Integrand size = 22, antiderivative size = 357 \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}-\frac {11 b c-5 a d}{30 a^2 x \left (a+b x^4\right )^{3/2}}-\frac {7 (11 b c-5 a d)}{60 a^3 x \sqrt {a+b x^4}}+\frac {7 (11 b c-5 a d) \sqrt {a+b x^4}}{20 a^4 x}-\frac {7 \sqrt {b} (11 b c-5 a d) x \sqrt {a+b x^4}}{20 a^4 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {7 \sqrt [4]{b} (11 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 a^{15/4} \sqrt {a+b x^4}}-\frac {7 \sqrt [4]{b} (11 b c-5 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{40 a^{15/4} \sqrt {a+b x^4}} \] Output:
-1/5*c/a/x^5/(b*x^4+a)^(3/2)-1/30*(-5*a*d+11*b*c)/a^2/x/(b*x^4+a)^(3/2)-7/ 60*(-5*a*d+11*b*c)/a^3/x/(b*x^4+a)^(1/2)+7/20*(-5*a*d+11*b*c)*(b*x^4+a)^(1 /2)/a^4/x-7/20*b^(1/2)*(-5*a*d+11*b*c)*x*(b*x^4+a)^(1/2)/a^4/(a^(1/2)+b^(1 /2)*x^2)+7/20*b^(1/4)*(-5*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*x^4+a)^(1/2)-7/40*b^(1/4)*(-5*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*ar ctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(15/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.23 \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\frac {-a^2 c+(11 b c-5 a d) x^4 \left (a+b x^4\right ) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{2},\frac {3}{4},-\frac {b x^4}{a}\right )}{5 a^3 x^5 \left (a+b x^4\right )^{3/2}} \] Input:
Integrate[(c + d*x^4)/(x^6*(a + b*x^4)^(5/2)),x]
Output:
(-(a^2*c) + (11*b*c - 5*a*d)*x^4*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*Hypergeom etric2F1[-1/4, 5/2, 3/4, -((b*x^4)/a)])/(5*a^3*x^5*(a + b*x^4)^(3/2))
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, 847, 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.
| \(\displaystyle \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(11 b c-5 a d) \int \frac {1}{x^2 \left (b x^4+a\right )^{5/2}}dx}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \int \frac {1}{x^2 \left (b x^4+a\right )^{3/2}}dx}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \left (\frac {3 \int \frac {1}{x^2 \sqrt {b x^4+a}}dx}{2 a}+\frac {1}{2 a x \sqrt {a+b x^4}}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \left (\frac {3 \left (\frac {b \int \frac {x^2}{\sqrt {b x^4+a}}dx}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{2 a}+\frac {1}{2 a x \sqrt {a+b x^4}}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \left (\frac {3 \left (\frac {b \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{2 a}+\frac {1}{2 a x \sqrt {a+b x^4}}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \left (\frac {3 \left (\frac {b \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{2 a}+\frac {1}{2 a x \sqrt {a+b x^4}}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \left (\frac {3 \left (\frac {b \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{2 a}+\frac {1}{2 a x \sqrt {a+b x^4}}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle -\frac {(11 b c-5 a d) \left (\frac {7 \left (\frac {3 \left (\frac {b \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}\right )}{a}-\frac {\sqrt {a+b x^4}}{a x}\right )}{2 a}+\frac {1}{2 a x \sqrt {a+b x^4}}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^4\right )^{3/2}}\right )}{5 a}-\frac {c}{5 a x^5 \left (a+b x^4\right )^{3/2}}\) |
Input:
Int[(c + d*x^4)/(x^6*(a + b*x^4)^(5/2)),x]
Output:
-1/5*c/(a*x^5*(a + b*x^4)^(3/2)) - ((11*b*c - 5*a*d)*(1/(6*a*x*(a + b*x^4) ^(3/2)) + (7*(1/(2*a*x*Sqrt[a + b*x^4]) + (3*(-(Sqrt[a + b*x^4]/(a*x)) + ( b*(-((-((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*ArcT an[(b^(1/4)*x)/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^4]))/Sqrt[b]) + (a^(1 /4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*El lipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^4]))) /a))/(2*a)))/(6*a)))/(5*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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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 = 3.79 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.68
