Integrand size = 22, antiderivative size = 355 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=-\frac {c}{a x \left (a+b x^4\right )^{5/2}}-\frac {(11 b c-a d) x^3}{10 a^2 \left (a+b x^4\right )^{5/2}}-\frac {7 (11 b c-a d) x^3}{60 a^3 \left (a+b x^4\right )^{3/2}}-\frac {7 (11 b c-a d) x^3}{40 a^4 \sqrt {a+b x^4}}+\frac {7 (11 b c-a d) x \sqrt {a+b x^4}}{40 a^4 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {7 (11 b c-a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{40 a^{15/4} b^{3/4} \sqrt {a+b x^4}}+\frac {7 (11 b c-a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{80 a^{15/4} b^{3/4} \sqrt {a+b x^4}} \] Output:
-c/a/x/(b*x^4+a)^(5/2)-1/10*(-a*d+11*b*c)*x^3/a^2/(b*x^4+a)^(5/2)-7/60*(-a *d+11*b*c)*x^3/a^3/(b*x^4+a)^(3/2)-7/40*(-a*d+11*b*c)*x^3/a^4/(b*x^4+a)^(1 /2)+7/40*(-a*d+11*b*c)*x*(b*x^4+a)^(1/2)/a^4/b^(1/2)/(a^(1/2)+b^(1/2)*x^2) -7/40*(-a*d+11*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)*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(15/4) /b^(3/4)/(b*x^4+a)^(1/2)+7/80*(-a*d+11*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^(15/4)/b^(3/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.24 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=-\frac {c}{a x \left (a+b x^4\right )^{5/2}}-\frac {(11 b c-a d) x^3 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},-\frac {b x^4}{a}\right )}{3 a^4 \sqrt {a+b x^4}} \] Input:
Integrate[(c + d*x^4)/(x^2*(a + b*x^4)^(7/2)),x]
Output:
-(c/(a*x*(a + b*x^4)^(5/2))) - ((11*b*c - a*d)*x^3*Sqrt[1 + (b*x^4)/a]*Hyp ergeometric2F1[3/4, 7/2, 7/4, -((b*x^4)/a)])/(3*a^4*Sqrt[a + b*x^4])
Time = 0.70 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {955, 819, 819, 819, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(11 b c-a d) \int \frac {x^2}{\left (b x^4+a\right )^{7/2}}dx}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \int \frac {x^2}{\left (b x^4+a\right )^{5/2}}dx}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \left (\frac {\int \frac {x^2}{\left (b x^4+a\right )^{3/2}}dx}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\int \frac {x^2}{\sqrt {b x^4+a}}dx}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle -\frac {(11 b c-a d) \left (\frac {7 \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{10 a}+\frac {x^3}{10 a \left (a+b x^4\right )^{5/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{5/2}}\) |
Input:
Int[(c + d*x^4)/(x^2*(a + b*x^4)^(7/2)),x]
Output:
-(c/(a*x*(a + b*x^4)^(5/2))) - ((11*b*c - a*d)*(x^3/(10*a*(a + b*x^4)^(5/2 )) + (7*(x^3/(6*a*(a + b*x^4)^(3/2)) + (x^3/(2*a*Sqrt[a + b*x^4]) - (-((-( (x*Sqrt[a + b*x^4])/(Sqrt[a] + Sqrt[b]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[b] *x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1 /4)*x)/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^4]))/Sqrt[b]) + (a^(1/4)*(Sqr t[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])/(2*b^(3/4)*Sqrt[a + b*x^4]))/(2*a))/( 2*a)))/(10*a)))/a
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Result contains complex when optimal does not.
