Integrand size = 17, antiderivative size = 149 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=-\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \]
-3/2*e^2*(-a*e^2+c*d^2)*x/a/c^2-1/2*d*e^3*x^2/a/c-1/2*(-c*d*x+a*e)*(e*x+d) ^3/a/c/(c*x^2+a)+1/2*(-3*a^2*e^4+6*a*c*d^2*e^2+c^2*d^4)*arctan(x*c^(1/2)/a ^(1/2))/a^(3/2)/c^(5/2)+2*d*e^3*ln(c*x^2+a)/c^2
Time = 0.09 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {e^4 x}{c^2}+\frac {c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)}{2 a c^2 \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \]
(e^4*x)/c^2 + (c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x) )/(2*a*c^2*(a + c*x^2)) + ((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(S qrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*x^2])/c^2
Time = 0.31 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {495, 657, 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.
\(\displaystyle \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 495 |
\(\displaystyle \frac {\int \frac {(d+e x)^2 \left (c d^2-2 c e x d+3 a e^2\right )}{c x^2+a}dx}{2 a c}-\frac {(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
\(\Big \downarrow \) 657 |
\(\displaystyle \frac {\int \left (-2 d x e^3-3 \left (d^2-\frac {a e^2}{c}\right ) e^2+\frac {c^2 d^4+6 a c e^2 d^2+8 a c e^3 x d-3 a^2 e^4}{c \left (c x^2+a\right )}\right )dx}{2 a c}-\frac {(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )}{\sqrt {a} c^{3/2}}-3 e^2 x \left (d^2-\frac {a e^2}{c}\right )+\frac {4 a d e^3 \log \left (a+c x^2\right )}{c}-d e^3 x^2}{2 a c}-\frac {(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )}\) |
-1/2*((a*e - c*d*x)*(d + e*x)^3)/(a*c*(a + c*x^2)) + (-3*e^2*(d^2 - (a*e^2 )/c)*x - d*e^3*x^2 + ((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sqrt[c ]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (4*a*d*e^3*Log[a + c*x^2])/c)/(2*a*c)
3.6.6.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (a*d - b*c*x)*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Time = 2.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {e^{4} x}{c^{2}}-\frac {\frac {-\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 a}-2 d e \left (e^{2} a -c \,d^{2}\right )}{c \,x^{2}+a}+\frac {-4 d \,e^{3} a \ln \left (c \,x^{2}+a \right )+\frac {\left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{c^{2}}\) | \(137\) |
risch | \(\frac {e^{4} x}{c^{2}}+\frac {\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 a}+2 d e \left (e^{2} a -c \,d^{2}\right )}{c^{2} \left (c \,x^{2}+a \right )}+\frac {2 \ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {\ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}}{4 c^{3} a^{2}}+\frac {2 \ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}-\frac {\ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}}{4 c^{3} a^{2}}\) | \(445\) |
e^4*x/c^2-1/c^2*((-1/2*(a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)/a*x-2*d*e*(a*e^2-c* d^2))/(c*x^2+a)+1/2/a*(-4*d*e^3*a*ln(c*x^2+a)+(3*a^2*e^4-6*a*c*d^2*e^2-c^2 *d^4)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))))
Time = 0.48 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.91 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\left [\frac {4 \, a^{2} c^{2} e^{4} x^{3} - 8 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} + {\left (a c^{2} d^{4} + 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4} + {\left (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (a c^{3} d^{4} - 6 \, a^{2} c^{2} d^{2} e^{2} + 3 \, a^{3} c e^{4}\right )} x + 8 \, {\left (a^{2} c^{2} d e^{3} x^{2} + a^{3} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac {2 \, a^{2} c^{2} e^{4} x^{3} - 4 \, a^{2} c^{2} d^{3} e + 4 \, a^{3} c d e^{3} + {\left (a c^{2} d^{4} + 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4} + {\left (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (a c^{3} d^{4} - 6 \, a^{2} c^{2} d^{2} e^{2} + 3 \, a^{3} c e^{4}\right )} x + 4 \, {\left (a^{2} c^{2} d e^{3} x^{2} + a^{3} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \]
[1/4*(4*a^2*c^2*e^4*x^3 - 8*a^2*c^2*d^3*e + 8*a^3*c*d*e^3 + (a*c^2*d^4 + 6 *a^2*c*d^2*e^2 - 3*a^3*e^4 + (c^3*d^4 + 6*a*c^2*d^2*e^2 - 3*a^2*c*e^4)*x^2 )*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(a*c^3*d^4 - 6*a^2*c^2*d^2*e^2 + 3*a^3*c*e^4)*x + 8*(a^2*c^2*d*e^3*x^2 + a^3*c*d*e^3) *log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3), 1/2*(2*a^2*c^2*e^4*x^3 - 4*a^2*c ^2*d^3*e + 4*a^3*c*d*e^3 + (a*c^2*d^4 + 6*a^2*c*d^2*e^2 - 3*a^3*e^4 + (c^3 *d^4 + 6*a*c^2*d^2*e^2 - 3*a^2*c*e^4)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (a*c^3*d^4 - 6*a^2*c^2*d^2*e^2 + 3*a^3*c*e^4)*x + 4*(a^2*c^2*d*e^3*x^2 + a^3*c*d*e^3)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (138) = 276\).