| method | result | size |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{5 a^{3} x^{5}}-\frac {\left (5 a d -13 c b \right ) \sqrt {b \,x^{4}+a}}{5 a^{4} x}-\frac {x^{3} \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{6 a^{3} b \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {b \,x^{3} \left (3 a d -5 c b \right )}{4 a^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {i \left (\frac {b \left (5 a d -13 c b \right )}{5 a^{4}}+\frac {b \left (3 a d -5 c b \right )}{4 a^{4}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(243\) |
| default | \(c \left (-\frac {\sqrt {b \,x^{4}+a}}{5 a^{3} x^{5}}+\frac {13 b \sqrt {b \,x^{4}+a}}{5 a^{4} x}+\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a^{3} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {5 b^{2} x^{3}}{4 a^{4} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {77 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{20 a^{\frac {7}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a^{2} b \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {3 b \,x^{3}}{4 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{a^{3} x}+\frac {7 i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(357\) |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (5 a d \,x^{4}-13 b c \,x^{4}+a c \right )}{5 a^{4} x^{5}}+\frac {b \left (\frac {i \left (5 a d -13 c b \right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}-5 a \left (a d -2 c b \right ) \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )-5 a^{2} \left (a d -c b \right ) \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\right )}{5 a^{4}}\) | \(440\) |
Input:
int((d*x^4+c)/x^6/(b*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/5/a^3*c*(b*x^4+a)^(1/2)/x^5-1/5/a^4*(5*a*d-13*b*c)*(b*x^4+a)^(1/2)/x-1/ 6/a^3*x^3/b*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^2-1/4*b/a^4*x^3*(3*a*d-5*b *c)/((x^4+a/b)*b)^(1/2)+I*(1/5*b/a^4*(5*a*d-13*b*c)+1/4*b/a^4*(3*a*d-5*b*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 = 273, normalized size of antiderivative = 0.76 \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\frac {21 \, {\left ({\left (11 \, b^{3} c - 5 \, a b^{2} d\right )} x^{13} + 2 \, {\left (11 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{9} + {\left (11 \, a^{2} b c - 5 \, a^{3} d\right )} x^{5}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 21 \, {\left ({\left (11 \, b^{3} c - 5 \, a b^{2} d\right )} x^{13} + 2 \, {\left (11 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{9} + {\left (11 \, a^{2} b c - 5 \, a^{3} d\right )} x^{5}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (21 \, {\left (11 \, b^{3} c - 5 \, a b^{2} d\right )} x^{12} + 35 \, {\left (11 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{8} + 12 \, {\left (11 \, a^{2} b c - 5 \, a^{3} d\right )} x^{4} - 12 \, a^{3} c\right )} \sqrt {b x^{4} + a}}{60 \, {\left (a^{4} b^{2} x^{13} + 2 \, a^{5} b x^{9} + a^{6} x^{5}\right )}} \] Input:
integrate((d*x^4+c)/x^6/(b*x^4+a)^(5/2),x, algorithm="fricas")
Output:
1/60*(21*((11*b^3*c - 5*a*b^2*d)*x^13 + 2*(11*a*b^2*c - 5*a^2*b*d)*x^9 + ( 11*a^2*b*c - 5*a^3*d)*x^5)*sqrt(a)*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a) ^(1/4)), -1) - 21*((11*b^3*c - 5*a*b^2*d)*x^13 + 2*(11*a*b^2*c - 5*a^2*b*d )*x^9 + (11*a^2*b*c - 5*a^3*d)*x^5)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(arcsin (x*(-b/a)^(1/4)), -1) + (21*(11*b^3*c - 5*a*b^2*d)*x^12 + 35*(11*a*b^2*c - 5*a^2*b*d)*x^8 + 12*(11*a^2*b*c - 5*a^3*d)*x^4 - 12*a^3*c)*sqrt(b*x^4 + a ))/(a^4*b^2*x^13 + 2*a^5*b*x^9 + a^6*x^5)
Result contains complex when optimal does not.