Time = 3.33 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.69
| method | result | size |
| elliptic | \(\frac {x^{3} \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{10 a^{2} b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {x^{3} \left (7 a d -17 c b \right ) \sqrt {b \,x^{4}+a}}{60 a^{3} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3} \left (7 a d -37 c b \right )}{40 a^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {c \sqrt {b \,x^{4}+a}}{a^{4} x}+\frac {i \left (-\frac {7 a d -37 c b}{40 a^{4}}+\frac {b c}{a^{4}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(246\) |
| default | \(c \left (-\frac {x^{3} \sqrt {b \,x^{4}+a}}{10 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {17 x^{3} \sqrt {b \,x^{4}+a}}{60 a^{3} b \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {37 b \,x^{3}}{40 a^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{a^{4} x}+\frac {77 i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 a^{\frac {7}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {7 x^{3} \sqrt {b \,x^{4}+a}}{60 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {7 x^{3}}{40 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {7 i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\) | \(384\) |
| risch | \(-\frac {c \sqrt {b \,x^{4}+a}}{a^{4} x}+\frac {a^{3} \left (a d -c b \right ) \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{10 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {7 x^{3} \sqrt {b \,x^{4}+a}}{60 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {7 x^{3}}{40 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {7 i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+\frac {i c \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-a b c \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )-a^{2} b c \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )}{a^{4}}\) | \(593\) |
Input:
int((d*x^4+c)/x^2/(b*x^4+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/10/a^2*x^3/b^3*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^3+1/60/a^3*x^3*(7*a*d -17*b*c)/b^2*(b*x^4+a)^(1/2)/(x^4+a/b)^2+1/40/a^4*x^3*(7*a*d-37*b*c)/((x^4 +a/b)*b)^(1/2)-1/a^4*c*(b*x^4+a)^(1/2)/x+I*(-1/40/a^4*(7*a*d-37*b*c)+b/a^4 *c)*a^(1/2)/(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)/b^(1/2)*(EllipticF(x*(I/a^(1/2 )*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.94 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=-\frac {21 \, {\left ({\left (11 \, b^{4} c - a b^{3} d\right )} x^{13} + 3 \, {\left (11 \, a b^{3} c - a^{2} b^{2} d\right )} x^{9} + 3 \, {\left (11 \, a^{2} b^{2} c - a^{3} b d\right )} x^{5} + {\left (11 \, a^{3} b c - a^{4} d\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 21 \, {\left ({\left (11 \, b^{4} c - a b^{3} d\right )} x^{13} + 3 \, {\left (11 \, a b^{3} c - a^{2} b^{2} d\right )} x^{9} + 3 \, {\left (11 \, a^{2} b^{2} c - a^{3} b d\right )} x^{5} + {\left (11 \, a^{3} b c - a^{4} d\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (21 \, {\left (11 \, b^{4} c - a b^{3} d\right )} x^{12} + 56 \, {\left (11 \, a b^{3} c - a^{2} b^{2} d\right )} x^{8} + 120 \, a^{3} b c + 47 \, {\left (11 \, a^{2} b^{2} c - a^{3} b d\right )} x^{4}\right )} \sqrt {b x^{4} + a}}{120 \, {\left (a^{4} b^{4} x^{13} + 3 \, a^{5} b^{3} x^{9} + 3 \, a^{6} b^{2} x^{5} + a^{7} b x\right )}} \] Input:
integrate((d*x^4+c)/x^2/(b*x^4+a)^(7/2),x, algorithm="fricas")
Output:
-1/120*(21*((11*b^4*c - a*b^3*d)*x^13 + 3*(11*a*b^3*c - a^2*b^2*d)*x^9 + 3 *(11*a^2*b^2*c - a^3*b*d)*x^5 + (11*a^3*b*c - a^4*d)*x)*sqrt(a)*(-b/a)^(3/ 4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) - 21*((11*b^4*c - a*b^3*d)*x^13 + 3*(11*a*b^3*c - a^2*b^2*d)*x^9 + 3*(11*a^2*b^2*c - a^3*b*d)*x^5 + (11*a^ 3*b*c - a^4*d)*x)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + (21*(11*b^4*c - a*b^3*d)*x^12 + 56*(11*a*b^3*c - a^2*b^2*d)*x^8 + 12 0*a^3*b*c + 47*(11*a^2*b^2*c - a^3*b*d)*x^4)*sqrt(b*x^4 + a))/(a^4*b^4*x^1 3 + 3*a^5*b^3*x^9 + 3*a^6*b^2*x^5 + a^7*b*x)
Result contains complex when optimal does not.
