Integrand size = 30, antiderivative size = 168 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {c}{3 a^3 x^3}+\frac {3 b c-a d}{a^4 x}+\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}+\frac {\left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2} b^{3/2}} \]
-1/3*c/a^3/x^3+(-a*d+3*b*c)/a^4/x+1/4*(b^2*c/a^2-b*d/a+e-a*f/b)*x/a/(b*x^2 +a)^2+1/8*(a^3*f+3*a^2*b*e-7*a*b^2*d+11*b^3*c)*x/a^4/b/(b*x^2+a)+1/8*(a^3* f+3*a^2*b*e-15*a*b^2*d+35*b^3*c)*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)/b^(3/2)
Time = 0.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {-3 a^4 f x^4+105 b^4 c x^6+5 a b^3 x^4 \left (35 c-9 d x^2\right )+a^2 b^2 x^2 \left (56 c-75 d x^2+9 e x^4\right )+a^3 b \left (-8 c+3 x^2 \left (-8 d+5 e x^2+f x^4\right )\right )}{24 a^4 b x^3 \left (a+b x^2\right )^2}+\frac {\left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2} b^{3/2}} \]
(-3*a^4*f*x^4 + 105*b^4*c*x^6 + 5*a*b^3*x^4*(35*c - 9*d*x^2) + a^2*b^2*x^2 *(56*c - 75*d*x^2 + 9*e*x^4) + a^3*b*(-8*c + 3*x^2*(-8*d + 5*e*x^2 + f*x^4 )))/(24*a^4*b*x^3*(a + b*x^2)^2) + ((35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a ^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2)*b^(3/2))
Time = 0.51 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2336, 25, 1582, 1584, 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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int -\frac {\left (\frac {3 c b^2}{a^2}-\frac {3 d b}{a}+3 e+\frac {a f}{b}\right ) x^4-4 \left (\frac {b c}{a}-d\right ) x^2+4 c}{x^4 \left (b x^2+a\right )^2}dx}{4 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (\frac {3 c b^2}{a^2}-\frac {3 d b}{a}+3 e+\frac {a f}{b}\right ) x^4-4 \left (\frac {b c}{a}-d\right ) x^2+4 c}{x^4 \left (b x^2+a\right )^2}dx}{4 a}+\frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {\frac {\int \frac {b \left (f a^3+3 b e a^2-7 b^2 d a+11 b^3 c\right ) x^4-8 a b^2 (2 b c-a d) x^2+8 a^2 b^2 c}{x^4 \left (b x^2+a\right )}dx}{2 a^3 b^2}+\frac {x \left (a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right )}{2 a^3 b \left (a+b x^2\right )}}{4 a}+\frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\frac {\int \left (-\frac {8 (3 b c-a d) b^2}{x^2}+\frac {8 a c b^2}{x^4}+\frac {\left (f a^3+3 b e a^2-15 b^2 d a+35 b^3 c\right ) b}{b x^2+a}\right )dx}{2 a^3 b^2}+\frac {x \left (a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right )}{2 a^3 b \left (a+b x^2\right )}}{4 a}+\frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac {\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right )}{\sqrt {a}}+\frac {8 b^2 (3 b c-a d)}{x}-\frac {8 a b^2 c}{3 x^3}}{2 a^3 b^2}+\frac {x \left (a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right )}{2 a^3 b \left (a+b x^2\right )}}{4 a}\) |
(((b^2*c)/a^2 - (b*d)/a + e - (a*f)/b)*x)/(4*a*(a + b*x^2)^2) + (((11*b^3* c - 7*a*b^2*d + 3*a^2*b*e + a^3*f)*x)/(2*a^3*b*(a + b*x^2)) + ((-8*a*b^2*c )/(3*x^3) + (8*b^2*(3*b*c - a*d))/x + (Sqrt[b]*(35*b^3*c - 15*a*b^2*d + 3* a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a])/(2*a^3*b^2))/(4*a)
