - 7*a^5*d)*x^4)*(b*x^4 + a)^(3/4) /(a^6*b^4*x^23 + 4*a^7*b^3*x^19 + 6*a^8*b^2*x^15 + 4*a^9*b*x^11 + a^10*x^7 )
Timed out. \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x**4+c)/x**8/(b*x**4+a)**(17/4),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=-\frac {1}{819} \, {\left (\frac {7 \, {\left (9 \, b^{5} - \frac {65 \, {\left (b x^{4} + a\right )} b^{4}}{x^{4}} + \frac {234 \, {\left (b x^{4} + a\right )}^{2} b^{3}}{x^{8}} - \frac {1170 \, {\left (b x^{4} + a\right )}^{3} b^{2}}{x^{12}}\right )} x^{13}}{{\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{6}} - \frac {39 \, {\left (\frac {35 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b}{x^{3}} - \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}}}{x^{7}}\right )}}{a^{6}}\right )} c + \frac {1}{585} \, {\left (\frac {{\left (45 \, b^{4} - \frac {260 \, {\left (b x^{4} + a\right )} b^{3}}{x^{4}} + \frac {702 \, {\left (b x^{4} + a\right )}^{2} b^{2}}{x^{8}} - \frac {2340 \, {\left (b x^{4} + a\right )}^{3} b}{x^{12}}\right )} x^{13}}{{\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{5}} - \frac {195 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{a^{5} x^{3}}\right )} d \] Input:
integrate((d*x^4+c)/x^8/(b*x^4+a)^(17/4),x, algorithm="maxima")
Output:
-1/819*(7*(9*b^5 - 65*(b*x^4 + a)*b^4/x^4 + 234*(b*x^4 + a)^2*b^3/x^8 - 11 70*(b*x^4 + a)^3*b^2/x^12)*x^13/((b*x^4 + a)^(13/4)*a^6) - 39*(35*(b*x^4 + a)^(3/4)*b/x^3 - 3*(b*x^4 + a)^(7/4)/x^7)/a^6)*c + 1/585*((45*b^4 - 260*( b*x^4 + a)*b^3/x^4 + 702*(b*x^4 + a)^2*b^2/x^8 - 2340*(b*x^4 + a)^3*b/x^12 )*x^13/((b*x^4 + a)^(13/4)*a^5) - 195*(b*x^4 + a)^(3/4)/(a^5*x^3))*d
\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {17}{4}} x^{8}} \,d x } \] Input:
integrate((d*x^4+c)/x^8/(b*x^4+a)^(17/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(17/4)*x^8), x)
Time = 4.02 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.39 \[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=\frac {x\,\left (\frac {b\,\left (3584\,a\,d-10357\,b\,c\right )}{1638\,a^5}+\frac {a\,\left (\frac {4\,b^3\,c}{35\,a^6}-\frac {b^2\,\left (3584\,a\,d-10357\,b\,c\right )}{1170\,a^6}\right )}{b}\right )}{{\left (b\,x^4+a\right )}^{5/4}}-\frac {\frac {7\,a^5\,d-32\,a^4\,b\,c}{28\,a^6}-\frac {a\,\left (\frac {25\,b^2\,c}{91\,a^3}+\frac {17\,b\,\left (7\,a^5\,d-32\,a^4\,b\,c\right )}{364\,a^7}\right )}{b}}{x^3\,{\left (b\,x^4+a\right )}^{13/4}}-\frac {x^4\,\left (\frac {2\,b^2\,c}{7\,a^4}+\frac {4\,b\,\left (112\,a^4\,d-437\,a^3\,b\,c\right )}{819\,a^7}\right )+\frac {112\,a^4\,d-437\,a^3\,b\,c}{273\,a^6}}{x^3\,{\left (b\,x^4+a\right )}^{9/4}}-\frac {c\,{\left (b\,x^4+a\right )}^{3/4}}{7\,a^5\,x^7}-\frac {2048\,b\,x\,\left (7\,a\,d-20\,b\,c\right )}{4095\,a^6\,{\left (b\,x^4+a\right )}^{1/4}} \] Input:
int((c + d*x^4)/(x^8*(a + b*x^4)^(17/4)),x)
Output:
(x*((b*(3584*a*d - 10357*b*c))/(1638*a^5) + (a*((4*b^3*c)/(35*a^6) - (b^2* (3584*a*d - 10357*b*c))/(1170*a^6)))/b))/(a + b*x^4)^(5/4) - ((7*a^5*d - 3 2*a^4*b*c)/(28*a^6) - (a*((25*b^2*c)/(91*a^3) + (17*b*(7*a^5*d - 32*a^4*b* c))/(364*a^7)))/b)/(x^3*(a + b*x^4)^(13/4)) - (x^4*((2*b^2*c)/(7*a^4) + (4 *b*(112*a^4*d - 437*a^3*b*c))/(819*a^7)) + (112*a^4*d - 437*a^3*b*c)/(273* a^6))/(x^3*(a + b*x^4)^(9/4)) - (c*(a + b*x^4)^(3/4))/(7*a^5*x^7) - (2048* b*x*(7*a*d - 20*b*c))/(4095*a^6*(a + b*x^4)^(1/4))
\[ \int \frac {c+d x^4}{x^8 \left (a+b x^4\right )^{17/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4} x^{8}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} b \,x^{12}+6 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{16}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{3} x^{20}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{4} x^{24}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4} x^{4}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3} b \,x^{8}+6 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{12}+4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,b^{3} x^{16}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{4} x^{20}}d x \right ) d \] Input:
int((d*x^4+c)/x^8/(b*x^4+a)^(17/4),x)
Output:
int(1/((a + b*x**4)**(1/4)*a**4*x**8 + 4*(a + b*x**4)**(1/4)*a**3*b*x**12 + 6*(a + b*x**4)**(1/4)*a**2*b**2*x**16 + 4*(a + b*x**4)**(1/4)*a*b**3*x** 20 + (a + b*x**4)**(1/4)*b**4*x**24),x)*c + int(1/((a + b*x**4)**(1/4)*a** 4*x**4 + 4*(a + b*x**4)**(1/4)*a**3*b*x**8 + 6*(a + b*x**4)**(1/4)*a**2*b* *2*x**12 + 4*(a + b*x**4)**(1/4)*a*b**3*x**16 + (a + b*x**4)**(1/4)*b**4*x **20),x)*d