Integrand size = 22, antiderivative size = 238 \[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {a^3 (b c-a d) x}{10 b^5 \left (a+b x^4\right )^{5/2}}-\frac {a^2 (31 b c-41 a d) x}{60 b^5 \left (a+b x^4\right )^{3/2}}+\frac {a (41 b c-79 a d) x}{24 b^5 \sqrt {a+b x^4}}+\frac {(7 b c-26 a d) x \sqrt {a+b x^4}}{21 b^5}+\frac {d x^5 \sqrt {a+b x^4}}{7 b^4}-\frac {13 a^{3/4} (7 b c-17 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{112 b^{21/4} \sqrt {a+b x^4}} \] Output:
1/10*a^3*(-a*d+b*c)*x/b^5/(b*x^4+a)^(5/2)-1/60*a^2*(-41*a*d+31*b*c)*x/b^5/ (b*x^4+a)^(3/2)+1/24*a*(-79*a*d+41*b*c)*x/b^5/(b*x^4+a)^(1/2)+1/21*(-26*a* d+7*b*c)*x*(b*x^4+a)^(1/2)/b^5+1/7*d*x^5*(b*x^4+a)^(1/2)/b^4-13/112*a^(3/4 )*(-17*a*d+7*b*c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2 )^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/b^(21/4)/ (b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66 \[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {-3315 a^4 d x+13 a^2 b^2 x^5 \left (252 c-425 d x^4\right )+39 a^3 b x \left (35 c-204 d x^4\right )+5 a b^3 x^9 \left (455 c-136 d x^4\right )+40 b^4 x^{13} \left (7 c+3 d x^4\right )+195 a (-7 b c+17 a d) x \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )}{840 b^5 \left (a+b x^4\right )^{5/2}} \] Input:
Integrate[(x^16*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(-3315*a^4*d*x + 13*a^2*b^2*x^5*(252*c - 425*d*x^4) + 39*a^3*b*x*(35*c - 2 04*d*x^4) + 5*a*b^3*x^9*(455*c - 136*d*x^4) + 40*b^4*x^13*(7*c + 3*d*x^4) + 195*a*(-7*b*c + 17*a*d)*x*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometr ic2F1[1/4, 1/2, 5/4, -((b*x^4)/a)])/(840*b^5*(a + b*x^4)^(5/2))
Time = 0.55 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 817, 817, 817, 843, 761}
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.
| \(\displaystyle \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(7 b c-17 a d) \int \frac {x^{16}}{\left (b x^4+a\right )^{7/2}}dx}{7 b}+\frac {d x^{17}}{7 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(7 b c-17 a d) \left (\frac {13 \int \frac {x^{12}}{\left (b x^4+a\right )^{5/2}}dx}{10 b}-\frac {x^{13}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{7 b}+\frac {d x^{17}}{7 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(7 b c-17 a d) \left (\frac {13 \left (\frac {3 \int \frac {x^8}{\left (b x^4+a\right )^{3/2}}dx}{2 b}-\frac {x^9}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{7 b}+\frac {d x^{17}}{7 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(7 b c-17 a d) \left (\frac {13 \left (\frac {3 \left (\frac {5 \int \frac {x^4}{\sqrt {b x^4+a}}dx}{2 b}-\frac {x^5}{2 b \sqrt {a+b x^4}}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{7 b}+\frac {d x^{17}}{7 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(7 b c-17 a d) \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {x \sqrt {a+b x^4}}{3 b}-\frac {a \int \frac {1}{\sqrt {b x^4+a}}dx}{3 b}\right )}{2 b}-\frac {x^5}{2 b \sqrt {a+b x^4}}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{7 b}+\frac {d x^{17}}{7 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(7 b c-17 a d) \left (\frac {13 \left (\frac {3 \left (\frac {5 \left (\frac {x \sqrt {a+b x^4}}{3 b}-\frac {a^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 b^{5/4} \sqrt {a+b x^4}}\right )}{2 b}-\frac {x^5}{2 b \sqrt {a+b x^4}}\right )}{2 b}-\frac {x^9}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^{13}}{10 b \left (a+b x^4\right )^{5/2}}\right )}{7 b}+\frac {d x^{17}}{7 b \left (a+b x^4\right )^{5/2}}\) |
Input:
Int[(x^16*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(d*x^17)/(7*b*(a + b*x^4)^(5/2)) + ((7*b*c - 17*a*d)*(-1/10*x^13/(b*(a + b *x^4)^(5/2)) + (13*(-1/6*x^9/(b*(a + b*x^4)^(3/2)) + (3*(-1/2*x^5/(b*Sqrt[ a + b*x^4]) + (5*((x*Sqrt[a + b*x^4])/(3*b) - (a^(3/4)*(Sqrt[a] + Sqrt[b]* x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/ 4)*x)/a^(1/4)], 1/2])/(6*b^(5/4)*Sqrt[a + b*x^4])))/(2*b)))/(2*b)))/(10*b) ))/(7*b)
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Result contains complex when optimal does not.
