Integrand size = 20, antiderivative size = 474 \[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {d x \left (b^2-2 a c+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {d x \left (\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )+3 b c \left (b^2-8 a c\right ) x^2\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b \left (b^2-8 a c\right ) \sqrt {b^2-4 a c}\right ) d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {3 \sqrt {c} \left (b^3-8 a b c-\frac {b^4-10 a b^2 c+56 a^2 c^2}{\sqrt {b^2-4 a c}}\right ) d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a^2 \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {6 c^2 e \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
-1/4*e*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/4*d*x*(b*c*x^2-2*a*c+b ^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+3/2*c*e*(2*c*x^2+b)/(-4*a*c+b^2)^2/(c *x^4+b*x^2+a)+1/8*d*x*((-7*a*c+b^2)*(-4*a*c+3*b^2)+3*b*c*(-8*a*c+b^2)*x^2) /a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)-6*c^2*e*arctanh((2*c*x^2+b)/(-4*a*c+b^ 2)^(1/2))/(-4*a*c+b^2)^(5/2)+3/16*d*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^ 2)^(1/2))^(1/2))*c^(1/2)*(b^4-10*a*b^2*c+56*a^2*c^2+b*(-8*a*c+b^2)*(-4*a*c +b^2)^(1/2))/a^2/(-4*a*c+b^2)^(5/2)*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+3 /16*d*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(b^3- 8*a*b*c+(-56*a^2*c^2+10*a*b^2*c-b^4)/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^ 2*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 1.14 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{16} \left (\frac {4 a b e+8 a c x (d+e x)-4 b d x \left (b+c x^2\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {6 b^3 d x \left (b+c x^2\right )-2 a b c d x \left (25 b+24 c x^2\right )+8 a^2 c (3 b e+c x (7 d+6 e x))}{a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {3 \sqrt {2} \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {2} \sqrt {c} \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right ) d \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {48 c^2 e \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac {48 c^2 e \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}\right ) \]
((4*a*b*e + 8*a*c*x*(d + e*x) - 4*b*d*x*(b + c*x^2))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (6*b^3*d*x*(b + c*x^2) - 2*a*b*c*d*x*(25*b + 24*c*x^ 2) + 8*a^2*c*(3*b*e + c*x*(7*d + 6*e*x)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b ^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt [b - Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a *c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]] ) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(5/2) - (48*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/16
Time = 0.85 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {2202, 27, 1405, 25, 1432, 1086, 1086, 1083, 219, 1492, 27, 1480, 218}
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.
| \(\displaystyle \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {d}{\left (c x^4+b x^2+a\right )^3}dx+\int \frac {e x}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \int \frac {1}{\left (c x^4+b x^2+a\right )^3}dx+e \int \frac {x}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle d \left (\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\int -\frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}\right )+e \int \frac {x}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (\frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+e \int \frac {x}{\left (c x^4+b x^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle d \left (\frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \int \frac {1}{\left (c x^4+b x^2+a\right )^3}dx^2\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle d \left (\frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \int \frac {1}{\left (c x^4+b x^2+a\right )^2}dx^2}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle d \left (\frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (-\frac {2 c \int \frac {1}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle d \left (\frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{b^2-4 a c}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\int \frac {3 b^2+5 c x^2 b-14 a c}{\left (c x^4+b x^2+a\right )^2}dx}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle d \left (\frac {\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {3 \left (b^4-9 a c b^2+c \left (b^2-8 a c\right ) x^2 b+28 a^2 c^2\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\frac {3 \int \frac {b^4-9 a c b^2+c \left (b^2-8 a c\right ) x^2 b+28 a^2 c^2}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle d \left (\frac {\frac {3 \left (\frac {1}{2} c \left (\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}+b \left (b^2-8 a c\right )\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (b \left (b^2-8 a c\right )-\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx\right )}{2 a \left (b^2-4 a c\right )}+\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle d \left (\frac {\frac {3 \left (\frac {\sqrt {c} \left (\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}+b \left (b^2-8 a c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b \left (b^2-8 a c\right )-\frac {56 a^2 c^2-10 a b^2 c+b^4}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{2 a \left (b^2-4 a c\right )}+\frac {x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 a \left (b^2-4 a c\right )}+\frac {x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )+\frac {1}{2} e \left (-\frac {3 c \left (\frac {4 c \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )}{b^2-4 a c}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\right )\) |
d*((x*(b^2 - 2*a*c + b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((x*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b*c*(b^2 - 8*a*c)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (3*((Sqrt[c]*(b*(b^2 - 8*a*c) + (b^4 - 10* a*b^2*c + 56*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*( b*(b^2 - 8*a*c) - (b^4 - 10*a*b^2*c + 56*a^2*c^2)/Sqrt[b^2 - 4*a*c])*ArcTa n[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt [b^2 - 4*a*c]])))/(2*a*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c))) + (e*(-1/2*(b + 2*c*x^2)/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*c*(-((b + 2*c*x^2)/( (b^2 - 4*a*c)*(a + b*x^2 + c*x^4))) + (4*c*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)))/(b^2 - 4*a*c)))/2
3.1.52.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), 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^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.53 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {-\frac {3 b \,c^{2} d \left (8 a c -b^{2}\right ) x^{7}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {3 c^{3} e \,x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {c d \left (28 a^{2} c^{2}-49 a \,b^{2} c +6 b^{4}\right ) x^{5}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {9 b \,c^{2} e \,x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {b d \left (4 a^{2} c^{2}+20 a \,b^{2} c -3 b^{4}\right ) x^{3}}{8 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (5 a c +b^{2}\right ) c e \,x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {d \left (44 a^{2} c^{2}-37 a \,b^{2} c +5 b^{4}\right ) x}{8 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a}+\frac {b \left (10 a c -b^{2}\right ) e}{64 a^{2} c^{2}-32 a \,b^{2} c +4 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-\frac {b c d \left (8 a c -b^{2}\right ) \textit {\_R}^{2}}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {16 c^{2} e \textit {\_R}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {d \left (28 a^{2} c^{2}-9 a \,b^{2} c +b^{4}\right )}{a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}\right )}{16}\) | \(499\) |
