Integrand size = 20, antiderivative size = 117 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{a^4 x}+\frac {b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac {b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}+\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2}} \]
-1/3*A/a^3/x^3+(3*A*b-B*a)/a^4/x+1/4*b*(A*b-B*a)*x/a^3/(b*x^2+a)^2+1/8*b*( 11*A*b-7*B*a)*x/a^4/(b*x^2+a)+5/8*(7*A*b-3*B*a)*arctan(x*b^(1/2)/a^(1/2))* b^(1/2)/a^(9/2)
Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {105 A b^3 x^6+a^2 b x^2 \left (56 A-75 B x^2\right )+5 a b^2 x^4 \left (35 A-9 B x^2\right )-8 a^3 \left (A+3 B x^2\right )}{24 a^4 x^3 \left (a+b x^2\right )^2}+\frac {5 \sqrt {b} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2}} \]
(105*A*b^3*x^6 + a^2*b*x^2*(56*A - 75*B*x^2) + 5*a*b^2*x^4*(35*A - 9*B*x^2 ) - 8*a^3*(A + 3*B*x^2))/(24*a^4*x^3*(a + b*x^2)^2) + (5*Sqrt[b]*(7*A*b - 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(9/2))
Time = 0.38 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {361, 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.
\(\displaystyle \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 361 |
\(\displaystyle \frac {b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}-\frac {1}{4} b \int -\frac {\frac {3 (A b-a B) x^4}{a^3}-\frac {4 (A b-a B) x^2}{a^2 b}+\frac {4 A}{a b}}{x^4 \left (b x^2+a\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{4} b \int \frac {\frac {3 (A b-a B) x^4}{a^3}-\frac {4 (A b-a B) x^2}{a^2 b}+\frac {4 A}{a b}}{x^4 \left (b x^2+a\right )^2}dx+\frac {b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 1582 |
\(\displaystyle \frac {1}{4} b \left (\frac {\int \frac {\frac {b^2 (11 A b-7 a B) x^4}{a}-8 b (2 A b-a B) x^2+8 a A b}{x^4 \left (b x^2+a\right )}dx}{2 a^3 b^2}+\frac {x (11 A b-7 a B)}{2 a^4 \left (a+b x^2\right )}\right )+\frac {b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {1}{4} b \left (\frac {\int \left (-\frac {5 (3 a B-7 A b) b^2}{a \left (b x^2+a\right )}+\frac {8 (a B-3 A b) b}{a x^2}+\frac {8 A b}{x^4}\right )dx}{2 a^3 b^2}+\frac {x (11 A b-7 a B)}{2 a^4 \left (a+b x^2\right )}\right )+\frac {b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}+\frac {1}{4} b \left (\frac {x (11 A b-7 a B)}{2 a^4 \left (a+b x^2\right )}+\frac {\frac {5 b^{3/2} (7 A b-3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {8 b (3 A b-a B)}{a x}-\frac {8 A b}{3 x^3}}{2 a^3 b^2}\right )\) |
(b*(A*b - a*B)*x)/(4*a^3*(a + b*x^2)^2) + (b*(((11*A*b - 7*a*B)*x)/(2*a^4* (a + b*x^2)) + ((-8*A*b)/(3*x^3) + (8*b*(3*A*b - a*B))/(a*x) + (5*b^(3/2)* (7*A*b - 3*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))/(2*a^3*b^2)))/4
3.2.5.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[x^m*(a + b*x^2)^(p + 1)*E xpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)] - ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ILtQ[m/ 2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
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]
Time = 2.54 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {A}{3 a^{3} x^{3}}-\frac {-3 A b +B a}{a^{4} x}+\frac {b \left (\frac {\left (\frac {11}{8} b^{2} A -\frac {7}{8} a b B \right ) x^{3}+\frac {a \left (13 A b -9 B a \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (7 A b -3 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}\) | \(98\) |
risch | \(\frac {\frac {5 b^{2} \left (7 A b -3 B a \right ) x^{6}}{8 a^{4}}+\frac {25 b \left (7 A b -3 B a \right ) x^{4}}{24 a^{3}}+\frac {\left (7 A b -3 B a \right ) x^{2}}{3 a^{2}}-\frac {A}{3 a}}{x^{3} \left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} \textit {\_Z}^{2}+49 A^{2} b^{3}-42 A B a \,b^{2}+9 B^{2} a^{2} b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9}+98 A^{2} b^{3}-84 A B a \,b^{2}+18 B^{2} a^{2} b \right ) x +\left (-7 A \,a^{5} b +3 B \,a^{6}\right ) \textit {\_R} \right )\right )}{16}\) | \(172\) |
