Integrand size = 22, antiderivative size = 117 \[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {c \sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac {(12 b c-13 a d) \sqrt [4]{a+b x^4}}{117 a^2 x^9}-\frac {8 b (12 b c-13 a d) \sqrt [4]{a+b x^4}}{585 a^3 x^5}+\frac {32 b^2 (12 b c-13 a d) \sqrt [4]{a+b x^4}}{585 a^4 x} \] Output:
-1/13*c*(b*x^4+a)^(1/4)/a/x^13+1/117*(-13*a*d+12*b*c)*(b*x^4+a)^(1/4)/a^2/ x^9-8/585*b*(-13*a*d+12*b*c)*(b*x^4+a)^(1/4)/a^3/x^5+32/585*b^2*(-13*a*d+1 2*b*c)*(b*x^4+a)^(1/4)/a^4/x
Time = 0.59 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-45 a^3 c+60 a^2 b c x^4-65 a^3 d x^4-96 a b^2 c x^8+104 a^2 b d x^8+384 b^3 c x^{12}-416 a b^2 d x^{12}\right )}{585 a^4 x^{13}} \] Input:
Integrate[(c + d*x^4)/(x^14*(a + b*x^4)^(3/4)),x]
Output:
((a + b*x^4)^(1/4)*(-45*a^3*c + 60*a^2*b*c*x^4 - 65*a^3*d*x^4 - 96*a*b^2*c *x^8 + 104*a^2*b*d*x^8 + 384*b^3*c*x^12 - 416*a*b^2*d*x^12))/(585*a^4*x^13 )
Time = 0.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {955, 803, 803, 796}
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.
\(\displaystyle \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {(12 b c-13 a d) \int \frac {1}{x^{10} \left (b x^4+a\right )^{3/4}}dx}{13 a}-\frac {c \sqrt [4]{a+b x^4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {(12 b c-13 a d) \left (-\frac {8 b \int \frac {1}{x^6 \left (b x^4+a\right )^{3/4}}dx}{9 a}-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}\right )}{13 a}-\frac {c \sqrt [4]{a+b x^4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {(12 b c-13 a d) \left (-\frac {8 b \left (-\frac {4 b \int \frac {1}{x^2 \left (b x^4+a\right )^{3/4}}dx}{5 a}-\frac {\sqrt [4]{a+b x^4}}{5 a x^5}\right )}{9 a}-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}\right )}{13 a}-\frac {c \sqrt [4]{a+b x^4}}{13 a x^{13}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {\left (-\frac {8 b \left (\frac {4 b \sqrt [4]{a+b x^4}}{5 a^2 x}-\frac {\sqrt [4]{a+b x^4}}{5 a x^5}\right )}{9 a}-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}\right ) (12 b c-13 a d)}{13 a}-\frac {c \sqrt [4]{a+b x^4}}{13 a x^{13}}\) |
Input:
Int[(c + d*x^4)/(x^14*(a + b*x^4)^(3/4)),x]
Output:
-1/13*(c*(a + b*x^4)^(1/4))/(a*x^13) - ((12*b*c - 13*a*d)*(-1/9*(a + b*x^4 )^(1/4)/(a*x^9) - (8*b*(-1/5*(a + b*x^4)^(1/4)/(a*x^5) + (4*b*(a + b*x^4)^ (1/4))/(5*a^2*x)))/(9*a)))/(13*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
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 = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (\frac {13 d \,x^{4}}{9}+c \right ) a^{3}-\frac {4 b \,x^{4} \left (\frac {26 d \,x^{4}}{15}+c \right ) a^{2}}{3}+\frac {32 \left (\frac {13 d \,x^{4}}{3}+c \right ) b^{2} x^{8} a}{15}-\frac {128 b^{3} c \,x^{12}}{15}\right )}{13 x^{13} a^{4}}\) | \(74\) |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{12}-384 b^{3} c \,x^{12}-104 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+65 a^{3} d \,x^{4}-60 a^{2} b c \,x^{4}+45 c \,a^{3}\right )}{585 x^{13} a^{4}}\) | \(83\) |
trager | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{12}-384 b^{3} c \,x^{12}-104 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+65 a^{3} d \,x^{4}-60 a^{2} b c \,x^{4}+45 c \,a^{3}\right )}{585 x^{13} a^{4}}\) | \(83\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{12}-384 b^{3} c \,x^{12}-104 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+65 a^{3} d \,x^{4}-60 a^{2} b c \,x^{4}+45 c \,a^{3}\right )}{585 x^{13} a^{4}}\) | \(83\) |
orering | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{12}-384 b^{3} c \,x^{12}-104 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+65 a^{3} d \,x^{4}-60 a^{2} b c \,x^{4}+45 c \,a^{3}\right )}{585 x^{13} a^{4}}\) | \(83\) |
Input:
int((d*x^4+c)/x^14/(b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)
Output:
-1/13*(b*x^4+a)^(1/4)*((13/9*d*x^4+c)*a^3-4/3*b*x^4*(26/15*d*x^4+c)*a^2+32 /15*(13/3*d*x^4+c)*b^2*x^8*a-128/15*b^3*c*x^12)/x^13/a^4
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70 \[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=\frac {{\left (32 \, {\left (12 \, b^{3} c - 13 \, a b^{2} d\right )} x^{12} - 8 \, {\left (12 \, a b^{2} c - 13 \, a^{2} b d\right )} x^{8} + 5 \, {\left (12 \, a^{2} b c - 13 \, a^{3} d\right )} x^{4} - 45 \, a^{3} c\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{585 \, a^{4} x^{13}} \] Input:
integrate((d*x^4+c)/x^14/(b*x^4+a)^(3/4),x, algorithm="fricas")
Output:
1/585*(32*(12*b^3*c - 13*a*b^2*d)*x^12 - 8*(12*a*b^2*c - 13*a^2*b*d)*x^8 + 5*(12*a^2*b*c - 13*a^3*d)*x^4 - 45*a^3*c)*(b*x^4 + a)^(1/4)/(a^4*x^13)
Leaf count of result is larger than twice the leaf count of optimal. 1120 vs. \(2 (110) = 220\).