Time = 0.77 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.70 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\left (\frac {2 d e^{3}}{c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log {\left (x + \frac {- 4 a^{2} c^{2} \cdot \left (\frac {2 d e^{3}}{c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \left (\frac {2 d e^{3}}{c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log {\left (x + \frac {- 4 a^{2} c^{2} \cdot \left (\frac {2 d e^{3}}{c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \frac {4 a^{2} d e^{3} - 4 a c d^{3} e + x \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} + \frac {e^{4} x}{c^{2}} \]
(2*d*e**3/c**2 - sqrt(-a**3*c**5)*(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d* *4)/(4*a**3*c**5))*log(x + (-4*a**2*c**2*(2*d*e**3/c**2 - sqrt(-a**3*c**5) *(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**5)) + 8*a**2*d*e** 3)/(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d**4)) + (2*d*e**3/c**2 + sqrt(-a **3*c**5)*(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**5))*log(x + (-4*a**2*c**2*(2*d*e**3/c**2 + sqrt(-a**3*c**5)*(3*a**2*e**4 - 6*a*c*d* *2*e**2 - c**2*d**4)/(4*a**3*c**5)) + 8*a**2*d*e**3)/(3*a**2*e**4 - 6*a*c* d**2*e**2 - c**2*d**4)) + (4*a**2*d*e**3 - 4*a*c*d**3*e + x*(a**2*e**4 - 6 *a*c*d**2*e**2 + c**2*d**4))/(2*a**2*c**2 + 2*a*c**3*x**2) + e**4*x/c**2
Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} + \frac {2 \, d e^{3} \log \left (c x^{2} + a\right )}{c^{2}} - \frac {4 \, a c d^{3} e - 4 \, a^{2} d e^{3} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} \]
e^4*x/c^2 + 2*d*e^3*log(c*x^2 + a)/c^2 - 1/2*(4*a*c*d^3*e - 4*a^2*d*e^3 - (c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*x)/(a*c^3*x^2 + a^2*c^2) + 1/2*(c^2*d^ 4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2)
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {e^{4} x}{c^{2}} + \frac {2 \, d e^{3} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} - \frac {4 \, a c d^{3} e - 4 \, a^{2} d e^{3} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \, {\left (c x^{2} + a\right )} a c^{2}} \]
e^4*x/c^2 + 2*d*e^3*log(c*x^2 + a)/c^2 + 1/2*(c^2*d^4 + 6*a*c*d^2*e^2 - 3* a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2) - 1/2*(4*a*c*d^3*e - 4*a^ 2*d*e^3 - (c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*x)/((c*x^2 + a)*a*c^2)
Time = 9.59 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {x\,\left (a^2\,e^4-6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,a}+2\,a\,d\,e^3-2\,c\,d^3\,e}{c^3\,x^2+a\,c^2}+\frac {e^4\,x}{c^2}+\frac {2\,d\,e^3\,\ln \left (c\,x^2+a\right )}{c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-3\,a^2\,e^4+6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,a^{3/2}\,c^{5/2}} \]