Integrand size = 18, antiderivative size = 97 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\frac {c+d x}{4 a \left (a+b x^2\right )^2}+\frac {4 c+3 d x}{8 a^2 \left (a+b x^2\right )}+\frac {3 d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}+\frac {c \log (x)}{a^3}-\frac {c \log \left (a+b x^2\right )}{2 a^3} \] Output:
1/4*(d*x+c)/a/(b*x^2+a)^2+1/8*(3*d*x+4*c)/a^2/(b*x^2+a)+3/8*d*arctan(b^(1/ 2)*x/a^(1/2))/a^(5/2)/b^(1/2)+c*ln(x)/a^3-1/2*c*ln(b*x^2+a)/a^3
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\frac {\frac {a \left (b x^2 (4 c+3 d x)+a (6 c+5 d x)\right )}{\left (a+b x^2\right )^2}+\frac {3 \sqrt {a} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+8 c \log (x)-4 c \log \left (a+b x^2\right )}{8 a^3} \] Input:
Integrate[(c + d*x)/(x*(a + b*x^2)^3),x]
Output:
((a*(b*x^2*(4*c + 3*d*x) + a*(6*c + 5*d*x)))/(a + b*x^2)^2 + (3*Sqrt[a]*d* ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 8*c*Log[x] - 4*c*Log[a + b*x^2])/(8 *a^3)
Time = 0.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {532, 25, 532, 25, 523, 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.
\(\displaystyle \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {c+d x}{4 a \left (a+b x^2\right )^2}-\frac {\int -\frac {4 c+3 d x}{x \left (b x^2+a\right )^2}dx}{4 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {4 c+3 d x}{x \left (b x^2+a\right )^2}dx}{4 a}+\frac {c+d x}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {4 c+3 d x}{2 a \left (a+b x^2\right )}-\frac {\int -\frac {8 c+3 d x}{x \left (b x^2+a\right )}dx}{2 a}}{4 a}+\frac {c+d x}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {8 c+3 d x}{x \left (b x^2+a\right )}dx}{2 a}+\frac {4 c+3 d x}{2 a \left (a+b x^2\right )}}{4 a}+\frac {c+d x}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {\frac {\int \left (\frac {8 c}{a x}+\frac {3 a d-8 b c x}{a \left (b x^2+a\right )}\right )dx}{2 a}+\frac {4 c+3 d x}{2 a \left (a+b x^2\right )}}{4 a}+\frac {c+d x}{4 a \left (a+b x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\frac {3 d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}-\frac {4 c \log \left (a+b x^2\right )}{a}+\frac {8 c \log (x)}{a}}{2 a}+\frac {4 c+3 d x}{2 a \left (a+b x^2\right )}}{4 a}+\frac {c+d x}{4 a \left (a+b x^2\right )^2}\) |
Input:
Int[(c + d*x)/(x*(a + b*x^2)^3),x]
Output:
(c + d*x)/(4*a*(a + b*x^2)^2) + ((4*c + 3*d*x)/(2*a*(a + b*x^2)) + ((3*d*A rcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) + (8*c*Log[x])/a - (4*c*Log[ a + b*x^2])/a)/(2*a))/(4*a)
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {c \ln \left (x \right )}{a^{3}}+\frac {\frac {\frac {3}{8} a b d \,x^{3}+\frac {1}{2} a b c \,x^{2}+\frac {5}{8} a^{2} d x +\frac {3}{4} a^{2} c}{\left (b \,x^{2}+a \right )^{2}}-\frac {c \ln \left (b \,x^{2}+a \right )}{2}+\frac {3 a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}}{a^{3}}\) | \(83\) |
