Integrand size = 22, antiderivative size = 205 \[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=-\frac {a^2 (b c-a d) x}{10 b^4 \left (a+b x^4\right )^{5/2}}+\frac {a (21 b c-31 a d) x}{60 b^4 \left (a+b x^4\right )^{3/2}}-\frac {(15 b c-41 a d) x}{24 b^4 \sqrt {a+b x^4}}+\frac {d x \sqrt {a+b x^4}}{3 b^4}+\frac {(3 b c-13 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{16 \sqrt [4]{a} b^{17/4} \sqrt {a+b x^4}} \] Output:
-1/10*a^2*(-a*d+b*c)*x/b^4/(b*x^4+a)^(5/2)+1/60*a*(-31*a*d+21*b*c)*x/b^4/( b*x^4+a)^(3/2)-1/24*(-41*a*d+15*b*c)*x/b^4/(b*x^4+a)^(1/2)+1/3*d*x*(b*x^4+ a)^(1/2)/b^4+1/16*(-13*a*d+3*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))/a^(1/4)/b^(17/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.64 \[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {x \left (195 a^3 d-9 a^2 b \left (5 c-52 d x^4\right )+5 b^3 x^8 \left (-15 c+8 d x^4\right )+a b^2 x^4 \left (-108 c+325 d x^4\right )+15 (3 b c-13 a d) \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{120 b^4 \left (a+b x^4\right )^{5/2}} \] Input:
Integrate[(x^12*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(x*(195*a^3*d - 9*a^2*b*(5*c - 52*d*x^4) + 5*b^3*x^8*(-15*c + 8*d*x^4) + a *b^2*x^4*(-108*c + 325*d*x^4) + 15*(3*b*c - 13*a*d)*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]))/(120*b^4*(a + b*x^4)^(5/2))
Time = 0.48 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {959, 817, 817, 817, 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.
| \(\displaystyle \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(3 b c-13 a d) \int \frac {x^{12}}{\left (b x^4+a\right )^{7/2}}dx}{3 b}+\frac {d x^{13}}{3 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(3 b c-13 a d) \left (\frac {9 \int \frac {x^8}{\left (b x^4+a\right )^{5/2}}dx}{10 b}-\frac {x^9}{10 b \left (a+b x^4\right )^{5/2}}\right )}{3 b}+\frac {d x^{13}}{3 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(3 b c-13 a d) \left (\frac {9 \left (\frac {5 \int \frac {x^4}{\left (b x^4+a\right )^{3/2}}dx}{6 b}-\frac {x^5}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^9}{10 b \left (a+b x^4\right )^{5/2}}\right )}{3 b}+\frac {d x^{13}}{3 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(3 b c-13 a d) \left (\frac {9 \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {b x^4+a}}dx}{2 b}-\frac {x}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^9}{10 b \left (a+b x^4\right )^{5/2}}\right )}{3 b}+\frac {d x^{13}}{3 b \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(3 b c-13 a d) \left (\frac {9 \left (\frac {5 \left (\frac {\left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} b^{5/4} \sqrt {a+b x^4}}-\frac {x}{2 b \sqrt {a+b x^4}}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 b}-\frac {x^9}{10 b \left (a+b x^4\right )^{5/2}}\right )}{3 b}+\frac {d x^{13}}{3 b \left (a+b x^4\right )^{5/2}}\) |
Input:
Int[(x^12*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(d*x^13)/(3*b*(a + b*x^4)^(5/2)) + ((3*b*c - 13*a*d)*(-1/10*x^9/(b*(a + b* x^4)^(5/2)) + (9*(-1/6*x^5/(b*(a + b*x^4)^(3/2)) + (5*(-1/2*x/(b*Sqrt[a + b*x^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])/(4*a^(1/4)*b^(5/4)*Sqr t[a + b*x^4])))/(6*b)))/(10*b)))/(3*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[((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 = 6.96 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.09
| method | result | size |
| elliptic | \(\frac {a^{2} x \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{10 b^{7} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {a x \left (31 a d -21 c b \right ) \sqrt {b \,x^{4}+a}}{60 b^{6} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x \left (41 a d -15 c b \right )}{24 b^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {d x \sqrt {b \,x^{4}+a}}{3 b^{4}}+\frac {\left (-\frac {3 a d -c b}{b^{4}}+\frac {41 a d -15 c b}{24 b^{4}}-\frac {d a}{3 b^{4}}\right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(224\) |
| default | \(c \left (-\frac {a^{2} x \sqrt {b \,x^{4}+a}}{10 b^{6} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {7 a x \sqrt {b \,x^{4}+a}}{20 b^{5} \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {5 x}{8 b^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 b^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {a^{3} x \sqrt {b \,x^{4}+a}}{10 b^{7} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {31 a^{2} x \sqrt {b \,x^{4}+a}}{60 b^{6} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {41 a x}{24 b^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {x \sqrt {b \,x^{4}+a}}{3 b^{4}}-\frac {13 a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 b^{4} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(323\) |
| risch | \(\frac {d x \sqrt {b \,x^{4}+a}}{3 b^{4}}-\frac {\frac {10 a d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-9 a \left (2 a d -c b \right ) \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+3 a^{2} \left (4 a d -3 c b \right ) \left (\frac {x \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {5 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-3 a^{3} \left (a d -c b \right ) \left (\frac {x \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {3 x \sqrt {b \,x^{4}+a}}{20 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {3 x}{8 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{8 a^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {3 c b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}}{3 b^{4}}\) | \(573\) |
