\(\int \frac {1}{\sqrt {x} (a+b x^2+c x^4)^3} \, dx\) [947]

Optimal result

Integrand size = 20, antiderivative size = 658 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {\sqrt {x} \left (7 b^4-55 a b^2 c+60 a^2 c^2+b c \left (7 b^2-52 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}}+\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4}}-\frac {3 c^{3/4} \left (7 b^4-66 a b^2 c+280 a^2 c^2+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32 \sqrt [4]{2} a^2 \left (b^2-4 a c\right )^{5/2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4}} \] Output:

1/4*x^(1/2)*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/16*x^(1 
/2)*(7*b^4-55*a*b^2*c+60*c^2*a^2+b*c*(-52*a*c+7*b^2)*x^2)/a^2/(-4*a*c+b^2) 
^2/(c*x^4+b*x^2+a)+3/64*c^(3/4)*(7*b^4-66*a*b^2*c+280*c^2*a^2-b*(-52*a*c+7 
*b^2)*(-4*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^ 
(1/2))^(1/4))*2^(3/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4) 
-3/64*c^(3/4)*(7*b^4-66*a*b^2*c+280*c^2*a^2+b*(-52*a*c+7*b^2)*(-4*a*c+b^2) 
^(1/2))*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(3 
/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)+3/64*c^(3/4)*(7*b 
^4-66*a*b^2*c+280*c^2*a^2-b*(-52*a*c+7*b^2)*(-4*a*c+b^2)^(1/2))*arctanh(2^ 
(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/a^2/(-4*a*c+b 
^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)-3/64*c^(3/4)*(7*b^4-66*a*b^2*c+280 
*c^2*a^2+b*(-52*a*c+7*b^2)*(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x^( 
1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b+(-4 
*a*c+b^2)^(1/2))^(3/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 \sqrt {x} \left (92 a^3 c^2+7 b^3 x^2 \left (b+c x^2\right )^2+a^2 c \left (-79 b^2-8 b c x^2+60 c^2 x^4\right )+a b \left (11 b^3-44 b^2 c x^2-107 b c^2 x^4-52 c^3 x^6\right )\right )}{\left (a+b x^2+c x^4\right )^2}+3 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {7 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )-59 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+140 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+7 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-52 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{64 a^2 \left (b^2-4 a c\right )^2} \] Input:

Integrate[1/(Sqrt[x]*(a + b*x^2 + c*x^4)^3),x]
 

Output:

((4*Sqrt[x]*(92*a^3*c^2 + 7*b^3*x^2*(b + c*x^2)^2 + a^2*c*(-79*b^2 - 8*b*c 
*x^2 + 60*c^2*x^4) + a*b*(11*b^3 - 44*b^2*c*x^2 - 107*b*c^2*x^4 - 52*c^3*x 
^6)))/(a + b*x^2 + c*x^4)^2 + 3*RootSum[a + b*#1^4 + c*#1^8 & , (7*b^4*Log 
[Sqrt[x] - #1] - 59*a*b^2*c*Log[Sqrt[x] - #1] + 140*a^2*c^2*Log[Sqrt[x] - 
#1] + 7*b^3*c*Log[Sqrt[x] - #1]*#1^4 - 52*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/ 
(b*#1^3 + 2*c*#1^7) & ])/(64*a^2*(b^2 - 4*a*c)^2)
 

Rubi [A] (verified)

Time = 2.39 (sec) , antiderivative size = 573, normalized size of antiderivative = 0.87, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {1435, 1683, 25, 1760, 27, 1752, 756, 218, 221}

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.

\(\displaystyle \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1435

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

\(\Big \downarrow \) 1683

\(\displaystyle 2 \left (\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int -\frac {7 b^2+11 c x^2 b-30 a c}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 a \left (b^2-4 a c\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {\int \frac {7 b^2+11 c x^2 b-30 a c}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 1760

\(\displaystyle 2 \left (\frac {\frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 \left (7 b^4-59 a c b^2+c \left (7 b^2-52 a c\right ) x^2 b+140 a^2 c^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\frac {3 \int \frac {7 b^4-59 a c b^2+c \left (7 b^2-52 a c\right ) x^2 b+140 a^2 c^2}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 1752

\(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {c \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}-\frac {c \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 756

\(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {c \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}\right )}{2 \sqrt {b^2-4 a c}}-\frac {c \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}\right )}{2 \sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {c \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \left (-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{2 \sqrt {b^2-4 a c}}-\frac {c \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \left (-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{\sqrt {-\sqrt {b^2-4 a c}-b}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{2 \sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle 2 \left (\frac {\frac {3 \left (\frac {c \left (280 a^2 c^2-66 a b^2 c+b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{2 \sqrt {b^2-4 a c}}-\frac {c \left (280 a^2 c^2-66 a b^2 c-b \left (7 b^2-52 a c\right ) \sqrt {b^2-4 a c}+7 b^4\right ) \left (-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt [4]{2} \sqrt [4]{c} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4}}\right )}{2 \sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (60 a^2 c^2+b c x^2 \left (7 b^2-52 a c\right )-55 a b^2 c+7 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\)