risch | \(\frac {\frac {3 d b \,x^{3}}{8 a^{2}}+\frac {b c \,x^{2}}{2 a^{2}}+\frac {5 d x}{8 a}+\frac {3 c}{4 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \ln \left (\left (-d a b -8 \sqrt {-a b}\, b c \right ) x -\sqrt {-a b}\, a d +8 a b c \right ) \sqrt {-a b}\, d}{16 b \,a^{3}}-\frac {\ln \left (\left (-d a b -8 \sqrt {-a b}\, b c \right ) x -\sqrt {-a b}\, a d +8 a b c \right ) c}{2 a^{3}}-\frac {3 \ln \left (\left (-d a b +8 \sqrt {-a b}\, b c \right ) x +\sqrt {-a b}\, a d +8 a b c \right ) \sqrt {-a b}\, d}{16 b \,a^{3}}-\frac {\ln \left (\left (-d a b +8 \sqrt {-a b}\, b c \right ) x +\sqrt {-a b}\, a d +8 a b c \right ) c}{2 a^{3}}+\frac {c \ln \left (x \right )}{a^{3}}\) | \(233\) |
Input:
int((d*x+c)/x/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
c*ln(x)/a^3+1/a^3*((3/8*a*b*d*x^3+1/2*a*b*c*x^2+5/8*a^2*d*x+3/4*a^2*c)/(b* x^2+a)^2-1/2*c*ln(b*x^2+a)+3/8*a*d/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (81) = 162\).
Time = 0.12 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.73 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\left [\frac {6 \, a b^{2} d x^{3} + 8 \, a b^{2} c x^{2} + 10 \, a^{2} b d x + 12 \, a^{2} b c - 3 \, {\left (b^{2} d x^{4} + 2 \, a b d x^{2} + a^{2} d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 8 \, {\left (b^{3} c x^{4} + 2 \, a b^{2} c x^{2} + a^{2} b c\right )} \log \left (b x^{2} + a\right ) + 16 \, {\left (b^{3} c x^{4} + 2 \, a b^{2} c x^{2} + a^{2} b c\right )} \log \left (x\right )}{16 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}, \frac {3 \, a b^{2} d x^{3} + 4 \, a b^{2} c x^{2} + 5 \, a^{2} b d x + 6 \, a^{2} b c + 3 \, {\left (b^{2} d x^{4} + 2 \, a b d x^{2} + a^{2} d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 4 \, {\left (b^{3} c x^{4} + 2 \, a b^{2} c x^{2} + a^{2} b c\right )} \log \left (b x^{2} + a\right ) + 8 \, {\left (b^{3} c x^{4} + 2 \, a b^{2} c x^{2} + a^{2} b c\right )} \log \left (x\right )}{8 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}}\right ] \] Input:
integrate((d*x+c)/x/(b*x^2+a)^3,x, algorithm="fricas")
Output:
[1/16*(6*a*b^2*d*x^3 + 8*a*b^2*c*x^2 + 10*a^2*b*d*x + 12*a^2*b*c - 3*(b^2* d*x^4 + 2*a*b*d*x^2 + a^2*d)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/( b*x^2 + a)) - 8*(b^3*c*x^4 + 2*a*b^2*c*x^2 + a^2*b*c)*log(b*x^2 + a) + 16* (b^3*c*x^4 + 2*a*b^2*c*x^2 + a^2*b*c)*log(x))/(a^3*b^3*x^4 + 2*a^4*b^2*x^2 + a^5*b), 1/8*(3*a*b^2*d*x^3 + 4*a*b^2*c*x^2 + 5*a^2*b*d*x + 6*a^2*b*c + 3*(b^2*d*x^4 + 2*a*b*d*x^2 + a^2*d)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 4*(b ^3*c*x^4 + 2*a*b^2*c*x^2 + a^2*b*c)*log(b*x^2 + a) + 8*(b^3*c*x^4 + 2*a*b^ 2*c*x^2 + a^2*b*c)*log(x))/(a^3*b^3*x^4 + 2*a^4*b^2*x^2 + a^5*b)]
Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (90) = 180\).