Time = 6.26 (sec) , antiderivative size = 1120, normalized size of antiderivative = 9.57 \[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=\text {Too large to display} \] Input:
integrate((d*x**4+c)/x**14/(b*x**4+a)**(3/4),x)
Output:
-45*a**6*b**(37/4)*c*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x **12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b**11*x**20*g amma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) - 75*a**5*b**(41/4)*c*x**4*(a /(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a **6*b**10*x**16*gamma(3/4) + 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b* *12*x**24*gamma(3/4)) - 51*a**4*b**(45/4)*c*x**8*(a/(b*x**4) + 1)**(1/4)*g amma(-13/4)/(256*a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3 /4) + 768*a**5*b**11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) + 5*a**4*b**(17/4)*d*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8 *gamma(3/4) + 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/ 4)) + 231*a**3*b**(49/4)*c*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256 *a**7*b**9*x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b **11*x**20*gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) + 2*a**3*b**(21/4 )*d*x**4*(a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) + 128*a**4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)) + 924*a **2*b**(53/4)*c*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-13/4)/(256*a**7*b**9* x**12*gamma(3/4) + 768*a**6*b**10*x**16*gamma(3/4) + 768*a**5*b**11*x**20* gamma(3/4) + 256*a**4*b**12*x**24*gamma(3/4)) + 21*a**2*b**(25/4)*d*x**8*( a/(b*x**4) + 1)**(1/4)*gamma(-9/4)/(64*a**5*b**4*x**8*gamma(3/4) + 128*a** 4*b**5*x**12*gamma(3/4) + 64*a**3*b**6*x**16*gamma(3/4)) + 1056*a*b**(5...
Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=-\frac {{\left (\frac {45 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}}{x} - \frac {18 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b}{x^{5}} + \frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}}}{x^{9}}\right )} d}{45 \, a^{3}} + \frac {{\left (\frac {195 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{3}}{x} - \frac {117 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{2}}{x^{5}} + \frac {65 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b}{x^{9}} - \frac {15 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}}}{x^{13}}\right )} c}{195 \, a^{4}} \] Input:
integrate((d*x^4+c)/x^14/(b*x^4+a)^(3/4),x, algorithm="maxima")
Output:
-1/45*(45*(b*x^4 + a)^(1/4)*b^2/x - 18*(b*x^4 + a)^(5/4)*b/x^5 + 5*(b*x^4 + a)^(9/4)/x^9)*d/a^3 + 1/195*(195*(b*x^4 + a)^(1/4)*b^3/x - 117*(b*x^4 + a)^(5/4)*b^2/x^5 + 65*(b*x^4 + a)^(9/4)*b/x^9 - 15*(b*x^4 + a)^(13/4)/x^13 )*c/a^4
\[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{14}} \,d x } \] Input:
integrate((d*x^4+c)/x^14/(b*x^4+a)^(3/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(3/4)*x^14), x)
Time = 3.59 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (384\,b^3\,c-416\,a\,b^2\,d\right )}{585\,a^4\,x}-\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (96\,b^2\,c-104\,a\,b\,d\right )}{585\,a^3\,x^5}-\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (13\,a\,d-12\,b\,c\right )}{117\,a^2\,x^9}-\frac {c\,{\left (b\,x^4+a\right )}^{1/4}}{13\,a\,x^{13}} \] Input:
int((c + d*x^4)/(x^14*(a + b*x^4)^(3/4)),x)
Output:
((a + b*x^4)^(1/4)*(384*b^3*c - 416*a*b^2*d))/(585*a^4*x) - ((a + b*x^4)^( 1/4)*(96*b^2*c - 104*a*b*d))/(585*a^3*x^5) - ((a + b*x^4)^(1/4)*(13*a*d - 12*b*c))/(117*a^2*x^9) - (c*(a + b*x^4)^(1/4))/(13*a*x^13)
\[ \int \frac {c+d x^4}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{14}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{10}}d x \right ) d \] Input:
int((d*x^4+c)/x^14/(b*x^4+a)^(3/4),x)
Output:
int(1/((a + b*x**4)**(3/4)*x**14),x)*c + int(1/((a + b*x**4)**(3/4)*x**10) ,x)*d