-1/3*A/a^3/x^3-(-3*A*b+B*a)/a^4/x+1/a^4*b*(((11/8*b^2*A-7/8*a*b*B)*x^3+1/8 *a*(13*A*b-9*B*a)*x)/(b*x^2+a)^2+5/8*(7*A*b-3*B*a)/(a*b)^(1/2)*arctan(b*x/ (a*b)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.15 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\left [-\frac {30 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 50 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 16 \, A a^{3} + 16 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{48 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}, -\frac {15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{24 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}\right ] \]
[-1/48*(30*(3*B*a*b^2 - 7*A*b^3)*x^6 + 50*(3*B*a^2*b - 7*A*a*b^2)*x^4 + 16 *A*a^3 + 16*(3*B*a^3 - 7*A*a^2*b)*x^2 + 15*((3*B*a*b^2 - 7*A*b^3)*x^7 + 2* (3*B*a^2*b - 7*A*a*b^2)*x^5 + (3*B*a^3 - 7*A*a^2*b)*x^3)*sqrt(-b/a)*log((b *x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^ 6*x^3), -1/24*(15*(3*B*a*b^2 - 7*A*b^3)*x^6 + 25*(3*B*a^2*b - 7*A*a*b^2)*x ^4 + 8*A*a^3 + 8*(3*B*a^3 - 7*A*a^2*b)*x^2 + 15*((3*B*a*b^2 - 7*A*b^3)*x^7 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^5 + (3*B*a^3 - 7*A*a^2*b)*x^3)*sqrt(b/a)*ar ctan(x*sqrt(b/a)))/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3)]
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (109) = 218\).
Time = 0.42 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.93 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {5 \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log {\left (- \frac {5 a^{5} \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} - \frac {5 \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log {\left (\frac {5 a^{5} \sqrt {- \frac {b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} + \frac {- 8 A a^{3} + x^{6} \cdot \left (105 A b^{3} - 45 B a b^{2}\right ) + x^{4} \cdot \left (175 A a b^{2} - 75 B a^{2} b\right ) + x^{2} \cdot \left (56 A a^{2} b - 24 B a^{3}\right )}{24 a^{6} x^{3} + 48 a^{5} b x^{5} + 24 a^{4} b^{2} x^{7}} \]
5*sqrt(-b/a**9)*(-7*A*b + 3*B*a)*log(-5*a**5*sqrt(-b/a**9)*(-7*A*b + 3*B*a )/(-35*A*b**2 + 15*B*a*b) + x)/16 - 5*sqrt(-b/a**9)*(-7*A*b + 3*B*a)*log(5 *a**5*sqrt(-b/a**9)*(-7*A*b + 3*B*a)/(-35*A*b**2 + 15*B*a*b) + x)/16 + (-8 *A*a**3 + x**6*(105*A*b**3 - 45*B*a*b**2) + x**4*(175*A*a*b**2 - 75*B*a**2 *b) + x**2*(56*A*a**2*b - 24*B*a**3))/(24*a**6*x**3 + 48*a**5*b*x**5 + 24* a**4*b**2*x**7)
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}}{24 \, {\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} - \frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} \]
-1/24*(15*(3*B*a*b^2 - 7*A*b^3)*x^6 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^4 + 8*A *a^3 + 8*(3*B*a^3 - 7*A*a^2*b)*x^2)/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3) - 5/8*(3*B*a*b - 7*A*b^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4)
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=-\frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} - \frac {7 \, B a b^{2} x^{3} - 11 \, A b^{3} x^{3} + 9 \, B a^{2} b x - 13 \, A a b^{2} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{4}} - \frac {3 \, B a x^{2} - 9 \, A b x^{2} + A a}{3 \, a^{4} x^{3}} \]
-5/8*(3*B*a*b - 7*A*b^2)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/8*(7*B* a*b^2*x^3 - 11*A*b^3*x^3 + 9*B*a^2*b*x - 13*A*a*b^2*x)/((b*x^2 + a)^2*a^4) - 1/3*(3*B*a*x^2 - 9*A*b*x^2 + A*a)/(a^4*x^3)
Time = 5.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx=\frac {\frac {x^2\,\left (7\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {A}{3\,a}+\frac {5\,b^2\,x^6\,\left (7\,A\,b-3\,B\,a\right )}{8\,a^4}+\frac {25\,b\,x^4\,\left (7\,A\,b-3\,B\,a\right )}{24\,a^3}}{a^2\,x^3+2\,a\,b\,x^5+b^2\,x^7}+\frac {5\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (7\,A\,b-3\,B\,a\right )}{8\,a^{9/2}} \]