\(\int \frac {(c+d x+e x^2+f x^3) \sqrt {a+b x^4}}{x^6} \, dx\) [50]

Optimal result

Integrand size = 30, antiderivative size = 368 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\frac {2 e \sqrt {a+b x^4}}{3 x^3}-\frac {2 b c \sqrt {a+b x^4}}{5 \sqrt {a} x \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (c+5 e x^2\right ) \sqrt {a+b x^4}}{5 x^5}-\frac {\left (d+2 f x^2\right ) \sqrt {a+b x^4}}{4 x^4}+\frac {1}{2} \sqrt {b} f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {2 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{3/4} \sqrt {a+b x^4}}+\frac {b^{3/4} \left (3 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}} \] Output:

2/3*e*(b*x^4+a)^(1/2)/x^3-2/5*b*c*(b*x^4+a)^(1/2)/a^(1/2)/x/(a^(1/2)+b^(1/ 
2)*x^2)-1/5*(5*e*x^2+c)*(b*x^4+a)^(1/2)/x^5-1/4*(2*f*x^2+d)*(b*x^4+a)^(1/2 
)/x^4+1/2*b^(1/2)*f*arctanh(b^(1/2)*x^2/(b*x^4+a)^(1/2))-1/4*b*d*arctanh(( 
b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-2/5*b^(5/4)*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^(3/4)/(b*x^4+a)^(1/2)+1/15*b^(3/4)*(3*b^(1/2)*c+5*a^(1 
/2)*e)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*Inv 
erseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(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 = 10.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.49 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=-\frac {\sqrt {a+b x^4} \left (12 a c \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\frac {b x^4}{a}\right )+5 x \left (3 a d \sqrt {1+\frac {b x^4}{a}}+6 a f x^2 \sqrt {1+\frac {b x^4}{a}}-6 \sqrt {a} \sqrt {b} f x^4 \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )+3 b d x^4 \text {arctanh}\left (\sqrt {1+\frac {b x^4}{a}}\right )+4 a e x \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\frac {b x^4}{a}\right )\right )\right )}{60 a x^5 \sqrt {1+\frac {b x^4}{a}}} \] Input:

Integrate[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^6,x]
 

Output:

-1/60*(Sqrt[a + b*x^4]*(12*a*c*Hypergeometric2F1[-5/4, -1/2, -1/4, -((b*x^ 
4)/a)] + 5*x*(3*a*d*Sqrt[1 + (b*x^4)/a] + 6*a*f*x^2*Sqrt[1 + (b*x^4)/a] - 
6*Sqrt[a]*Sqrt[b]*f*x^4*ArcSinh[(Sqrt[b]*x^2)/Sqrt[a]] + 3*b*d*x^4*ArcTanh 
[Sqrt[1 + (b*x^4)/a]] + 4*a*e*x*Hypergeometric2F1[-3/4, -1/2, 1/4, -((b*x^ 
4)/a)])))/(a*x^5*Sqrt[1 + (b*x^4)/a])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2364, 27, 2372, 2009}

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.

\(\displaystyle \int \frac {\sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right )}{x^6} \, dx\)

\(\Big \downarrow \) 2364

\(\displaystyle -2 b \int -\frac {30 f x^3+20 e x^2+15 d x+12 c}{60 x^2 \sqrt {b x^4+a}}dx-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} b \int \frac {30 f x^3+20 e x^2+15 d x+12 c}{x^2 \sqrt {b x^4+a}}dx-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\)

\(\Big \downarrow \) 2372

\(\displaystyle \frac {1}{30} b \int \left (\frac {20 e x^2+12 c}{x^2 \sqrt {b x^4+a}}+\frac {30 f x^2+15 d}{x \sqrt {b x^4+a}}\right )dx-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{30} b \left (\frac {2 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+3 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{a^{3/4} \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {12 \sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} \sqrt {a+b x^4}}-\frac {15 d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {15 f \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{\sqrt {b}}-\frac {12 c \sqrt {a+b x^4}}{a x}+\frac {12 \sqrt {b} c x \sqrt {a+b x^4}}{a \left (\sqrt {a}+\sqrt {b} x^2\right )}\right )-\frac {1}{60} \sqrt {a+b x^4} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )\)

Input:

Int[((c + d*x + e*x^2 + f*x^3)*Sqrt[a + b*x^4])/x^6,x]
 

Output:

-1/60*(((12*c)/x^5 + (15*d)/x^4 + (20*e)/x^3 + (30*f)/x^2)*Sqrt[a + b*x^4] 
) + (b*((-12*c*Sqrt[a + b*x^4])/(a*x) + (12*Sqrt[b]*c*x*Sqrt[a + b*x^4])/( 
a*(Sqrt[a] + Sqrt[b]*x^2)) + (15*f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]]) 
/Sqrt[b] - (15*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(2*Sqrt[a]) - (12*b^(1/ 
4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*E 
llipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(3/4)*Sqrt[a + b*x^4]) + 
(2*(3*Sqrt[b]*c + 5*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(S 
qrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a 
^(3/4)*b^(1/4)*Sqrt[a + b*x^4])))/30
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2364