Input:
int(x^12*(d*x^4+c)/(b*x^4+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/10*a^2*x/b^7*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^3-1/60*a*x*(31*a*d-21*b *c)/b^6*(b*x^4+a)^(1/2)/(x^4+a/b)^2+1/24/b^4*x*(41*a*d-15*b*c)/((x^4+a/b)* b)^(1/2)+1/3*d*x*(b*x^4+a)^(1/2)/b^4+(-1/b^4*(3*a*d-b*c)+1/24/b^4*(41*a*d- 15*b*c)-1/3/b^4*d*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 = 0.09 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.12 \[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {15 \, {\left ({\left (3 \, b^{4} c - 13 \, a b^{3} d\right )} x^{12} + 3 \, {\left (3 \, a b^{3} c - 13 \, a^{2} b^{2} d\right )} x^{8} + 3 \, a^{3} b c - 13 \, a^{4} d + 3 \, {\left (3 \, a^{2} b^{2} c - 13 \, a^{3} b d\right )} x^{4}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (40 \, a b^{3} d x^{13} - 25 \, {\left (3 \, a b^{3} c - 13 \, a^{2} b^{2} d\right )} x^{9} - 36 \, {\left (3 \, a^{2} b^{2} c - 13 \, a^{3} b d\right )} x^{5} - 15 \, {\left (3 \, a^{3} b c - 13 \, a^{4} d\right )} x\right )} \sqrt {b x^{4} + a}}{120 \, {\left (a b^{7} x^{12} + 3 \, a^{2} b^{6} x^{8} + 3 \, a^{3} b^{5} x^{4} + a^{4} b^{4}\right )}} \] Input:
integrate(x^12*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="fricas")
Output:
1/120*(15*((3*b^4*c - 13*a*b^3*d)*x^12 + 3*(3*a*b^3*c - 13*a^2*b^2*d)*x^8 + 3*a^3*b*c - 13*a^4*d + 3*(3*a^2*b^2*c - 13*a^3*b*d)*x^4)*sqrt(b)*(-a/b)^ (3/4)*elliptic_f(arcsin((-a/b)^(1/4)/x), -1) + (40*a*b^3*d*x^13 - 25*(3*a* b^3*c - 13*a^2*b^2*d)*x^9 - 36*(3*a^2*b^2*c - 13*a^3*b*d)*x^5 - 15*(3*a^3* b*c - 13*a^4*d)*x)*sqrt(b*x^4 + a))/(a*b^7*x^12 + 3*a^2*b^6*x^8 + 3*a^3*b^ 5*x^4 + a^4*b^4)
Result contains complex when optimal does not.
Time = 175.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.39 \[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {c x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {13}{4}, \frac {7}{2} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} \Gamma \left (\frac {17}{4}\right )} + \frac {d x^{17} \Gamma \left (\frac {17}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {17}{4} \\ \frac {21}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} \Gamma \left (\frac {21}{4}\right )} \] Input:
integrate(x**12*(d*x**4+c)/(b*x**4+a)**(7/2),x)
Output:
c*x**13*gamma(13/4)*hyper((13/4, 7/2), (17/4,), b*x**4*exp_polar(I*pi)/a)/ (4*a**(7/2)*gamma(17/4)) + d*x**17*gamma(17/4)*hyper((7/2, 17/4), (21/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(7/2)*gamma(21/4))
\[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{12}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^12*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^12/(b*x^4 + a)^(7/2), x)
\[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{12}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^12*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^12/(b*x^4 + a)^(7/2), x)
Timed out. \[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int \frac {x^{12}\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/2}} \,d x \] Input:
int((x^12*(c + d*x^4))/(a + b*x^4)^(7/2),x)
Output:
int((x^12*(c + d*x^4))/(a + b*x^4)^(7/2), x)
\[ \int \frac {x^{12} \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {65 \sqrt {b \,x^{4}+a}\, a^{3} d x -15 \sqrt {b \,x^{4}+a}\, a^{2} b c x +117 \sqrt {b \,x^{4}+a}\, a^{2} b d \,x^{5}-27 \sqrt {b \,x^{4}+a}\, a \,b^{2} c \,x^{5}+65 \sqrt {b \,x^{4}+a}\, a \,b^{2} d \,x^{9}-15 \sqrt {b \,x^{4}+a}\, b^{3} c \,x^{9}+5 \sqrt {b \,x^{4}+a}\, b^{3} d \,x^{13}-65 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{7} d +15 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{6} b c -195 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{6} b d \,x^{4}+45 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} b^{2} c \,x^{4}-195 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} b^{2} d \,x^{8}+45 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{4} b^{3} c \,x^{8}-65 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{4} b^{3} d \,x^{12}+15 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{3} b^{4} c \,x^{12}}{15 b^{4} \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )} \] Input:
int(x^12*(d*x^4+c)/(b*x^4+a)^(7/2),x)
Output:
(65*sqrt(a + b*x**4)*a**3*d*x - 15*sqrt(a + b*x**4)*a**2*b*c*x + 117*sqrt( a + b*x**4)*a**2*b*d*x**5 - 27*sqrt(a + b*x**4)*a*b**2*c*x**5 + 65*sqrt(a + b*x**4)*a*b**2*d*x**9 - 15*sqrt(a + b*x**4)*b**3*c*x**9 + 5*sqrt(a + b*x **4)*b**3*d*x**13 - 65*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*d + 15*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*c - 195*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*d*x**4 + 45*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**2*c*x**4 - 195*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**2*d*x**8 + 4 5*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**3*c*x**8 - 65*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 **3*d*x**12 + 15*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**3*b**4*c*x**12)/(15*b**4*(a**3 + 3*a**2*b*x**4 + 3*a*b**2*x**8 + b**3*x**12))