| default | \(64 c^{3} \left (\frac {\frac {-\frac {3 \left (24 a^{2} c^{2} \sqrt {-4 a c +b^{2}}-10 a \,b^{2} c \sqrt {-4 a c +b^{2}}+b^{4} \sqrt {-4 a c +b^{2}}+32 a^{2} b \,c^{2}-12 a \,b^{3} c +b^{5}\right ) d \,x^{3}}{64 a^{2} c^{3}}+\frac {3 e \left (4 a c -b^{2}\right ) x^{2}}{8 c^{2}}-\frac {d \left (-20 \sqrt {-4 a c +b^{2}}\, a b c +5 \sqrt {-4 a c +b^{2}}\, b^{3}+176 a^{2} c^{2}-64 a \,b^{2} c +5 b^{4}\right ) x}{64 a \,c^{3}}+\frac {e \left (-16 a c \sqrt {-4 a c +b^{2}}+4 b^{2} \sqrt {-4 a c +b^{2}}+12 a b c -3 b^{3}\right )}{16 c^{3}}}{\left (x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}}+\frac {-\frac {3 \sqrt {-4 a c +b^{2}}\, a^{2} c e \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4}+\frac {3 \left (56 \sqrt {-4 a c +b^{2}}\, a^{2} c^{2} d -10 \sqrt {-4 a c +b^{2}}\, a \,b^{2} c d +\sqrt {-4 a c +b^{2}}\, b^{4} d +32 a^{2} b \,c^{2} d -12 a \,b^{3} c d +b^{5} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{64 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{a^{2} c^{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (4 a c -b^{2}\right )}+\frac {\frac {-\frac {3 \left (-24 a^{2} c^{2} \sqrt {-4 a c +b^{2}}+10 a \,b^{2} c \sqrt {-4 a c +b^{2}}-b^{4} \sqrt {-4 a c +b^{2}}+32 a^{2} b \,c^{2}-12 a \,b^{3} c +b^{5}\right ) d \,x^{3}}{64 a^{2} c^{3}}+\frac {3 e \left (4 a c -b^{2}\right ) x^{2}}{8 c^{2}}-\frac {d \left (20 \sqrt {-4 a c +b^{2}}\, a b c -5 \sqrt {-4 a c +b^{2}}\, b^{3}+176 a^{2} c^{2}-64 a \,b^{2} c +5 b^{4}\right ) x}{64 a \,c^{3}}+\frac {e \left (16 a c \sqrt {-4 a c +b^{2}}-4 b^{2} \sqrt {-4 a c +b^{2}}+12 a b c -3 b^{3}\right )}{16 c^{3}}}{\left (x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right )^{2}}+\frac {\frac {3 \sqrt {-4 a c +b^{2}}\, a^{2} c e \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4}+\frac {3 \left (56 \sqrt {-4 a c +b^{2}}\, a^{2} c^{2} d -10 \sqrt {-4 a c +b^{2}}\, a \,b^{2} c d +\sqrt {-4 a c +b^{2}}\, b^{4} d -32 a^{2} b \,c^{2} d +12 a \,b^{3} c d -b^{5} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{64 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{a^{2} c^{2}}}{16 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \left (4 a c -b^{2}\right )}\right )\) | \(889\) |
(-3/8*b*c^2*d*(8*a*c-b^2)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+3*c^3*e/(16*a ^2*c^2-8*a*b^2*c+b^4)*x^6+1/8/a^2*c*d*(28*a^2*c^2-49*a*b^2*c+6*b^4)/(16*a^ 2*c^2-8*a*b^2*c+b^4)*x^5+9/2*b*c^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-1/8*b* d*(4*a^2*c^2+20*a*b^2*c-3*b^4)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+(5*a*c+b ^2)*c*e/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2+1/8*d*(44*a^2*c^2-37*a*b^2*c+5*b^4) /(16*a^2*c^2-8*a*b^2*c+b^4)/a*x+1/4*b*(10*a*c-b^2)*e/(16*a^2*c^2-8*a*b^2*c +b^4))/(c*x^4+b*x^2+a)^2+3/16*sum((-b*c*d*(8*a*c-b^2)/a^2/(16*a^2*c^2-8*a* b^2*c+b^4)*_R^2+16*c^2*e/(16*a^2*c^2-8*a*b^2*c+b^4)*_R+d*(28*a^2*c^2-9*a*b ^2*c+b^4)/a^2/(16*a^2*c^2-8*a*b^2*c+b^4))/(2*_R^3*c+_R*b)*ln(x-_R),_R=Root Of(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \]
1/8*(24*a^2*c^3*e*x^6 + 36*a^2*b*c^2*e*x^4 + 3*(b^3*c^2 - 8*a*b*c^3)*d*x^7 + (6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d*x^5 + (3*b^5 - 20*a*b^3*c - 4*a ^2*b*c^2)*d*x^3 + 8*(a^2*b^2*c + 5*a^3*c^2)*e*x^2 + (5*a*b^4 - 37*a^2*b^2* c + 44*a^3*c^2)*d*x - 2*(a^2*b^3 - 10*a^3*b*c)*e)/((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) - 3/8*integrate(-(1 6*a^2*c^2*e*x + (b^3*c - 8*a*b*c^2)*d*x^2 + (b^4 - 9*a*b^2*c + 28*a^2*c^2) *d)/(c*x^4 + b*x^2 + a), x)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 3389 vs. \(2 (420) = 840\).