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u 
 = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p   Int[x^(m + 
n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} 
, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 
, 0]
 

rule 2372

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo 
dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, 
 j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, 
 n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 
] &&  !PolyQ[Pq, x^(n/2)]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.15 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (24 b c \,x^{4}+30 a f \,x^{3}+20 a e \,x^{2}+15 a d x +12 a c \right )}{60 x^{5} a}+\frac {b \left (\frac {20 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {12 i c \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {15 \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {15 a f \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{\sqrt {b}}\right )}{30 a}\)	\(280\)
elliptic	\(-\frac {c \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {d \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {e \sqrt {b \,x^{4}+a}}{3 x^{3}}-\frac {f \sqrt {b \,x^{4}+a}}{2 x^{2}}-\frac {2 b c \sqrt {b \,x^{4}+a}}{5 a x}+\frac {2 b e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {f \sqrt {b}\, \ln \left (2 \sqrt {b}\, x^{2}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {2 i b^{\frac {3}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {b d \,\operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4 \sqrt {a}}\)	\(298\)
default	\(c \left (-\frac {\sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {2 b \sqrt {b \,x^{4}+a}}{5 a x}+\frac {2 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 \sqrt {a}}+\frac {b \sqrt {b \,x^{4}+a}}{4 a}\right )+e \left (-\frac {\sqrt {b \,x^{4}+a}}{3 x^{3}}+\frac {2 b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \,x^{2} \sqrt {b \,x^{4}+a}}{2 a}+\frac {\sqrt {b}\, \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{2}\right )\)	\(346\)

Input:

int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/60*(b*x^4+a)^(1/2)*(24*b*c*x^4+30*a*f*x^3+20*a*e*x^2+15*a*d*x+12*a*c)/x 
^5/a+1/30*b/a*(20*a*e/(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)+12*I*c*b^(1/2)*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 
)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2) 
)^(1/2),I))-15/2*a^(1/2)*d*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)+15*a*f* 
ln(b^(1/2)*x^2+(b*x^4+a)^(1/2))/b^(1/2))
 

Fricas [F]

\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}} \,d x } \] Input:

integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^6,x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.88 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.59 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\frac {\sqrt {a} c \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} f}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {\sqrt {b} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} - \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 \sqrt {a}} - \frac {b f x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \] Input:

integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(1/2)/x**6,x)
 

Output:

sqrt(a)*c*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi)/ 
a)/(4*x**5*gamma(-1/4)) + sqrt(a)*e*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,) 
, b*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/4)) - sqrt(a)*f/(2*x**2*sqrt(1 
 + b*x**4/a)) - sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(4*x**2) + sqrt(b)*f*asinh( 
sqrt(b)*x**2/sqrt(a))/2 - b*d*asinh(sqrt(a)/(sqrt(b)*x**2))/(4*sqrt(a)) - 
b*f*x**2/(2*sqrt(a)*sqrt(1 + b*x**4/a))
 

Maxima [F]

\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}} \,d x } \] Input:

integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^6,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)
 

Giac [F]

\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{6}} \,d x } \] Input:

integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^6,x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^4 + a)*(f*x^3 + e*x^2 + d*x + c)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^6} \,d x \] Input:

int(((a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3))/x^6,x)
 

Output:

int(((a + b*x^4)^(1/2)*(c + d*x + e*x^2 + f*x^3))/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^6} \, dx=\frac {-8 \sqrt {b \,x^{4}+a}\, a c -6 \sqrt {b \,x^{4}+a}\, a d x -24 \sqrt {b \,x^{4}+a}\, a e \,x^{2}-12 \sqrt {b \,x^{4}+a}\, a f \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {a}\right ) b d \,x^{5}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {a}\right ) b d \,x^{5}-6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}-\sqrt {b}\, x^{2}\right ) a f \,x^{5}+6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{4}+a}+\sqrt {b}\, x^{2}\right ) a f \,x^{5}-16 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{10}+a \,x^{6}}d x \right ) a^{2} c \,x^{5}-48 \left (\int \frac {\sqrt {b \,x^{4}+a}}{b \,x^{8}+a \,x^{4}}d x \right ) a^{2} e \,x^{5}}{24 a \,x^{5}} \] Input:

int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(1/2)/x^6,x)
 

Output:

( - 8*sqrt(a + b*x**4)*a*c - 6*sqrt(a + b*x**4)*a*d*x - 24*sqrt(a + b*x**4 
)*a*e*x**2 - 12*sqrt(a + b*x**4)*a*f*x**3 + 3*sqrt(a)*log(sqrt(a + b*x**4) 
 - sqrt(a))*b*d*x**5 - 3*sqrt(a)*log(sqrt(a + b*x**4) + sqrt(a))*b*d*x**5 
- 6*sqrt(b)*log(sqrt(a + b*x**4) - sqrt(b)*x**2)*a*f*x**5 + 6*sqrt(b)*log( 
sqrt(a + b*x**4) + sqrt(b)*x**2)*a*f*x**5 - 16*int(sqrt(a + b*x**4)/(a*x** 
6 + b*x**10),x)*a**2*c*x**5 - 48*int(sqrt(a + b*x**4)/(a*x**4 + b*x**8),x) 
*a**2*e*x**5)/(24*a*x**5)