Time = 10.13 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.21
| method | result | size |
| elliptic | \(-\frac {a^{3} x \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{10 b^{8} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {a^{2} x \left (41 a d -31 c b \right ) \sqrt {b \,x^{4}+a}}{60 b^{7} \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {a x \left (79 a d -41 c b \right )}{24 b^{5} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {d \,x^{5} \sqrt {b \,x^{4}+a}}{7 b^{4}}+\frac {\left (-\frac {3 a d -c b}{b^{4}}-\frac {5 d a}{7 b^{4}}\right ) x \sqrt {b \,x^{4}+a}}{3 b}+\frac {\left (\frac {3 a \left (2 a d -c b \right )}{b^{5}}-\frac {a \left (79 a d -41 c b \right )}{24 b^{5}}-\frac {\left (-\frac {3 a d -c b}{b^{4}}-\frac {5 d a}{7 b^{4}}\right ) a}{3 b}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(289\) |
| default | \(c \left (\frac {a^{3} x \sqrt {b \,x^{4}+a}}{10 b^{7} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {31 a^{2} x \sqrt {b \,x^{4}+a}}{60 b^{6} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {41 a x}{24 b^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x \sqrt {b \,x^{4}+a}}{3 b^{4}}-\frac {13 a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 b^{4} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {a^{4} x \sqrt {b \,x^{4}+a}}{10 b^{8} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {41 a^{3} x \sqrt {b \,x^{4}+a}}{60 b^{7} \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {79 a^{2} x}{24 b^{5} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b^{4}}-\frac {26 a x \sqrt {b \,x^{4}+a}}{21 b^{5}}+\frac {221 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{56 b^{5} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(364\) |
| risch | \(-\frac {x \left (-3 d b \,x^{4}+26 a d -7 c b \right ) \sqrt {b \,x^{4}+a}}{21 b^{5}}+\frac {a \left (\frac {152 a d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-42 a \left (5 a d -3 c b \right ) \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+21 a^{2} \left (5 a d -4 c b \right ) \left (\frac {x \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {5 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-21 a^{3} \left (a d -c b \right ) \left (\frac {x \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {3 x \sqrt {b \,x^{4}+a}}{20 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {3 x}{8 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 a^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {70 c b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )}{21 b^{5}}\) | \(589\) |
Input:
int(x^16*(d*x^4+c)/(b*x^4+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
-1/10*a^3*x/b^8*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^3+1/60*a^2*x*(41*a*d-3 1*b*c)/b^7*(b*x^4+a)^(1/2)/(x^4+a/b)^2-1/24/b^5*a*x*(79*a*d-41*b*c)/((x^4+ a/b)*b)^(1/2)+1/7*d*x^5*(b*x^4+a)^(1/2)/b^4+1/3*(-1/b^4*(3*a*d-b*c)-5/7/b^ 4*d*a)/b*x*(b*x^4+a)^(1/2)+(3*a*(2*a*d-b*c)/b^5-1/24/b^5*a*(79*a*d-41*b*c) -1/3*(-1/b^4*(3*a*d-b*c)-5/7/b^4*d*a)/b*a)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/ a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2) *EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)
Time = 0.10 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.03 \[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=-\frac {195 \, {\left ({\left (7 \, b^{4} c - 17 \, a b^{3} d\right )} x^{12} + 3 \, {\left (7 \, a b^{3} c - 17 \, a^{2} b^{2} d\right )} x^{8} + 7 \, a^{3} b c - 17 \, a^{4} d + 3 \, {\left (7 \, a^{2} b^{2} c - 17 \, a^{3} b d\right )} x^{4}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (120 \, b^{4} d x^{17} + 40 \, {\left (7 \, b^{4} c - 17 \, a b^{3} d\right )} x^{13} + 325 \, {\left (7 \, a b^{3} c - 17 \, a^{2} b^{2} d\right )} x^{9} + 468 \, {\left (7 \, a^{2} b^{2} c - 17 \, a^{3} b d\right )} x^{5} + 195 \, {\left (7 \, a^{3} b c - 17 \, a^{4} d\right )} x\right )} \sqrt {b x^{4} + a}}{840 \, {\left (b^{8} x^{12} + 3 \, a b^{7} x^{8} + 3 \, a^{2} b^{6} x^{4} + a^{3} b^{5}\right )}} \] Input:
integrate(x^16*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="fricas")
Output:
-1/840*(195*((7*b^4*c - 17*a*b^3*d)*x^12 + 3*(7*a*b^3*c - 17*a^2*b^2*d)*x^ 8 + 7*a^3*b*c - 17*a^4*d + 3*(7*a^2*b^2*c - 17*a^3*b*d)*x^4)*sqrt(b)*(-a/b )^(3/4)*elliptic_f(arcsin((-a/b)^(1/4)/x), -1) - (120*b^4*d*x^17 + 40*(7*b ^4*c - 17*a*b^3*d)*x^13 + 325*(7*a*b^3*c - 17*a^2*b^2*d)*x^9 + 468*(7*a^2* b^2*c - 17*a^3*b*d)*x^5 + 195*(7*a^3*b*c - 17*a^4*d)*x)*sqrt(b*x^4 + a))/( b^8*x^12 + 3*a*b^7*x^8 + 3*a^2*b^6*x^4 + a^3*b^5)
Timed out. \[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(x**16*(d*x**4+c)/(b*x**4+a)**(7/2),x)
Output:
Timed out
\[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{16}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^16*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^16/(b*x^4 + a)^(7/2), x)
\[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{16}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^16*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^16/(b*x^4 + a)^(7/2), x)
Timed out. \[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int \frac {x^{16}\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/2}} \,d x \] Input:
int((x^16*(c + d*x^4))/(a + b*x^4)^(7/2),x)
Output:
int((x^16*(c + d*x^4))/(a + b*x^4)^(7/2), x)
\[ \int \frac {x^{16} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {-1105 \sqrt {b \,x^{4}+a}\, a^{4} d x +455 \sqrt {b \,x^{4}+a}\, a^{3} b c x -1989 \sqrt {b \,x^{4}+a}\, a^{3} b d \,x^{5}+819 \sqrt {b \,x^{4}+a}\, a^{2} b^{2} c \,x^{5}-1105 \sqrt {b \,x^{4}+a}\, a^{2} b^{2} d \,x^{9}+455 \sqrt {b \,x^{4}+a}\, a \,b^{3} c \,x^{9}-85 \sqrt {b \,x^{4}+a}\, a \,b^{3} d \,x^{13}+35 \sqrt {b \,x^{4}+a}\, b^{4} c \,x^{13}+15 \sqrt {b \,x^{4}+a}\, b^{4} d \,x^{17}+1105 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{8} d -455 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{7} b c +3315 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{7} b d \,x^{4}-1365 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{6} b^{2} c \,x^{4}+3315 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{6} b^{2} d \,x^{8}-1365 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} b^{3} c \,x^{8}+1105 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} b^{3} d \,x^{12}-455 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{4} b^{4} c \,x^{12}}{105 b^{5} \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )} \] Input:
int(x^16*(d*x^4+c)/(b*x^4+a)^(7/2),x)
Output:
( - 1105*sqrt(a + b*x**4)*a**4*d*x + 455*sqrt(a + b*x**4)*a**3*b*c*x - 198 9*sqrt(a + b*x**4)*a**3*b*d*x**5 + 819*sqrt(a + b*x**4)*a**2*b**2*c*x**5 - 1105*sqrt(a + b*x**4)*a**2*b**2*d*x**9 + 455*sqrt(a + b*x**4)*a*b**3*c*x* *9 - 85*sqrt(a + b*x**4)*a*b**3*d*x**13 + 35*sqrt(a + b*x**4)*b**4*c*x**13 + 15*sqrt(a + b*x**4)*b**4*d*x**17 + 1105*int(sqrt(a + b*x**4)/(a**4 + 4* a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**8*d - 455*int(sqrt(a + b*x**4)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b* *3*x**12 + b**4*x**16),x)*a**7*b*c + 3315*int(sqrt(a + b*x**4)/(a**4 + 4*a **3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**7*b*d*x **4 - 1365*int(sqrt(a + b*x**4)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**6*b**2*c*x**4 + 3315*int(sqrt(a + b*x* *4)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16 ),x)*a**6*b**2*d*x**8 - 1365*int(sqrt(a + b*x**4)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**5*b**3*c*x**8 + 1105 *int(sqrt(a + b*x**4)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3* x**12 + b**4*x**16),x)*a**5*b**3*d*x**12 - 455*int(sqrt(a + b*x**4)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**4* b**4*c*x**12)/(105*b**5*(a**3 + 3*a**2*b*x**4 + 3*a*b**2*x**8 + b**3*x**12 ))