Input:

Int[1/(Sqrt[x]*(a + b*x^2 + c*x^4)^3),x]
 

Output:

2*((Sqrt[x]*(b^2 - 2*a*c + b*c*x^2))/(8*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4 
)^2) + ((Sqrt[x]*(7*b^4 - 55*a*b^2*c + 60*a^2*c^2 + b*c*(7*b^2 - 52*a*c)*x 
^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*(-1/2*(c*(7*b^4 - 66*a*b 
^2*c + 280*a^2*c^2 - b*(7*b^2 - 52*a*c)*Sqrt[b^2 - 4*a*c])*(-(ArcTan[(2^(1 
/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b 
- Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqr 
t[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4))))/ 
Sqrt[b^2 - 4*a*c] + (c*(7*b^4 - 66*a*b^2*c + 280*a^2*c^2 + b*(7*b^2 - 52*a 
*c)*Sqrt[b^2 - 4*a*c])*(-(ArcTan[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 
- 4*a*c])^(1/4)]/(2^(1/4)*c^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))) - ArcTa 
nh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2^(1/4)*c^(1 
/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))))/(2*Sqrt[b^2 - 4*a*c])))/(4*a*(b^2 - 
4*a*c)))/(8*a*(b^2 - 4*a*c)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 1435

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> With[{k = Denominator[m]}, Simp[k/d   Subst[Int[x^(k*(m + 1) - 1)*(a + b 
*(x^(2*k)/d^2) + c*(x^(4*k)/d^4))^p, x], x, (d*x)^(1/k)], x]] /; FreeQ[{a, 
b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
 

rule 1683

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(- 
x)*(b^2 - 2*a*c + b*c*x^n)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 
 n*(p + 1)*(b^2 - 4*a*c) + b*c*(n*(2*p + 3) + 1)*x^n)*(a + b*x^n + c*x^(2*n 
))^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4 
*a*c, 0] && ILtQ[p, -1]
 

rule 1752

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x 
_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) 
   Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 
 - 4*a*c] ||  !IGtQ[n/2, 0])
 

rule 1760

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Simp[(-x)*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)* 
((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/ 
(a*n*(p + 1)*(b^2 - 4*a*c))   Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c* 
d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n 
+ c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n 
] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.25 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.48

method	result	size
derivativedivides	\(\frac {\frac {\left (92 a^{2} c^{2}-79 a \,b^{2} c +11 b^{4}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {b \left (8 a^{2} c^{2}+44 a \,b^{2} c -7 b^{4}\right ) x^{\frac {5}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (60 a^{2} c^{2}-107 a \,b^{2} c +14 b^{4}\right ) x^{\frac {9}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \,c^{2} \left (52 a c -7 b^{2}\right ) x^{\frac {13}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c \left (-52 a c +7 b^{2}\right ) \textit {\_R}^{4}+140 a^{2} c^{2}-59 a \,b^{2} c +7 b^{4}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(316\)
default	\(\frac {\frac {\left (92 a^{2} c^{2}-79 a \,b^{2} c +11 b^{4}\right ) \sqrt {x}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {b \left (8 a^{2} c^{2}+44 a \,b^{2} c -7 b^{4}\right ) x^{\frac {5}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (60 a^{2} c^{2}-107 a \,b^{2} c +14 b^{4}\right ) x^{\frac {9}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \,c^{2} \left (52 a c -7 b^{2}\right ) x^{\frac {13}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c \left (-52 a c +7 b^{2}\right ) \textit {\_R}^{4}+140 a^{2} c^{2}-59 a \,b^{2} c +7 b^{4}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}\right )}{64 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\)	\(316\)

Input:

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

Output:

2*(1/32*(92*a^2*c^2-79*a*b^2*c+11*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^(1/2 
)-1/32*b*(8*a^2*c^2+44*a*b^2*c-7*b^4)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/ 
2)+1/32/a^2*c*(60*a^2*c^2-107*a*b^2*c+14*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x 
^(9/2)-1/32*b*c^2*(52*a*c-7*b^2)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(13/2))/ 
(c*x^4+b*x^2+a)^2+3/64/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*sum((b*c*(-52*a*c+7* 
b^2)*_R^4+140*a^2*c^2-59*a*b^2*c+7*b^4)/(2*_R^7*c+_R^3*b)*ln(x^(1/2)-_R),_ 
R=RootOf(_Z^8*c+_Z^4*b+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17801 vs. \(2 (554) = 1108\).