Time = 78.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.25 \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\frac {c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {5}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} x \Gamma \left (\frac {3}{4}\right )} \] Input:
integrate((d*x**4+c)/x**6/(b*x**4+a)**(5/2),x)
Output:
c*gamma(-5/4)*hyper((-5/4, 5/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*a** (5/2)*x**5*gamma(-1/4)) + d*gamma(-1/4)*hyper((-1/4, 5/2), (3/4,), b*x**4* exp_polar(I*pi)/a)/(4*a**(5/2)*x*gamma(3/4))
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {5}{2}} x^{6}} \,d x } \] Input:
integrate((d*x^4+c)/x^6/(b*x^4+a)^(5/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(5/2)*x^6), x)
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {5}{2}} x^{6}} \,d x } \] Input:
integrate((d*x^4+c)/x^6/(b*x^4+a)^(5/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(5/2)*x^6), x)
Timed out. \[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\int \frac {d\,x^4+c}{x^6\,{\left (b\,x^4+a\right )}^{5/2}} \,d x \] Input:
int((c + d*x^4)/(x^6*(a + b*x^4)^(5/2)),x)
Output:
int((c + d*x^4)/(x^6*(a + b*x^4)^(5/2)), x)
\[ \int \frac {c+d x^4}{x^6 \left (a+b x^4\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, d -5 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{14}+3 a^{2} b \,x^{10}+a^{3} x^{6}}d x \right ) a^{3} d \,x^{5}+11 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{14}+3 a^{2} b \,x^{10}+a^{3} x^{6}}d x \right ) a^{2} b c \,x^{5}-10 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{14}+3 a^{2} b \,x^{10}+a^{3} x^{6}}d x \right ) a^{2} b d \,x^{9}+22 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{14}+3 a^{2} b \,x^{10}+a^{3} x^{6}}d x \right ) a \,b^{2} c \,x^{9}-5 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{14}+3 a^{2} b \,x^{10}+a^{3} x^{6}}d x \right ) a \,b^{2} d \,x^{13}+11 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{18}+3 a \,b^{2} x^{14}+3 a^{2} b \,x^{10}+a^{3} x^{6}}d x \right ) b^{3} c \,x^{13}}{11 b \,x^{5} \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )} \] Input:
int((d*x^4+c)/x^6/(b*x^4+a)^(5/2),x)
Output:
( - sqrt(a + b*x**4)*d - 5*int(sqrt(a + b*x**4)/(a**3*x**6 + 3*a**2*b*x**1 0 + 3*a*b**2*x**14 + b**3*x**18),x)*a**3*d*x**5 + 11*int(sqrt(a + b*x**4)/ (a**3*x**6 + 3*a**2*b*x**10 + 3*a*b**2*x**14 + b**3*x**18),x)*a**2*b*c*x** 5 - 10*int(sqrt(a + b*x**4)/(a**3*x**6 + 3*a**2*b*x**10 + 3*a*b**2*x**14 + b**3*x**18),x)*a**2*b*d*x**9 + 22*int(sqrt(a + b*x**4)/(a**3*x**6 + 3*a** 2*b*x**10 + 3*a*b**2*x**14 + b**3*x**18),x)*a*b**2*c*x**9 - 5*int(sqrt(a + b*x**4)/(a**3*x**6 + 3*a**2*b*x**10 + 3*a*b**2*x**14 + b**3*x**18),x)*a*b **2*d*x**13 + 11*int(sqrt(a + b*x**4)/(a**3*x**6 + 3*a**2*b*x**10 + 3*a*b* *2*x**14 + b**3*x**18),x)*b**3*c*x**13)/(11*b*x**5*(a**2 + 2*a*b*x**4 + b* *2*x**8))