Integrand size = 20, antiderivative size = 658 \[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x^{3/2} \left (5 b^4-45 a b^2 c+52 a^2 c^2+b c \left (5 b^2-44 a c\right ) x^2\right )}{16 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [4]{c} \left (5 b^4-54 a b^2 c+520 a^2 c^2+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{32\ 2^{3/4} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt [4]{-b+\sqrt {b^2-4 a c}}} \]
1/4*x^(3/2)*(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^(3 /2)*(5*b^4-45*a*b^2*c+52*a^2*c^2+b*c*(-44*a*c+5*b^2)*x^2)/a^2/(-4*a*c+b^2) ^2/(c*x^4+b*x^2+a)-1/64*c^(1/4)*arctan(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c +b^2)^(1/2))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2-b*(-44*a*c+5*b^2)*(-4*a* c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4) +1/64*c^(1/4)*arctanh(2^(1/4)*c^(1/4)*x^(1/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4 ))*(5*b^4-54*a*b^2*c+520*a^2*c^2-b*(-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^( 1/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b-(-4*a*c+b^2)^(1/2))^(1/4)+1/64*c^(1/4)*arc tan(2^(1/4)*c^(1/4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^4-54*a*b^2 *c+520*a^2*c^2+b*(-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b ^2)^(5/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4)-1/64*c^(1/4)*arctanh(2^(1/4)*c^(1/ 4)*x^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*(5*b^4-54*a*b^2*c+520*a^2*c^2+b* (-44*a*c+5*b^2)*(-4*a*c+b^2)^(1/2))*2^(1/4)/a^2/(-4*a*c+b^2)^(5/2)/(-b+(-4 *a*c+b^2)^(1/2))^(1/4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.44 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 x^{3/2} \left (84 a^3 c^2+5 b^3 x^2 \left (b+c x^2\right )^2+a^2 c \left (-69 b^2-8 b c x^2+52 c^2 x^4\right )+a b \left (9 b^3-36 b^2 c x^2-89 b c^2 x^4-44 c^3 x^6\right )\right )}{\left (a+b x^2+c x^4\right )^2}+\text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {5 b^4 \log \left (\sqrt {x}-\text {$\#$1}\right )-49 a b^2 c \log \left (\sqrt {x}-\text {$\#$1}\right )+260 a^2 c^2 \log \left (\sqrt {x}-\text {$\#$1}\right )+5 b^3 c \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4-44 a b c^2 \log \left (\sqrt {x}-\text {$\#$1}\right ) \text {$\#$1}^4}{b \text {$\#$1}+2 c \text {$\#$1}^5}\&\right ]}{64 a^2 \left (b^2-4 a c\right )^2} \]
((4*x^(3/2)*(84*a^3*c^2 + 5*b^3*x^2*(b + c*x^2)^2 + a^2*c*(-69*b^2 - 8*b*c *x^2 + 52*c^2*x^4) + a*b*(9*b^3 - 36*b^2*c*x^2 - 89*b*c^2*x^4 - 44*c^3*x^6 )))/(a + b*x^2 + c*x^4)^2 + RootSum[a + b*#1^4 + c*#1^8 & , (5*b^4*Log[Sqr t[x] - #1] - 49*a*b^2*c*Log[Sqrt[x] - #1] + 260*a^2*c^2*Log[Sqrt[x] - #1] + 5*b^3*c*Log[Sqrt[x] - #1]*#1^4 - 44*a*b*c^2*Log[Sqrt[x] - #1]*#1^4)/(b*# 1 + 2*c*#1^5) & ])/(64*a^2*(b^2 - 4*a*c)^2)
Time = 1.41 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1435, 1702, 25, 1824, 25, 1834, 27, 827, 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.