Time = 1.16 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.26 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\left (- \frac {c}{2 a^{3}} - \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right ) \log {\left (x + \frac {- 2048 a^{6} b c \left (- \frac {c}{2 a^{3}} - \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right )^{2} + 48 a^{4} d^{2} \left (- \frac {c}{2 a^{3}} - \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right ) + 1024 a^{3} b c^{2} \left (- \frac {c}{2 a^{3}} - \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right ) - 48 a c d^{2} + 1024 b c^{3}}{9 a d^{3} + 576 b c^{2} d} \right )} + \left (- \frac {c}{2 a^{3}} + \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right ) \log {\left (x + \frac {- 2048 a^{6} b c \left (- \frac {c}{2 a^{3}} + \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right )^{2} + 48 a^{4} d^{2} \left (- \frac {c}{2 a^{3}} + \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right ) + 1024 a^{3} b c^{2} \left (- \frac {c}{2 a^{3}} + \frac {3 d \sqrt {- a^{7} b}}{16 a^{6} b}\right ) - 48 a c d^{2} + 1024 b c^{3}}{9 a d^{3} + 576 b c^{2} d} \right )} + \frac {6 a c + 5 a d x + 4 b c x^{2} + 3 b d x^{3}}{8 a^{4} + 16 a^{3} b x^{2} + 8 a^{2} b^{2} x^{4}} + \frac {c \log {\left (x \right )}}{a^{3}} \] Input:
integrate((d*x+c)/x/(b*x**2+a)**3,x)
Output:
(-c/(2*a**3) - 3*d*sqrt(-a**7*b)/(16*a**6*b))*log(x + (-2048*a**6*b*c*(-c/ (2*a**3) - 3*d*sqrt(-a**7*b)/(16*a**6*b))**2 + 48*a**4*d**2*(-c/(2*a**3) - 3*d*sqrt(-a**7*b)/(16*a**6*b)) + 1024*a**3*b*c**2*(-c/(2*a**3) - 3*d*sqrt (-a**7*b)/(16*a**6*b)) - 48*a*c*d**2 + 1024*b*c**3)/(9*a*d**3 + 576*b*c**2 *d)) + (-c/(2*a**3) + 3*d*sqrt(-a**7*b)/(16*a**6*b))*log(x + (-2048*a**6*b *c*(-c/(2*a**3) + 3*d*sqrt(-a**7*b)/(16*a**6*b))**2 + 48*a**4*d**2*(-c/(2* a**3) + 3*d*sqrt(-a**7*b)/(16*a**6*b)) + 1024*a**3*b*c**2*(-c/(2*a**3) + 3 *d*sqrt(-a**7*b)/(16*a**6*b)) - 48*a*c*d**2 + 1024*b*c**3)/(9*a*d**3 + 576 *b*c**2*d)) + (6*a*c + 5*a*d*x + 4*b*c*x**2 + 3*b*d*x**3)/(8*a**4 + 16*a** 3*b*x**2 + 8*a**2*b**2*x**4) + c*log(x)/a**3
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\frac {3 \, b d x^{3} + 4 \, b c x^{2} + 5 \, a d x + 6 \, a c}{8 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} + \frac {3 \, d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} - \frac {c \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {c \log \left (x\right )}{a^{3}} \] Input:
integrate((d*x+c)/x/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/8*(3*b*d*x^3 + 4*b*c*x^2 + 5*a*d*x + 6*a*c)/(a^2*b^2*x^4 + 2*a^3*b*x^2 + a^4) + 3/8*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/2*c*log(b*x^2 + a) /a^3 + c*log(x)/a^3
Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\frac {3 \, d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} - \frac {c \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {3 \, a b d x^{3} + 4 \, a b c x^{2} + 5 \, a^{2} d x + 6 \, a^{2} c}{8 \, {\left (b x^{2} + a\right )}^{2} a^{3}} \] Input:
integrate((d*x+c)/x/(b*x^2+a)^3,x, algorithm="giac")
Output:
3/8*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/2*c*log(b*x^2 + a)/a^3 + c *log(abs(x))/a^3 + 1/8*(3*a*b*d*x^3 + 4*a*b*c*x^2 + 5*a^2*d*x + 6*a^2*c)/( (b*x^2 + a)^2*a^3)
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.04 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\frac {\frac {3\,c}{4\,a}+\frac {5\,d\,x}{8\,a}+\frac {b\,c\,x^2}{2\,a^2}+\frac {3\,b\,d\,x^3}{8\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {c\,\ln \left (x\right )}{a^3}+\frac {\ln \left (a\,d\,\sqrt {-a^7\,b}-8\,a^4\,b\,c+a^4\,b\,d\,x+8\,b\,c\,x\,\sqrt {-a^7\,b}\right )\,\left (3\,d\,\sqrt {-a^7\,b}-8\,a^3\,b\,c\right )}{16\,a^6\,b}-\frac {\ln \left (a\,d\,\sqrt {-a^7\,b}+8\,a^4\,b\,c-a^4\,b\,d\,x+8\,b\,c\,x\,\sqrt {-a^7\,b}\right )\,\left (3\,d\,\sqrt {-a^7\,b}+8\,a^3\,b\,c\right )}{16\,a^6\,b} \] Input:
int((c + d*x)/(x*(a + b*x^2)^3),x)
Output:
((3*c)/(4*a) + (5*d*x)/(8*a) + (b*c*x^2)/(2*a^2) + (3*b*d*x^3)/(8*a^2))/(a ^2 + b^2*x^4 + 2*a*b*x^2) + (c*log(x))/a^3 + (log(a*d*(-a^7*b)^(1/2) - 8*a ^4*b*c + a^4*b*d*x + 8*b*c*x*(-a^7*b)^(1/2))*(3*d*(-a^7*b)^(1/2) - 8*a^3*b *c))/(16*a^6*b) - (log(a*d*(-a^7*b)^(1/2) + 8*a^4*b*c - a^4*b*d*x + 8*b*c* x*(-a^7*b)^(1/2))*(3*d*(-a^7*b)^(1/2) + 8*a^3*b*c))/(16*a^6*b)
Time = 0.18 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.23 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^3} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} d +6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a b d \,x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} d \,x^{4}-4 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b c -8 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{2} c \,x^{2}-4 \,\mathrm {log}\left (b \,x^{2}+a \right ) b^{3} c \,x^{4}+8 \,\mathrm {log}\left (x \right ) a^{2} b c +16 \,\mathrm {log}\left (x \right ) a \,b^{2} c \,x^{2}+8 \,\mathrm {log}\left (x \right ) b^{3} c \,x^{4}+4 a^{2} b c +5 a^{2} b d x +3 a \,b^{2} d \,x^{3}-2 b^{3} c \,x^{4}}{8 a^{3} b \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((d*x+c)/x/(b*x^2+a)^3,x)
Output:
(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*d + 6*sqrt(b)*sqrt(a )*atan((b*x)/(sqrt(b)*sqrt(a)))*a*b*d*x**2 + 3*sqrt(b)*sqrt(a)*atan((b*x)/ (sqrt(b)*sqrt(a)))*b**2*d*x**4 - 4*log(a + b*x**2)*a**2*b*c - 8*log(a + b* x**2)*a*b**2*c*x**2 - 4*log(a + b*x**2)*b**3*c*x**4 + 8*log(x)*a**2*b*c + 16*log(x)*a*b**2*c*x**2 + 8*log(x)*b**3*c*x**4 + 4*a**2*b*c + 5*a**2*b*d*x + 3*a*b**2*d*x**3 - 2*b**3*c*x**4)/(8*a**3*b*(a**2 + 2*a*b*x**2 + b**2*x* *4))