Time = 44.14 (sec) , antiderivative size = 17801, normalized size of antiderivative = 27.05 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/x**(1/2)/(c*x**4+b*x**2+a)**3,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{3} \sqrt {x}} \,d x } \] Input:

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

Output:

1/16*(3*(7*b^4*c^2 - 59*a*b^2*c^3 + 140*a^2*c^4)*x^(17/2) + (42*b^5*c - 34 
7*a*b^3*c^2 + 788*a^2*b*c^3)*x^(13/2) + (21*b^6 - 121*a*b^4*c - 41*a^2*b^2 
*c^2 + 900*a^3*c^3)*x^(9/2) + (49*a*b^5 - 398*a^2*b^3*c + 832*a^3*b*c^2)*x 
^(5/2) + 32*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*sqrt(x))/(a^5*b^4 - 8*a^6 
*b^2*c + 16*a^7*c^2 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^8 + 2*( 
a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^6 + (a^3*b^6 - 6*a^4*b^4*c + 3 
2*a^6*c^3)*x^4 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2) - integrate 
(3/32*((7*b^4*c - 59*a*b^2*c^2 + 140*a^2*c^3)*x^(7/2) + (7*b^5 - 66*a*b^3* 
c + 192*a^2*b*c^2)*x^(3/2))/(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + (a^3*b^4 
*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^4 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c 
^2)*x^2), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{3} \sqrt {x}} \,d x } \] Input:

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

Output:

integrate(1/((c*x^4 + b*x^2 + a)^3*sqrt(x)), x)
 

Mupad [B] (verification not implemented)

Time = 23.43 (sec) , antiderivative size = 60099, normalized size of antiderivative = 91.34 \[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((x^(9/2)*(14*b^4*c + 60*a^2*c^3 - 107*a*b^2*c^2))/(16*a^2*(b^4 + 16*a^2*c 
^2 - 8*a*b^2*c)) + (x^(1/2)*(11*b^4 + 92*a^2*c^2 - 79*a*b^2*c))/(16*a*(b^4 
 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^(5/2)*(8*a^2*b*c^2 - 7*b^5 + 44*a*b^3*c)) 
/(16*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^(13/2)*(52*a*c - 7*b^2 
))/(16*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2 
*x^8 + 2*a*b*x^2 + 2*b*c*x^6) - atan((((((9*x^(1/2)*(1546704997025054720*a 
^19*b*c^19 - 822083584*a^4*b^31*c^4 + 50851741696*a^5*b^29*c^5 - 147367709 
9008*a^6*b^27*c^6 + 26523687976960*a^7*b^25*c^7 - 331351626612736*a^8*b^23 
*c^8 + 3041476258824192*a^9*b^21*c^9 - 21176692735213568*a^10*b^19*c^10 + 
113812892427485184*a^11*b^17*c^11 - 475720885626470400*a^12*b^15*c^12 + 15 
45406748670558208*a^13*b^13*c^13 - 3867206695260258304*a^14*b^11*c^14 + 73 
15227880965799936*a^15*b^9*c^15 - 10117494892562219008*a^16*b^7*c^16 + 965 
0897342106173440*a^17*b^5*c^17 - 5672002255696429056*a^18*b^3*c^18))/(4194 
304*(a^8*b^24 + 16777216*a^20*c^12 - 48*a^9*b^22*c + 1056*a^10*b^20*c^2 - 
14080*a^11*b^18*c^3 + 126720*a^12*b^16*c^4 - 811008*a^13*b^14*c^5 + 378470 
4*a^14*b^12*c^6 - 12976128*a^15*b^10*c^7 + 32440320*a^16*b^8*c^8 - 5767168 
0*a^17*b^6*c^9 + 69206016*a^18*b^4*c^10 - 50331648*a^19*b^2*c^11)) - (3*(- 
(81*(2401*b^39 - 2401*b^14*(-(4*a*c - b^2)^25)^(1/2) - 2405416566784000*a^ 
19*b*c^19 + 7445060*a^2*b^35*c^2 - 180851965*a^3*b^33*c^3 + 3112544495*a^4 
*b^31*c^4 - 40302663491*a^5*b^29*c^5 + 406936342200*a^6*b^27*c^6 - 3276...
 

Reduce [F]

\[ \int \frac {1}{\sqrt {x} \left (a+b x^2+c x^4\right )^3} \, dx=\int \frac {1}{\sqrt {x}\, \left (c \,x^{4}+b \,x^{2}+a \right )^{3}}d x \] Input:

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

Output:

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