Time = 3.11 (sec) , antiderivative size = 3389, normalized size of antiderivative = 7.15 \[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
3/32*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 17*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b ^7*c - 2*b^8*c + 116*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 26*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 34*a*b^6*c^2 + 2*b^7*c^2 - 368*sqrt(2)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 128*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 13*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^ 4*c^3 - 232*a^2*b^4*c^3 - 30*a*b^5*c^3 + 448*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^4 + 224*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 64*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 736*a^3*b^2*c^ 4 + 176*a^2*b^3*c^4 - 112*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^5 - 896*a^4*c^5 - 352*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt( b^2 - 4*a*c)*c)*b^7 + 15*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4 *a*c)*c)*a*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c )*c)*b^6*c - 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* a^2*b^3*c^2 - 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) *a*b^4*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 *c^2 + 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b *c^3 + 88*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^ 2*c^3 + 11*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*...
Time = 8.77 (sec) , antiderivative size = 4225, normalized size of antiderivative = 8.91 \[ \int \frac {d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \]
symsum(log(root(56371445760*a^11*b^8*c^6*z^4 - 503316480*a^8*b^14*c^3*z^4 + 47185920*a^7*b^16*c^2*z^4 - 171798691840*a^14*b^2*c^9*z^4 + 193273528320 *a^13*b^4*c^8*z^4 - 128849018880*a^12*b^6*c^7*z^4 - 16911433728*a^10*b^10* c^5*z^4 + 3523215360*a^9*b^12*c^4*z^4 - 2621440*a^6*b^18*c*z^4 + 687194767 36*a^15*c^10*z^4 + 65536*a^5*b^20*z^4 + 6936330240*a^8*b^3*c^8*d^2*z^2 + 2 464874496*a^6*b^7*c^6*d^2*z^2 - 3963617280*a^9*b*c^9*d^2*z^2 - 1509949440* a^9*b^2*c^8*e^2*z^2 - 5400428544*a^7*b^5*c^7*d^2*z^2 - 94464*a*b^17*c*d^2* z^2 + 754974720*a^8*b^4*c^7*e^2*z^2 - 730054656*a^5*b^9*c^5*d^2*z^2 - 1887 43680*a^7*b^6*c^6*e^2*z^2 + 146165760*a^4*b^11*c^4*d^2*z^2 + 23592960*a^6* b^8*c^5*e^2*z^2 - 19860480*a^3*b^13*c^3*d^2*z^2 - 1179648*a^5*b^10*c^4*e^2 *z^2 + 1771776*a^2*b^15*c^2*d^2*z^2 + 1207959552*a^10*c^9*e^2*z^2 + 2304*b ^19*d^2*z^2 - 428544*a*b^12*c^3*d^2*e*z + 1022754816*a^6*b^2*c^8*d^2*e*z - 642318336*a^5*b^4*c^7*d^2*e*z + 223395840*a^4*b^6*c^6*d^2*e*z - 46725120* a^3*b^8*c^5*d^2*e*z + 5930496*a^2*b^10*c^4*d^2*e*z - 693633024*a^7*c^9*d^2 *e*z + 13824*b^14*c^2*d^2*e*z + 34836480*a^4*b*c^8*d^2*e^2 - 435456*a*b^7* c^5*d^2*e^2 - 17418240*a^3*b^3*c^7*d^2*e^2 + 3919104*a^2*b^5*c^6*d^2*e^2 + 20736*b^9*c^4*d^2*e^2 - 27433728*a^3*b^2*c^8*d^4 + 6446304*a^2*b^4*c^7*d^ 4 - 734832*a*b^6*c^6*d^4 + 49787136*a^4*c^9*d^4 + 5308416*a^5*c^8*e^4 + 35 721*b^8*c^5*d^4, z, k)*(root(56371445760*a^11*b^8*c^6*z^4 - 503316480*a^8* b^14*c^3*z^4 + 47185920*a^7*b^16*c^2*z^4 - 171798691840*a^14*b^2*c^9*z^...