3.2.39.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Time = 3.51 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.90
| method | result | size |
| default | \(-\frac {c}{3 a^{3} x^{3}}-\frac {a d -3 b c}{a^{4} x}+\frac {\frac {\left (\frac {1}{8} f \,a^{3}+\frac {3}{8} a^{2} b e -\frac {7}{8} a \,b^{2} d +\frac {11}{8} b^{3} c \right ) x^{3}-\frac {a \left (f \,a^{3}-5 a^{2} b e +9 a \,b^{2} d -13 b^{3} c \right ) x}{8 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (f \,a^{3}+3 a^{2} b e -15 a \,b^{2} d +35 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 b \sqrt {a b}}}{a^{4}}\) | \(152\) |
| risch | \(\frac {\frac {\left (f \,a^{3}+3 a^{2} b e -15 a \,b^{2} d +35 b^{3} c \right ) x^{6}}{8 a^{4}}-\frac {\left (3 f \,a^{3}-15 a^{2} b e +75 a \,b^{2} d -175 b^{3} c \right ) x^{4}}{24 a^{3} b}-\frac {\left (3 a d -7 b c \right ) x^{2}}{3 a^{2}}-\frac {c}{3 a}}{x^{3} \left (b \,x^{2}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} b^{3} \textit {\_Z}^{2}+a^{6} f^{2}+6 a^{5} b e f -30 a^{4} b^{2} d f +9 a^{4} b^{2} e^{2}+70 a^{3} b^{3} c f -90 a^{3} b^{3} d e +210 a^{2} b^{4} c e +225 a^{2} b^{4} d^{2}-1050 a c d \,b^{5}+1225 c^{2} b^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9} b^{3}+2 a^{6} f^{2}+12 a^{5} b e f -60 a^{4} b^{2} d f +18 a^{4} b^{2} e^{2}+140 a^{3} b^{3} c f -180 a^{3} b^{3} d e +420 a^{2} b^{4} c e +450 a^{2} b^{4} d^{2}-2100 a c d \,b^{5}+2450 c^{2} b^{6}\right ) x +\left (-f \,a^{8} b -3 b^{2} e \,a^{7}+15 b^{3} d \,a^{6}-35 b^{4} c \,a^{5}\right ) \textit {\_R} \right )\right )}{16}\) | \(370\) |
-1/3*c/a^3/x^3-(a*d-3*b*c)/a^4/x+1/a^4*(((1/8*f*a^3+3/8*a^2*b*e-7/8*a*b^2* d+11/8*b^3*c)*x^3-1/8*a*(a^3*f-5*a^2*b*e+9*a*b^2*d-13*b^3*c)/b*x)/(b*x^2+a )^2+1/8*(a^3*f+3*a^2*b*e-15*a*b^2*d+35*b^3*c)/b/(a*b)^(1/2)*arctan(b*x/(a* b)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.39 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\left [-\frac {16 \, a^{4} b^{2} c - 6 \, {\left (35 \, a b^{5} c - 15 \, a^{2} b^{4} d + 3 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{6} - 2 \, {\left (175 \, a^{2} b^{4} c - 75 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{4} - 16 \, {\left (7 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{2} + 3 \, {\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \, {\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} + {\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{48 \, {\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}, -\frac {8 \, a^{4} b^{2} c - 3 \, {\left (35 \, a b^{5} c - 15 \, a^{2} b^{4} d + 3 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{6} - {\left (175 \, a^{2} b^{4} c - 75 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{4} - 8 \, {\left (7 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{2} - 3 \, {\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \, {\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} + {\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{24 \, {\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}\right ] \]
