Integrand size = 22, antiderivative size = 114 \[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=-\frac {c}{11 a x^{11} \sqrt [4]{a+b x^4}}-\frac {12 b c-11 a d}{11 a^2 x^7 \sqrt [4]{a+b x^4}}+\frac {8 (12 b c-11 a d) \left (a+b x^4\right )^{3/4}}{77 a^3 x^7}-\frac {32 b (12 b c-11 a d) \left (a+b x^4\right )^{3/4}}{231 a^4 x^3} \] Output:
-1/11*c/a/x^11/(b*x^4+a)^(1/4)-1/11*(-11*a*d+12*b*c)/a^2/x^7/(b*x^4+a)^(1/ 4)+8/77*(-11*a*d+12*b*c)*(b*x^4+a)^(3/4)/a^3/x^7-32/231*b*(-11*a*d+12*b*c) *(b*x^4+a)^(3/4)/a^4/x^3
Time = 0.88 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=\frac {-21 a^3 c+36 a^2 b c x^4-33 a^3 d x^4-96 a b^2 c x^8+88 a^2 b d x^8-384 b^3 c x^{12}+352 a b^2 d x^{12}}{231 a^4 x^{11} \sqrt [4]{a+b x^4}} \] Input:
Integrate[(c + d*x^4)/(x^12*(a + b*x^4)^(5/4)),x]
Output:
(-21*a^3*c + 36*a^2*b*c*x^4 - 33*a^3*d*x^4 - 96*a*b^2*c*x^8 + 88*a^2*b*d*x ^8 - 384*b^3*c*x^12 + 352*a*b^2*d*x^12)/(231*a^4*x^11*(a + b*x^4)^(1/4))
Time = 0.39 (sec) , antiderivative size = 111, 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, 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.
\(\displaystyle \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {(12 b c-11 a d) \int \frac {1}{x^8 \left (b x^4+a\right )^{5/4}}dx}{11 a}-\frac {c}{11 a x^{11} \sqrt [4]{a+b x^4}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {(12 b c-11 a d) \left (-\frac {8 b \int \frac {1}{x^4 \left (b x^4+a\right )^{5/4}}dx}{7 a}-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}\right )}{11 a}-\frac {c}{11 a x^{11} \sqrt [4]{a+b x^4}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {(12 b c-11 a d) \left (-\frac {8 b \left (-\frac {4 b \int \frac {1}{\left (b x^4+a\right )^{5/4}}dx}{3 a}-\frac {1}{3 a x^3 \sqrt [4]{a+b x^4}}\right )}{7 a}-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}\right )}{11 a}-\frac {c}{11 a x^{11} \sqrt [4]{a+b x^4}}\) |
\(\Big \downarrow \) 746 |
\(\displaystyle -\frac {\left (-\frac {8 b \left (-\frac {4 b x}{3 a^2 \sqrt [4]{a+b x^4}}-\frac {1}{3 a x^3 \sqrt [4]{a+b x^4}}\right )}{7 a}-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}\right ) (12 b c-11 a d)}{11 a}-\frac {c}{11 a x^{11} \sqrt [4]{a+b x^4}}\) |
Input:
Int[(c + d*x^4)/(x^12*(a + b*x^4)^(5/4)),x]
Output:
-1/11*c/(a*x^11*(a + b*x^4)^(1/4)) - ((12*b*c - 11*a*d)*(-1/7*1/(a*x^7*(a + b*x^4)^(1/4)) - (8*b*(-1/3*1/(a*x^3*(a + b*x^4)^(1/4)) - (4*b*x)/(3*a^2* (a + b*x^4)^(1/4))))/(7*a)))/(11*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[(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 = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {\left (\frac {11 d \,x^{4}}{7}+c \right ) a^{3}-\frac {12 b \,x^{4} \left (\frac {22 d \,x^{4}}{9}+c \right ) a^{2}}{7}+\frac {32 b^{2} x^{8} \left (-\frac {11 d \,x^{4}}{3}+c \right ) a}{7}+\frac {128 b^{3} c \,x^{12}}{7}}{11 \left (b \,x^{4}+a \right )^{\frac {1}{4}} x^{11} a^{4}}\) | \(74\) |
gosper | \(-\frac {-352 a \,b^{2} d \,x^{12}+384 b^{3} c \,x^{12}-88 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+33 a^{3} d \,x^{4}-36 a^{2} b c \,x^{4}+21 c \,a^{3}}{231 x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}}\) | \(83\) |