| \(\displaystyle \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1435 |
| \(\displaystyle 2 \int \frac {x}{\left (c x^4+b x^2+a\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 1702 |
| \(\displaystyle 2 \left (\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int -\frac {x \left (5 b^2+9 c x^2 b-26 a c\right )}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 a \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {x \left (5 b^2+9 c x^2 b-26 a c\right )}{\left (c x^4+b x^2+a\right )^2}d\sqrt {x}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1824 |
| \(\displaystyle 2 \left (\frac {\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {x \left (5 b^4-49 a c b^2+c \left (5 b^2-44 a c\right ) x^2 b+260 a^2 c^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {x \left (5 b^4-49 a c b^2+c \left (5 b^2-44 a c\right ) x^2 b+260 a^2 c^2\right )}{c x^4+b x^2+a}d\sqrt {x}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1834 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {c \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {2 x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}-\frac {c \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {2 x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{2 \sqrt {b^2-4 a c}}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {c \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {x}{2 c x^2+b-\sqrt {b^2-4 a c}}d\sqrt {x}}{\sqrt {b^2-4 a c}}-\frac {c \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \int \frac {x}{2 c x^2+b+\sqrt {b^2-4 a c}}d\sqrt {x}}{\sqrt {b^2-4 a c}}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {c \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {\sqrt {b^2-4 a c}-b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {c} x+\sqrt {-b-\sqrt {b^2-4 a c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{\sqrt {b^2-4 a c}}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {c \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {\sqrt {b^2-4 a c}-b}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\int \frac {1}{\sqrt {-b-\sqrt {b^2-4 a c}}-\sqrt {2} \sqrt {c} x}d\sqrt {x}}{2 \sqrt {2} \sqrt {c}}\right )}{\sqrt {b^2-4 a c}}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {c \left (520 a^2 c^2-54 a b^2 c+b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{\sqrt {b^2-4 a c}-b}}\right )}{\sqrt {b^2-4 a c}}-\frac {c \left (520 a^2 c^2-54 a b^2 c-b \left (5 b^2-44 a c\right ) \sqrt {b^2-4 a c}+5 b^4\right ) \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{2\ 2^{3/4} c^{3/4} \sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right )}{\sqrt {b^2-4 a c}}}{4 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (52 a^2 c^2+b c x^2 \left (5 b^2-44 a c\right )-45 a b^2 c+5 b^4\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{8 a \left (b^2-4 a c\right )}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{8 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
2*((x^(3/2)*(b^2 - 2*a*c + b*c*x^2))/(8*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4 )^2) + ((x^(3/2)*(5*b^4 - 45*a*b^2*c + 52*a^2*c^2 + b*c*(5*b^2 - 44*a*c)*x ^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (-((c*(5*b^4 - 54*a*b^2*c + 520*a^2*c^2 - b*(5*b^2 - 44*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*2^(3/4)*c^(3/4)*(-b - Sqrt [b^2 - 4*a*c])^(1/4)) - ArcTanh[(2^(1/4)*c^(1/4)*Sqrt[x])/(-b - Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4)*(-b - Sqrt[b^2 - 4*a*c])^(1/4))))/Sqrt[ b^2 - 4*a*c]) + (c*(5*b^4 - 54*a*b^2*c + 520*a^2*c^2 + b*(5*b^2 - 44*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*2^(3/4)*c^(3/4)*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) - ArcTanh[(2 ^(1/4)*c^(1/4)*Sqrt[x])/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(2*2^(3/4)*c^(3/4) *(-b + Sqrt[b^2 - 4*a*c])^(1/4))))/Sqrt[b^2 - 4*a*c])/(4*a*(b^2 - 4*a*c))) /(8*a*(b^2 - 4*a*c)))
3.11.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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]
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x _Symbol] :> Simp[(-(d*x)^(m + 1))*(b^2 - 2*a*c + b*c*x^n)*((a + b*x^n + c*x ^(2*n))^(p + 1)/(a*d*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b ^2 - 4*a*c)) Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[b^2*(m + n* (p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1) + b*c*(m + n*(2*p + 3) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c , 0] && IGtQ[n, 0] && ILtQ[p, -1]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^n + c*x^ (2*n))^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^m* (a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + c*(m + n*(2*p + 3) + 1)*(b*d - 2*a*e) *x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && IntegerQ[p]
Int[(((f_.)*(x_))^(m_.)*((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[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ [{a, b, c, d, e, f, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n , 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.19 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.49
| method | result | size |
| derivativedivides | \(\frac {\frac {3 \left (28 a^{2} c^{2}-23 a \,b^{2} c +3 b^{4}\right ) x^{\frac {3}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {b \left (8 a^{2} c^{2}+36 a \,b^{2} c -5 b^{4}\right ) x^{\frac {7}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (52 a^{2} c^{2}-89 a \,b^{2} c +10 b^{4}\right ) x^{\frac {11}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \,c^{2} \left (44 a c -5 b^{2}\right ) x^{\frac {15}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c \left (-44 a c +5 b^{2}\right ) \textit {\_R}^{6}+\left (260 a^{2} c^{2}-49 a \,b^{2} c +5 b^{4}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(321\) |
| default | \(\frac {\frac {3 \left (28 a^{2} c^{2}-23 a \,b^{2} c +3 b^{4}\right ) x^{\frac {3}{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}-\frac {b \left (8 a^{2} c^{2}+36 a \,b^{2} c -5 b^{4}\right ) x^{\frac {7}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (52 a^{2} c^{2}-89 a \,b^{2} c +10 b^{4}\right ) x^{\frac {11}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b \,c^{2} \left (44 a c -5 b^{2}\right ) x^{\frac {15}{2}}}{16 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (b c \left (-44 a c +5 b^{2}\right ) \textit {\_R}^{6}+\left (260 a^{2} c^{2}-49 a \,b^{2} c +5 b^{4}\right ) \textit {\_R}^{2}\right ) \ln \left (\sqrt {x}-\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{64 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(321\) |
2*(3/32*(28*a^2*c^2-23*a*b^2*c+3*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/a*x^(3/2) -1/32*b*(8*a^2*c^2+36*a*b^2*c-5*b^4)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2 )+1/32/a^2*c*(52*a^2*c^2-89*a*b^2*c+10*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^( 11/2)-1/32*b*c^2*(44*a*c-5*b^2)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(15/2))/( c*x^4+b*x^2+a)^2+1/64/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*sum((b*c*(-44*a*c+5*b ^2)*_R^6+(260*a^2*c^2-49*a*b^2*c+5*b^4)*_R^2)/(2*_R^7*c+_R^3*b)*ln(x^(1/2) -_R),_R=RootOf(_Z^8*c+_Z^4*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 23877 vs. \(2 (554) = 1108\).