Time = 112.65 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.23 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=\frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:
integrate((d*x**4+c)/x**2/(b*x**4+a)**(7/2),x)
Output:
c*gamma(-1/4)*hyper((-1/4, 7/2), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**( 7/2)*x*gamma(3/4)) + d*x**3*gamma(3/4)*hyper((3/4, 7/2), (7/4,), b*x**4*ex p_polar(I*pi)/a)/(4*a**(7/2)*gamma(7/4))
\[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{2}} x^{2}} \,d x } \] Input:
integrate((d*x^4+c)/x^2/(b*x^4+a)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(7/2)*x^2), x)
\[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {7}{2}} x^{2}} \,d x } \] Input:
integrate((d*x^4+c)/x^2/(b*x^4+a)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(7/2)*x^2), x)
Timed out. \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=\int \frac {d\,x^4+c}{x^2\,{\left (b\,x^4+a\right )}^{7/2}} \,d x \] Input:
int((c + d*x^4)/(x^2*(a + b*x^4)^(7/2)),x)
Output:
int((c + d*x^4)/(x^2*(a + b*x^4)^(7/2)), x)
\[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{7/2}} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, d -\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a^{4} d x +11 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a^{3} b c x -3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a^{3} b d \,x^{5}+33 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a^{2} b^{2} c \,x^{5}-3 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a^{2} b^{2} d \,x^{9}+33 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a \,b^{3} c \,x^{9}-\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) a \,b^{3} d \,x^{13}+11 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{4} x^{18}+4 a \,b^{3} x^{14}+6 a^{2} b^{2} x^{10}+4 a^{3} b \,x^{6}+a^{4} x^{2}}d x \right ) b^{4} c \,x^{13}}{11 b x \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )} \] Input:
int((d*x^4+c)/x^2/(b*x^4+a)^(7/2),x)
Output:
( - sqrt(a + b*x**4)*d - int(sqrt(a + b*x**4)/(a**4*x**2 + 4*a**3*b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x**14 + b**4*x**18),x)*a**4*d*x + 11*int(sqr t(a + b*x**4)/(a**4*x**2 + 4*a**3*b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x* *14 + b**4*x**18),x)*a**3*b*c*x - 3*int(sqrt(a + b*x**4)/(a**4*x**2 + 4*a* *3*b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x**14 + b**4*x**18),x)*a**3*b*d*x **5 + 33*int(sqrt(a + b*x**4)/(a**4*x**2 + 4*a**3*b*x**6 + 6*a**2*b**2*x** 10 + 4*a*b**3*x**14 + b**4*x**18),x)*a**2*b**2*c*x**5 - 3*int(sqrt(a + b*x **4)/(a**4*x**2 + 4*a**3*b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x**14 + b** 4*x**18),x)*a**2*b**2*d*x**9 + 33*int(sqrt(a + b*x**4)/(a**4*x**2 + 4*a**3 *b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x**14 + b**4*x**18),x)*a*b**3*c*x** 9 - int(sqrt(a + b*x**4)/(a**4*x**2 + 4*a**3*b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x**14 + b**4*x**18),x)*a*b**3*d*x**13 + 11*int(sqrt(a + b*x**4)/( a**4*x**2 + 4*a**3*b*x**6 + 6*a**2*b**2*x**10 + 4*a*b**3*x**14 + b**4*x**1 8),x)*b**4*c*x**13)/(11*b*x*(a**3 + 3*a**2*b*x**4 + 3*a*b**2*x**8 + b**3*x **12))