trager | \(-\frac {-352 a \,b^{2} d \,x^{12}+384 b^{3} c \,x^{12}-88 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+33 a^{3} d \,x^{4}-36 a^{2} b c \,x^{4}+21 c \,a^{3}}{231 x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}}\) | \(83\) |
orering | \(-\frac {-352 a \,b^{2} d \,x^{12}+384 b^{3} c \,x^{12}-88 a^{2} b d \,x^{8}+96 a \,b^{2} c \,x^{8}+33 a^{3} d \,x^{4}-36 a^{2} b c \,x^{4}+21 c \,a^{3}}{231 x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}}\) | \(83\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-121 a b d \,x^{8}+153 b^{2} c \,x^{8}+33 a^{2} d \,x^{4}-57 a b c \,x^{4}+21 a^{2} c \right )}{231 a^{4} x^{11}}+\frac {x \,b^{2} \left (a d -c b \right )}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{4}}\) | \(85\) |
Input:
int((d*x^4+c)/x^12/(b*x^4+a)^(5/4),x,method=_RETURNVERBOSE)
Output:
-1/11/(b*x^4+a)^(1/4)*((11/7*d*x^4+c)*a^3-12/7*b*x^4*(22/9*d*x^4+c)*a^2+32 /7*b^2*x^8*(-11/3*d*x^4+c)*a+128/7*b^3*c*x^12)/x^11/a^4
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=-\frac {{\left (32 \, {\left (12 \, b^{3} c - 11 \, a b^{2} d\right )} x^{12} + 8 \, {\left (12 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{8} - 3 \, {\left (12 \, a^{2} b c - 11 \, a^{3} d\right )} x^{4} + 21 \, a^{3} c\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{231 \, {\left (a^{4} b x^{15} + a^{5} x^{11}\right )}} \] Input:
integrate((d*x^4+c)/x^12/(b*x^4+a)^(5/4),x, algorithm="fricas")
Output:
-1/231*(32*(12*b^3*c - 11*a*b^2*d)*x^12 + 8*(12*a*b^2*c - 11*a^2*b*d)*x^8 - 3*(12*a^2*b*c - 11*a^3*d)*x^4 + 21*a^3*c)*(b*x^4 + a)^(3/4)/(a^4*b*x^15 + a^5*x^11)
Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (107) = 214\).
Time = 48.90 (sec) , antiderivative size = 920, normalized size of antiderivative = 8.07 \[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=\text {Too large to display} \] Input:
integrate((d*x**4+c)/x**12/(b*x**4+a)**(5/4),x)
Output:
c*(21*a**5*b**(39/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x **8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*ga mma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 6*a**4*b**(43/4)*x**4*(a/(b* x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b **10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x **20*gamma(5/4)) + 45*a**3*b**(47/4)*x**8*(a/(b*x**4) + 1)**(3/4)*gamma(-1 1/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 76 8*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 540*a** 2*b**(51/4)*x**12*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8 *gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma (5/4) + 256*a**4*b**12*x**20*gamma(5/4)) + 864*a*b**(55/4)*x**16*(a/(b*x** 4) + 1)**(3/4)*gamma(-11/4)/(256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**1 0*x**12*gamma(5/4) + 768*a**5*b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**2 0*gamma(5/4)) + 384*b**(59/4)*x**20*(a/(b*x**4) + 1)**(3/4)*gamma(-11/4)/( 256*a**7*b**9*x**8*gamma(5/4) + 768*a**6*b**10*x**12*gamma(5/4) + 768*a**5 *b**11*x**16*gamma(5/4) + 256*a**4*b**12*x**20*gamma(5/4))) + d*(-3*a**3*b **(19/4)*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(64*a**5*b**4*x**4*gamma(5/4) + 128*a**4*b**5*x**8*gamma(5/4) + 64*a**3*b**6*x**12*gamma(5/4)) + 5*a**2 *b**(23/4)*x**4*(a/(b*x**4) + 1)**(3/4)*gamma(-7/4)/(64*a**5*b**4*x**4*gam ma(5/4) + 128*a**4*b**5*x**8*gamma(5/4) + 64*a**3*b**6*x**12*gamma(5/4)...
