\(\int \frac {c+d x^4}{x^2 (a+b x^4)^{5/2}} \, dx\) [54]

Optimal result
Mathematica [C] (verified)
Rubi [A] (verified)
Maple [C] (verified)
Fricas [A] (verification not implemented)
Sympy [C] (verification not implemented)
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

Integrand size = 22, antiderivative size = 325 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=-\frac {c}{a x \left (a+b x^4\right )^{3/2}}-\frac {(7 b c-a d) x^3}{6 a^2 \left (a+b x^4\right )^{3/2}}-\frac {(7 b c-a d) x^3}{4 a^3 \sqrt {a+b x^4}}+\frac {(7 b c-a d) x \sqrt {a+b x^4}}{4 a^3 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {(7 b c-a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{11/4} b^{3/4} \sqrt {a+b x^4}}+\frac {(7 b c-a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 a^{11/4} b^{3/4} \sqrt {a+b x^4}} \] Output:

-c/a/x/(b*x^4+a)^(3/2)-1/6*(-a*d+7*b*c)*x^3/a^2/(b*x^4+a)^(3/2)-1/4*(-a*d+ 
7*b*c)*x^3/a^3/(b*x^4+a)^(1/2)+1/4*(-a*d+7*b*c)*x*(b*x^4+a)^(1/2)/a^3/b^(1 
/2)/(a^(1/2)+b^(1/2)*x^2)-1/4*(-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)*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4)) 
),1/2*2^(1/2))/a^(11/4)/b^(3/4)/(b*x^4+a)^(1/2)+1/8*(-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*a 
rctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(11/4)/b^(3/4)/(b*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.25 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=\frac {-3 a^2 c+(-7 b c+a d) x^4 \left (a+b x^4\right ) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},-\frac {b x^4}{a}\right )}{3 a^3 x \left (a+b x^4\right )^{3/2}} \] Input:

Integrate[(c + d*x^4)/(x^2*(a + b*x^4)^(5/2)),x]
 

Output:

(-3*a^2*c + (-7*b*c + a*d)*x^4*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*Hypergeomet 
ric2F1[3/4, 5/2, 7/4, -((b*x^4)/a)])/(3*a^3*x*(a + b*x^4)^(3/2))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.94, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {955, 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.

\(\displaystyle \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 955

\(\displaystyle -\frac {(7 b c-a d) \int \frac {x^2}{\left (b x^4+a\right )^{5/2}}dx}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(7 b c-a d) \left (\frac {\int \frac {x^2}{\left (b x^4+a\right )^{3/2}}dx}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

\(\Big \downarrow \) 819

\(\displaystyle -\frac {(7 b c-a d) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\int \frac {x^2}{\sqrt {b x^4+a}}dx}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

\(\Big \downarrow \) 834

\(\displaystyle -\frac {(7 b c-a d) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(7 b c-a d) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

\(\Big \downarrow \) 761

\(\displaystyle -\frac {(7 b c-a d) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

\(\Big \downarrow \) 1510

\(\displaystyle -\frac {(7 b c-a d) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}}{2 a}}{2 a}+\frac {x^3}{6 a \left (a+b x^4\right )^{3/2}}\right )}{a}-\frac {c}{a x \left (a+b x^4\right )^{3/2}}\)

Input:

Int[(c + d*x^4)/(x^2*(a + b*x^4)^(5/2)),x]
 

Output:

-(c/(a*x*(a + b*x^4)^(3/2))) - ((7*b*c - a*d)*(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] + Sq 
rt[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)*(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]) 
/(2*b^(3/4)*Sqrt[a + b*x^4]))/(2*a))/(2*a)))/a
 

Defintions of rubi rules used

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761
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]
 

rule 819
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]
 

rule 834
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]
 

rule 955
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]
 

rule 1510
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]
 
Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.63

method result size
elliptic \(\frac {x^{3} \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{6 a^{2} b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3} \left (a d -3 c b \right )}{4 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {c \sqrt {b \,x^{4}+a}}{a^{3} x}+\frac {i \left (-\frac {a d -3 c b}{4 a^{3}}+\frac {b c}{a^{3}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) \(204\)
default \(c \left (-\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a^{2} b \left (x^{4}+\frac {a}{b}\right )^{2}}-\frac {3 b \,x^{3}}{4 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{a^{3} x}+\frac {7 i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\) \(322\)
risch \(-\frac {c \sqrt {b \,x^{4}+a}}{a^{3} x}+\frac {a^{2} \left (a d -c b \right ) \left (\frac {x^{3} \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+\frac {i c \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-a b c \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )}{a^{3}}\) \(406\)

