Integrand size = 22, antiderivative size = 100 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=-\frac {a^2 (b c-a d)}{5 b^4 \left (a+b x^4\right )^{5/4}}+\frac {a (2 b c-3 a d)}{b^4 \sqrt [4]{a+b x^4}}+\frac {(b c-3 a d) \left (a+b x^4\right )^{3/4}}{3 b^4}+\frac {d \left (a+b x^4\right )^{7/4}}{7 b^4} \] Output:
-1/5*a^2*(-a*d+b*c)/b^4/(b*x^4+a)^(5/4)+a*(-3*a*d+2*b*c)/b^4/(b*x^4+a)^(1/ 4)+1/3*(-3*a*d+b*c)*(b*x^4+a)^(3/4)/b^4+1/7*d*(b*x^4+a)^(7/4)/b^4
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\frac {224 a^2 b c-384 a^3 d+280 a b^2 c x^4-480 a^2 b d x^4+35 b^3 c x^8-60 a b^2 d x^8+15 b^3 d x^{12}}{105 b^4 \left (a+b x^4\right )^{5/4}} \] Input:
Integrate[(x^11*(c + d*x^4))/(a + b*x^4)^(9/4),x]
Output:
(224*a^2*b*c - 384*a^3*d + 280*a*b^2*c*x^4 - 480*a^2*b*d*x^4 + 35*b^3*c*x^ 8 - 60*a*b^2*d*x^8 + 15*b^3*d*x^12)/(105*b^4*(a + b*x^4)^(5/4))
Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {948, 86, 2009}
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.
\(\displaystyle \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{4} \int \frac {x^8 \left (d x^4+c\right )}{\left (b x^4+a\right )^{9/4}}dx^4\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{4} \int \left (-\frac {(a d-b c) a^2}{b^3 \left (b x^4+a\right )^{9/4}}+\frac {(3 a d-2 b c) a}{b^3 \left (b x^4+a\right )^{5/4}}+\frac {d \left (b x^4+a\right )^{3/4}}{b^3}+\frac {b c-3 a d}{b^3 \sqrt [4]{b x^4+a}}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {4 a^2 (b c-a d)}{5 b^4 \left (a+b x^4\right )^{5/4}}+\frac {4 a (2 b c-3 a d)}{b^4 \sqrt [4]{a+b x^4}}+\frac {4 \left (a+b x^4\right )^{3/4} (b c-3 a d)}{3 b^4}+\frac {4 d \left (a+b x^4\right )^{7/4}}{7 b^4}\right )\) |
Input:
Int[(x^11*(c + d*x^4))/(a + b*x^4)^(9/4),x]
Output:
((-4*a^2*(b*c - a*d))/(5*b^4*(a + b*x^4)^(5/4)) + (4*a*(2*b*c - 3*a*d))/(b ^4*(a + b*x^4)^(1/4)) + (4*(b*c - 3*a*d)*(a + b*x^4)^(3/4))/(3*b^4) + (4*d *(a + b*x^4)^(7/4))/(7*b^4))/4
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(-\frac {128 \left (-\frac {35 \left (\frac {3 d \,x^{4}}{7}+c \right ) x^{8} b^{3}}{384}-\frac {35 \left (-\frac {3 d \,x^{4}}{14}+c \right ) x^{4} a \,b^{2}}{48}-\frac {7 \left (-\frac {15 d \,x^{4}}{7}+c \right ) a^{2} b}{12}+a^{3} d \right )}{35 \left (b \,x^{4}+a \right )^{\frac {5}{4}} b^{4}}\) | \(68\) |
gosper | \(-\frac {-15 b^{3} d \,x^{12}+60 a \,b^{2} d \,x^{8}-35 c \,b^{3} x^{8}+480 a^{2} b d \,x^{4}-280 a \,b^{2} c \,x^{4}+384 a^{3} d -224 a^{2} b c}{105 \left (b \,x^{4}+a \right )^{\frac {5}{4}} b^{4}}\) | \(77\) |
trager | \(-\frac {-15 b^{3} d \,x^{12}+60 a \,b^{2} d \,x^{8}-35 c \,b^{3} x^{8}+480 a^{2} b d \,x^{4}-280 a \,b^{2} c \,x^{4}+384 a^{3} d -224 a^{2} b c}{105 \left (b \,x^{4}+a \right )^{\frac {5}{4}} b^{4}}\) | \(77\) |
orering | \(-\frac {-15 b^{3} d \,x^{12}+60 a \,b^{2} d \,x^{8}-35 c \,b^{3} x^{8}+480 a^{2} b d \,x^{4}-280 a \,b^{2} c \,x^{4}+384 a^{3} d -224 a^{2} b c}{105 \left (b \,x^{4}+a \right )^{\frac {5}{4}} b^{4}}\) | \(77\) |