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.14 \[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=\frac {1}{21} \, d {\left (\frac {21 \, b^{2} x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}} + \frac {\frac {14 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b}{x^{3}} - \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}}}{x^{7}}}{a^{3}}\right )} - \frac {1}{77} \, c {\left (\frac {77 \, b^{3} x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{4}} + \frac {\frac {77 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{2}}{x^{3}} - \frac {33 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b}{x^{7}} + \frac {7 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}}}{x^{11}}}{a^{4}}\right )} \] Input:
integrate((d*x^4+c)/x^12/(b*x^4+a)^(5/4),x, algorithm="maxima")
Output:
1/21*d*(21*b^2*x/((b*x^4 + a)^(1/4)*a^3) + (14*(b*x^4 + a)^(3/4)*b/x^3 - 3 *(b*x^4 + a)^(7/4)/x^7)/a^3) - 1/77*c*(77*b^3*x/((b*x^4 + a)^(1/4)*a^4) + (77*(b*x^4 + a)^(3/4)*b^2/x^3 - 33*(b*x^4 + a)^(7/4)*b/x^7 + 7*(b*x^4 + a) ^(11/4)/x^11)/a^4)
\[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{12}} \,d x } \] Input:
integrate((d*x^4+c)/x^12/(b*x^4+a)^(5/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(5/4)*x^12), x)
Time = 4.00 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=\frac {b\,{\left (b\,x^4+a\right )}^{3/4}\,\left (121\,a\,d-153\,b\,c\right )}{231\,a^4\,x^3}-\frac {{\left (b\,x^4+a\right )}^{3/4}\,\left (11\,a^2\,d-19\,a\,b\,c\right )}{77\,a^4\,x^7}-\frac {x\,\left (b^3\,c-a\,b^2\,d\right )}{a^4\,{\left (b\,x^4+a\right )}^{1/4}}-\frac {c\,{\left (b\,x^4+a\right )}^{3/4}}{11\,a^2\,x^{11}} \] Input:
int((c + d*x^4)/(x^12*(a + b*x^4)^(5/4)),x)
Output:
(b*(a + b*x^4)^(3/4)*(121*a*d - 153*b*c))/(231*a^4*x^3) - ((a + b*x^4)^(3/ 4)*(11*a^2*d - 19*a*b*c))/(77*a^4*x^7) - (x*(b^3*c - a*b^2*d))/(a^4*(a + b *x^4)^(1/4)) - (c*(a + b*x^4)^(3/4))/(11*a^2*x^11)
\[ \int \frac {c+d x^4}{x^{12} \left (a+b x^4\right )^{5/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,x^{12}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b \,x^{16}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a \,x^{8}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b \,x^{12}}d x \right ) d \] Input:
int((d*x^4+c)/x^12/(b*x^4+a)^(5/4),x)
Output:
int(1/((a + b*x**4)**(1/4)*a*x**12 + (a + b*x**4)**(1/4)*b*x**16),x)*c + i nt(1/((a + b*x**4)**(1/4)*a*x**8 + (a + b*x**4)**(1/4)*b*x**12),x)*d