Input:

int((d*x^4+c)/x^2/(b*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/6/a^2*x^3/b^2*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^2+1/4/a^3*x^3*(a*d-3*b 
*c)/((x^4+a/b)*b)^(1/2)-1/a^3*c*(b*x^4+a)^(1/2)/x+I*(-1/4/a^3*(a*d-3*b*c)+ 
b/a^3*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))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=-\frac {3 \, {\left ({\left (7 \, b^{3} c - a b^{2} d\right )} x^{9} + 2 \, {\left (7 \, a b^{2} c - a^{2} b d\right )} x^{5} + {\left (7 \, a^{2} b c - a^{3} d\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 3 \, {\left ({\left (7 \, b^{3} c - a b^{2} d\right )} x^{9} + 2 \, {\left (7 \, a b^{2} c - a^{2} b d\right )} x^{5} + {\left (7 \, a^{2} b c - a^{3} d\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, {\left (7 \, b^{3} c - a b^{2} d\right )} x^{8} + 5 \, {\left (7 \, a b^{2} c - a^{2} b d\right )} x^{4} + 12 \, a^{2} b c\right )} \sqrt {b x^{4} + a}}{12 \, {\left (a^{3} b^{3} x^{9} + 2 \, a^{4} b^{2} x^{5} + a^{5} b x\right )}} \] Input:

integrate((d*x^4+c)/x^2/(b*x^4+a)^(5/2),x, algorithm="fricas")
 

Output:

-1/12*(3*((7*b^3*c - a*b^2*d)*x^9 + 2*(7*a*b^2*c - a^2*b*d)*x^5 + (7*a^2*b 
*c - a^3*d)*x)*sqrt(a)*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) 
 - 3*((7*b^3*c - a*b^2*d)*x^9 + 2*(7*a*b^2*c - a^2*b*d)*x^5 + (7*a^2*b*c - 
 a^3*d)*x)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + ( 
3*(7*b^3*c - a*b^2*d)*x^8 + 5*(7*a*b^2*c - a^2*b*d)*x^4 + 12*a^2*b*c)*sqrt 
(b*x^4 + a))/(a^3*b^3*x^9 + 2*a^4*b^2*x^5 + a^5*b*x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 30.98 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.25 \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=\frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

integrate((d*x**4+c)/x**2/(b*x**4+a)**(5/2),x)
 

Output:

c*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)) + d*x**3*gamma(3/4)*hyper((3/4, 5/2), (7/4,), b*x**4*ex 
p_polar(I*pi)/a)/(4*a**(5/2)*gamma(7/4))
 

Maxima [F]

\[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate((d*x^4+c)/x^2/(b*x^4+a)^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(5/2)*x^2), x)
 

Giac [F]

\[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:

integrate((d*x^4+c)/x^2/(b*x^4+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)/((b*x^4 + a)^(5/2)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=\int \frac {d\,x^4+c}{x^2\,{\left (b\,x^4+a\right )}^{5/2}} \,d x \] Input:

int((c + d*x^4)/(x^2*(a + b*x^4)^(5/2)),x)
 

Output:

int((c + d*x^4)/(x^2*(a + b*x^4)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {c+d x^4}{x^2 \left (a+b x^4\right )^{5/2}} \, dx=\frac {-\sqrt {b \,x^{4}+a}\, d -\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{14}+3 a \,b^{2} x^{10}+3 a^{2} b \,x^{6}+a^{3} x^{2}}d x \right ) a^{3} d x +7 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{14}+3 a \,b^{2} x^{10}+3 a^{2} b \,x^{6}+a^{3} x^{2}}d x \right ) a^{2} b c x -2 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{14}+3 a \,b^{2} x^{10}+3 a^{2} b \,x^{6}+a^{3} x^{2}}d x \right ) a^{2} b d \,x^{5}+14 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{14}+3 a \,b^{2} x^{10}+3 a^{2} b \,x^{6}+a^{3} x^{2}}d x \right ) a \,b^{2} c \,x^{5}-\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{14}+3 a \,b^{2} x^{10}+3 a^{2} b \,x^{6}+a^{3} x^{2}}d x \right ) a \,b^{2} d \,x^{9}+7 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{3} x^{14}+3 a \,b^{2} x^{10}+3 a^{2} b \,x^{6}+a^{3} x^{2}}d x \right ) b^{3} c \,x^{9}}{7 b x \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )} \] Input:

int((d*x^4+c)/x^2/(b*x^4+a)^(5/2),x)
 

Output:

( - sqrt(a + b*x**4)*d - int(sqrt(a + b*x**4)/(a**3*x**2 + 3*a**2*b*x**6 + 
 3*a*b**2*x**10 + b**3*x**14),x)*a**3*d*x + 7*int(sqrt(a + b*x**4)/(a**3*x 
**2 + 3*a**2*b*x**6 + 3*a*b**2*x**10 + b**3*x**14),x)*a**2*b*c*x - 2*int(s 
qrt(a + b*x**4)/(a**3*x**2 + 3*a**2*b*x**6 + 3*a*b**2*x**10 + b**3*x**14), 
x)*a**2*b*d*x**5 + 14*int(sqrt(a + b*x**4)/(a**3*x**2 + 3*a**2*b*x**6 + 3* 
a*b**2*x**10 + b**3*x**14),x)*a*b**2*c*x**5 - int(sqrt(a + b*x**4)/(a**3*x 
**2 + 3*a**2*b*x**6 + 3*a*b**2*x**10 + b**3*x**14),x)*a*b**2*d*x**9 + 7*in 
t(sqrt(a + b*x**4)/(a**3*x**2 + 3*a**2*b*x**6 + 3*a*b**2*x**10 + b**3*x**1 
4),x)*b**3*c*x**9)/(7*b*x*(a**2 + 2*a*b*x**4 + b**2*x**8))