[-1/48*(16*a^4*b^2*c - 6*(35*a*b^5*c - 15*a^2*b^4*d + 3*a^3*b^3*e + a^4*b^ 2*f)*x^6 - 2*(175*a^2*b^4*c - 75*a^3*b^3*d + 15*a^4*b^2*e - 3*a^5*b*f)*x^4 - 16*(7*a^3*b^3*c - 3*a^4*b^2*d)*x^2 + 3*((35*b^5*c - 15*a*b^4*d + 3*a^2* b^3*e + a^3*b^2*f)*x^7 + 2*(35*a*b^4*c - 15*a^2*b^3*d + 3*a^3*b^2*e + a^4* b*f)*x^5 + (35*a^2*b^3*c - 15*a^3*b^2*d + 3*a^4*b*e + a^5*f)*x^3)*sqrt(-a* b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^5*b^4*x^7 + 2*a^6*b^3 *x^5 + a^7*b^2*x^3), -1/24*(8*a^4*b^2*c - 3*(35*a*b^5*c - 15*a^2*b^4*d + 3 *a^3*b^3*e + a^4*b^2*f)*x^6 - (175*a^2*b^4*c - 75*a^3*b^3*d + 15*a^4*b^2*e - 3*a^5*b*f)*x^4 - 8*(7*a^3*b^3*c - 3*a^4*b^2*d)*x^2 - 3*((35*b^5*c - 15* a*b^4*d + 3*a^2*b^3*e + a^3*b^2*f)*x^7 + 2*(35*a*b^4*c - 15*a^2*b^3*d + 3* a^3*b^2*e + a^4*b*f)*x^5 + (35*a^2*b^3*c - 15*a^3*b^2*d + 3*a^4*b*e + a^5* f)*x^3)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^5*b^4*x^7 + 2*a^6*b^3*x^5 + a^ 7*b^2*x^3)]
Timed out. \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (35 \, b^{4} c - 15 \, a b^{3} d + 3 \, a^{2} b^{2} e + a^{3} b f\right )} x^{6} - 8 \, a^{3} b c + {\left (175 \, a b^{3} c - 75 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{4} + 8 \, {\left (7 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{2}}{24 \, {\left (a^{4} b^{3} x^{7} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{3}\right )}} + \frac {{\left (35 \, b^{3} c - 15 \, a b^{2} d + 3 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} b} \]
1/24*(3*(35*b^4*c - 15*a*b^3*d + 3*a^2*b^2*e + a^3*b*f)*x^6 - 8*a^3*b*c + (175*a*b^3*c - 75*a^2*b^2*d + 15*a^3*b*e - 3*a^4*f)*x^4 + 8*(7*a^2*b^2*c - 3*a^3*b*d)*x^2)/(a^4*b^3*x^7 + 2*a^5*b^2*x^5 + a^6*b*x^3) + 1/8*(35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4*b)
Time = 0.32 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {{\left (35 \, b^{3} c - 15 \, a b^{2} d + 3 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} b} + \frac {11 \, b^{4} c x^{3} - 7 \, a b^{3} d x^{3} + 3 \, a^{2} b^{2} e x^{3} + a^{3} b f x^{3} + 13 \, a b^{3} c x - 9 \, a^{2} b^{2} d x + 5 \, a^{3} b e x - a^{4} f x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{4} b} + \frac {9 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{4} x^{3}} \]
1/8*(35*b^3*c - 15*a*b^2*d + 3*a^2*b*e + a^3*f)*arctan(b*x/sqrt(a*b))/(sqr t(a*b)*a^4*b) + 1/8*(11*b^4*c*x^3 - 7*a*b^3*d*x^3 + 3*a^2*b^2*e*x^3 + a^3* b*f*x^3 + 13*a*b^3*c*x - 9*a^2*b^2*d*x + 5*a^3*b*e*x - a^4*f*x)/((b*x^2 + a)^2*a^4*b) + 1/3*(9*b*c*x^2 - 3*a*d*x^2 - a*c)/(a^4*x^3)
Time = 5.72 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (f\,a^3+3\,e\,a^2\,b-15\,d\,a\,b^2+35\,c\,b^3\right )}{8\,a^{9/2}\,b^{3/2}}-\frac {\frac {c}{3\,a}-\frac {x^6\,\left (f\,a^3+3\,e\,a^2\,b-15\,d\,a\,b^2+35\,c\,b^3\right )}{8\,a^4}+\frac {x^2\,\left (3\,a\,d-7\,b\,c\right )}{3\,a^2}-\frac {x^4\,\left (-3\,f\,a^3+15\,e\,a^2\,b-75\,d\,a\,b^2+175\,c\,b^3\right )}{24\,a^3\,b}}{a^2\,x^3+2\,a\,b\,x^5+b^2\,x^7} \]
(atan((b^(1/2)*x)/a^(1/2))*(35*b^3*c + a^3*f - 15*a*b^2*d + 3*a^2*b*e))/(8 *a^(9/2)*b^(3/2)) - (c/(3*a) - (x^6*(35*b^3*c + a^3*f - 15*a*b^2*d + 3*a^2 *b*e))/(8*a^4) + (x^2*(3*a*d - 7*b*c))/(3*a^2) - (x^4*(175*b^3*c - 3*a^3*f - 75*a*b^2*d + 15*a^2*b*e))/(24*a^3*b))/(a^2*x^3 + b^2*x^7 + 2*a*b*x^5)