risch | \(-\frac {\left (-3 d b \,x^{4}+18 a d -7 c b \right ) \left (b \,x^{4}+a \right )^{\frac {3}{4}}}{21 b^{4}}-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (15 a b d \,x^{4}-10 b^{2} c \,x^{4}+14 a^{2} d -9 a b c \right ) a}{5 b^{4} \left (x^{8} b^{2}+2 a \,x^{4} b +a^{2}\right )}\) | \(96\) |
Input:
int(x^11*(d*x^4+c)/(b*x^4+a)^(9/4),x,method=_RETURNVERBOSE)
Output:
-128/35*(-35/384*(3/7*d*x^4+c)*x^8*b^3-35/48*(-3/14*d*x^4+c)*x^4*a*b^2-7/1 2*(-15/7*d*x^4+c)*a^2*b+a^3*d)/(b*x^4+a)^(5/4)/b^4
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.99 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\frac {{\left (15 \, b^{3} d x^{12} + 5 \, {\left (7 \, b^{3} c - 12 \, a b^{2} d\right )} x^{8} + 40 \, {\left (7 \, a b^{2} c - 12 \, a^{2} b d\right )} x^{4} + 224 \, a^{2} b c - 384 \, a^{3} d\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{105 \, {\left (b^{6} x^{8} + 2 \, a b^{5} x^{4} + a^{2} b^{4}\right )}} \] Input:
integrate(x^11*(d*x^4+c)/(b*x^4+a)^(9/4),x, algorithm="fricas")
Output:
1/105*(15*b^3*d*x^12 + 5*(7*b^3*c - 12*a*b^2*d)*x^8 + 40*(7*a*b^2*c - 12*a ^2*b*d)*x^4 + 224*a^2*b*c - 384*a^3*d)*(b*x^4 + a)^(3/4)/(b^6*x^8 + 2*a*b^ 5*x^4 + a^2*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (88) = 176\).
Time = 1.48 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.38 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\begin {cases} - \frac {384 a^{3} d}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} + \frac {224 a^{2} b c}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} - \frac {480 a^{2} b d x^{4}}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} + \frac {280 a b^{2} c x^{4}}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} - \frac {60 a b^{2} d x^{8}}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} + \frac {35 b^{3} c x^{8}}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} + \frac {15 b^{3} d x^{12}}{105 a b^{4} \sqrt [4]{a + b x^{4}} + 105 b^{5} x^{4} \sqrt [4]{a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {\frac {c x^{12}}{12} + \frac {d x^{16}}{16}}{a^{\frac {9}{4}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**11*(d*x**4+c)/(b*x**4+a)**(9/4),x)
Output:
Piecewise((-384*a**3*d/(105*a*b**4*(a + b*x**4)**(1/4) + 105*b**5*x**4*(a + b*x**4)**(1/4)) + 224*a**2*b*c/(105*a*b**4*(a + b*x**4)**(1/4) + 105*b** 5*x**4*(a + b*x**4)**(1/4)) - 480*a**2*b*d*x**4/(105*a*b**4*(a + b*x**4)** (1/4) + 105*b**5*x**4*(a + b*x**4)**(1/4)) + 280*a*b**2*c*x**4/(105*a*b**4 *(a + b*x**4)**(1/4) + 105*b**5*x**4*(a + b*x**4)**(1/4)) - 60*a*b**2*d*x* *8/(105*a*b**4*(a + b*x**4)**(1/4) + 105*b**5*x**4*(a + b*x**4)**(1/4)) + 35*b**3*c*x**8/(105*a*b**4*(a + b*x**4)**(1/4) + 105*b**5*x**4*(a + b*x**4 )**(1/4)) + 15*b**3*d*x**12/(105*a*b**4*(a + b*x**4)**(1/4) + 105*b**5*x** 4*(a + b*x**4)**(1/4)), Ne(b, 0)), ((c*x**12/12 + d*x**16/16)/a**(9/4), Tr ue))