Time = 166.08 (sec) , antiderivative size = 23877, normalized size of antiderivative = 36.29 \[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {\sqrt {x}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
1/16*((5*b^3*c^2 - 44*a*b*c^3)*x^(15/2) + (10*b^4*c - 89*a*b^2*c^2 + 52*a^ 2*c^3)*x^(11/2) + (5*b^5 - 36*a*b^3*c - 8*a^2*b*c^2)*x^(7/2) + 3*(3*a*b^4 - 23*a^2*b^2*c + 28*a^3*c^2)*x^(3/2))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a ^4*c^4)*x^8 + a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^ 3*c^2 + 16*a^4*b*c^3)*x^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*( a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^2) - integrate(-1/32*((5*b^3*c - 4 4*a*b*c^2)*x^(5/2) + (5*b^4 - 49*a*b^2*c + 260*a^2*c^2)*sqrt(x))/(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2), x)
\[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {\sqrt {x}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
Time = 18.05 (sec) , antiderivative size = 46948, normalized size of antiderivative = 71.35 \[ \int \frac {\sqrt {x}}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
((x^(11/2)*(10*b^4*c + 52*a^2*c^3 - 89*a*b^2*c^2))/(16*(a^2*b^4 + 16*a^4*c ^2 - 8*a^3*b^2*c)) - (x^(7/2)*(8*a^2*b*c^2 - 5*b^5 + 36*a*b^3*c))/(16*a*(a *b^4 + 16*a^3*c^2 - 8*a^2*b^2*c)) + (3*x^(3/2)*(3*b^4 + 28*a^2*c^2 - 23*a* b^2*c))/(16*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (b*c^2*x^(15/2)*(44*a*c - 5*b^2))/(16*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*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(((((2097152000*a*b^33*c^4 + 46 6178856428188467200*a^17*b*c^20 - 151833804800*a^2*b^31*c^5 + 534002008064 0*a^3*b^29*c^6 - 120300087803904*a^4*b^27*c^7 + 1933149881761792*a^5*b^25* c^8 - 23398590986584064*a^6*b^23*c^9 + 219878252263505920*a^7*b^21*c^10 - 1631099300505190400*a^8*b^19*c^11 + 9625014804028588032*a^9*b^17*c^12 - 45 207702606568226816*a^10*b^15*c^13 + 168027072287612076032*a^11*b^13*c^14 - 487882094458626375680*a^12*b^11*c^15 + 1082673222923122114560*a^13*b^9*c^ 16 - 1771946621413479153664*a^14*b^7*c^17 + 2014068018680264916992*a^15*b^ 5*c^18 - 1418770116510434197504*a^16*b^3*c^19)/(268435456*(a^6*b^28 + 2684 35456*a^20*c^14 - 56*a^7*b^26*c + 1456*a^8*b^24*c^2 - 23296*a^9*b^22*c^3 + 256256*a^10*b^20*c^4 - 2050048*a^11*b^18*c^5 + 12300288*a^12*b^16*c^6 - 5 6229888*a^13*b^14*c^7 + 196804608*a^14*b^12*c^8 - 524812288*a^15*b^10*c^9 + 1049624576*a^16*b^8*c^10 - 1526726656*a^17*b^6*c^11 + 1526726656*a^18*b^ 4*c^12 - 939524096*a^19*b^2*c^13)) - (x^(1/2)*(-(625*b^37 - 625*b^12*(-(4* a*c - b^2)^25)^(1/2) + 11279020326912000*a^18*b*c^18 + 2168275*a^2*b^33...