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.18 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\frac {1}{35} \, d {\left (\frac {5 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}}}{b^{4}} - \frac {35 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a}{b^{4}} - \frac {105 \, a^{2}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{4}} + \frac {7 \, a^{3}}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{4}}\right )} + \frac {1}{15} \, c {\left (\frac {5 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b^{3}} + \frac {30 \, a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{3}} - \frac {3 \, a^{2}}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{3}}\right )} \] Input:
integrate(x^11*(d*x^4+c)/(b*x^4+a)^(9/4),x, algorithm="maxima")
Output:
1/35*d*(5*(b*x^4 + a)^(7/4)/b^4 - 35*(b*x^4 + a)^(3/4)*a/b^4 - 105*a^2/((b *x^4 + a)^(1/4)*b^4) + 7*a^3/((b*x^4 + a)^(5/4)*b^4)) + 1/15*c*(5*(b*x^4 + a)^(3/4)/b^3 + 30*a/((b*x^4 + a)^(1/4)*b^3) - 3*a^2/((b*x^4 + a)^(5/4)*b^ 3))
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.05 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\frac {10 \, {\left (b x^{4} + a\right )} a b c - a^{2} b c - 15 \, {\left (b x^{4} + a\right )} a^{2} d + a^{3} d}{5 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b^{4}} + \frac {7 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{25} c + 3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{24} d - 21 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a b^{24} d}{21 \, b^{28}} \] Input:
integrate(x^11*(d*x^4+c)/(b*x^4+a)^(9/4),x, algorithm="giac")
Output:
1/5*(10*(b*x^4 + a)*a*b*c - a^2*b*c - 15*(b*x^4 + a)*a^2*d + a^3*d)/((b*x^ 4 + a)^(5/4)*b^4) + 1/21*(7*(b*x^4 + a)^(3/4)*b^25*c + 3*(b*x^4 + a)^(7/4) *b^24*d - 21*(b*x^4 + a)^(3/4)*a*b^24*d)/b^28
Time = 3.64 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\frac {\frac {d\,{\left (b\,x^4+a\right )}^3}{7}+\frac {a^3\,d}{5}-a\,d\,{\left (b\,x^4+a\right )}^2+\frac {b\,c\,{\left (b\,x^4+a\right )}^2}{3}-3\,a^2\,d\,\left (b\,x^4+a\right )-\frac {a^2\,b\,c}{5}+2\,a\,b\,c\,\left (b\,x^4+a\right )}{b^4\,{\left (b\,x^4+a\right )}^{5/4}} \] Input:
int((x^11*(c + d*x^4))/(a + b*x^4)^(9/4),x)
Output:
((d*(a + b*x^4)^3)/7 + (a^3*d)/5 - a*d*(a + b*x^4)^2 + (b*c*(a + b*x^4)^2) /3 - 3*a^2*d*(a + b*x^4) - (a^2*b*c)/5 + 2*a*b*c*(a + b*x^4))/(b^4*(a + b* x^4)^(5/4))
\[ \int \frac {x^{11} \left (c+d x^4\right )}{\left (a+b x^4\right )^{9/4}} \, dx=\left (\int \frac {x^{15}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} x^{8}}d x \right ) d +\left (\int \frac {x^{11}}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{2}+2 \left (b \,x^{4}+a \right )^{\frac {1}{4}} a b \,x^{4}+\left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2} x^{8}}d x \right ) c \] Input:
int(x^11*(d*x^4+c)/(b*x^4+a)^(9/4),x)
Output:
int(x**15/((a + b*x**4)**(1/4)*a**2 + 2*(a + b*x**4)**(1/4)*a*b*x**4 + (a + b*x**4)**(1/4)*b**2*x**8),x)*d + int(x**11/((a + b*x**4)**(1/4)*a**2 + 2 *(a + b*x**4)**(1/4)*a*b*x**4 + (a + b*x**4)**(1/4)*b**2*x**8),x)*c