Integrand size = 18, antiderivative size = 73 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {d x}{b^2}+\frac {a (c+d x)}{2 b^2 \left (a+b x^2\right )}-\frac {3 \sqrt {a} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \] Output:
d*x/b^2+1/2*a*(d*x+c)/b^2/(b*x^2+a)-3/2*a^(1/2)*d*arctan(b^(1/2)*x/a^(1/2) )/b^(5/2)+1/2*c*ln(b*x^2+a)/b^2
Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {d x}{b^2}+\frac {a c+a d x}{2 b^2 \left (a+b x^2\right )}-\frac {3 \sqrt {a} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{5/2}}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \] Input:
Integrate[(x^3*(c + d*x))/(a + b*x^2)^2,x]
Output:
(d*x)/b^2 + (a*c + a*d*x)/(2*b^2*(a + b*x^2)) - (3*Sqrt[a]*d*ArcTan[(Sqrt[ b]*x)/Sqrt[a]])/(2*b^(5/2)) + (c*Log[a + b*x^2])/(2*b^2)
Time = 0.45 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {530, 2341, 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.
\(\displaystyle \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 530 |
\(\displaystyle \frac {a (c+d x)}{2 b^2 \left (a+b x^2\right )}-\frac {\int \frac {\frac {d a^2}{b^2}-\frac {2 d x^2 a}{b}-\frac {2 c x a}{b}}{b x^2+a}dx}{2 a}\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \frac {a (c+d x)}{2 b^2 \left (a+b x^2\right )}-\frac {\int \left (\frac {3 a^2 d-2 a b c x}{b^2 \left (b x^2+a\right )}-\frac {2 a d}{b^2}\right )dx}{2 a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a (c+d x)}{2 b^2 \left (a+b x^2\right )}-\frac {\frac {3 a^{3/2} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a c \log \left (a+b x^2\right )}{b^2}-\frac {2 a d x}{b^2}}{2 a}\) |
Input:
Int[(x^3*(c + d*x))/(a + b*x^2)^2,x]
Output:
(a*(c + d*x))/(2*b^2*(a + b*x^2)) - ((-2*a*d*x)/b^2 + (3*a^(3/2)*d*ArcTan[ (Sqrt[b]*x)/Sqrt[a]])/b^(5/2) - (a*c*Log[a + b*x^2])/b^2)/(2*a)
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[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[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {d x}{b^{2}}-\frac {\frac {-\frac {1}{2} a d x -\frac {1}{2} a c}{b \,x^{2}+a}-\frac {c \ln \left (b \,x^{2}+a \right )}{2}+\frac {3 a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{b^{2}}\) | \(63\) |
risch | \(\frac {d x}{b^{2}}+\frac {\frac {1}{2} a d x +\frac {1}{2} a c}{b^{2} \left (b \,x^{2}+a \right )}+\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) \sqrt {-a b}\, d}{4 b^{3}}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) c}{2 b^{2}}-\frac {3 \ln \left (\sqrt {-a b}\, x -a \right ) \sqrt {-a b}\, d}{4 b^{3}}+\frac {\ln \left (\sqrt {-a b}\, x -a \right ) c}{2 b^{2}}\) | \(121\) |
Input:
int(x^3*(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
d/b^2*x-1/b^2*((-1/2*a*d*x-1/2*a*c)/(b*x^2+a)-1/2*c*ln(b*x^2+a)+3/2*a*d/(a *b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.63 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, b d x^{3} + 6 \, a d x + 3 \, {\left (b d x^{2} + a d\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, a c + 2 \, {\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, \frac {2 \, b d x^{3} + 3 \, a d x - 3 \, {\left (b d x^{2} + a d\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + a c + {\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}}\right ] \] Input:
integrate(x^3*(d*x+c)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/4*(4*b*d*x^3 + 6*a*d*x + 3*(b*d*x^2 + a*d)*sqrt(-a/b)*log((b*x^2 - 2*b* x*sqrt(-a/b) - a)/(b*x^2 + a)) + 2*a*c + 2*(b*c*x^2 + a*c)*log(b*x^2 + a)) /(b^3*x^2 + a*b^2), 1/2*(2*b*d*x^3 + 3*a*d*x - 3*(b*d*x^2 + a*d)*sqrt(a/b) *arctan(b*x*sqrt(a/b)/a) + a*c + (b*c*x^2 + a*c)*log(b*x^2 + a))/(b^3*x^2 + a*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (68) = 136\).
Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.22 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=\left (\frac {c}{2 b^{2}} - \frac {3 d \sqrt {- a b^{5}}}{4 b^{5}}\right ) \log {\left (x + \frac {- 4 b^{2} \left (\frac {c}{2 b^{2}} - \frac {3 d \sqrt {- a b^{5}}}{4 b^{5}}\right ) + 2 c}{3 d} \right )} + \left (\frac {c}{2 b^{2}} + \frac {3 d \sqrt {- a b^{5}}}{4 b^{5}}\right ) \log {\left (x + \frac {- 4 b^{2} \left (\frac {c}{2 b^{2}} + \frac {3 d \sqrt {- a b^{5}}}{4 b^{5}}\right ) + 2 c}{3 d} \right )} + \frac {a c + a d x}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {d x}{b^{2}} \] Input:
integrate(x**3*(d*x+c)/(b*x**2+a)**2,x)
Output:
(c/(2*b**2) - 3*d*sqrt(-a*b**5)/(4*b**5))*log(x + (-4*b**2*(c/(2*b**2) - 3 *d*sqrt(-a*b**5)/(4*b**5)) + 2*c)/(3*d)) + (c/(2*b**2) + 3*d*sqrt(-a*b**5) /(4*b**5))*log(x + (-4*b**2*(c/(2*b**2) + 3*d*sqrt(-a*b**5)/(4*b**5)) + 2* c)/(3*d)) + (a*c + a*d*x)/(2*a*b**2 + 2*b**3*x**2) + d*x/b**2
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {3 \, a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {a d x + a c}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {d x}{b^{2}} + \frac {c \log \left (b x^{2} + a\right )}{2 \, b^{2}} \] Input:
integrate(x^3*(d*x+c)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
-3/2*a*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/2*(a*d*x + a*c)/(b^3*x^ 2 + a*b^2) + d*x/b^2 + 1/2*c*log(b*x^2 + a)/b^2
Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {3 \, a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {d x}{b^{2}} + \frac {c \log \left (b x^{2} + a\right )}{2 \, b^{2}} + \frac {a d x + a c}{2 \, {\left (b x^{2} + a\right )} b^{2}} \] Input:
integrate(x^3*(d*x+c)/(b*x^2+a)^2,x, algorithm="giac")
Output:
-3/2*a*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) + d*x/b^2 + 1/2*c*log(b*x^2 + a)/b^2 + 1/2*(a*d*x + a*c)/((b*x^2 + a)*b^2)
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {a\,c}{2}+\frac {a\,d\,x}{2}}{b^3\,x^2+a\,b^2}+\frac {c\,\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {d\,x}{b^2}-\frac {3\,\sqrt {a}\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{5/2}} \] Input:
int((x^3*(c + d*x))/(a + b*x^2)^2,x)
Output:
((a*c)/2 + (a*d*x)/2)/(a*b^2 + b^3*x^2) + (c*log(a + b*x^2))/(2*b^2) + (d* x)/b^2 - (3*a^(1/2)*d*atan((b^(1/2)*x)/a^(1/2)))/(2*b^(5/2))
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int \frac {x^3 (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {-3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a d -3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b d \,x^{2}+\mathrm {log}\left (b \,x^{2}+a \right ) a b c +\mathrm {log}\left (b \,x^{2}+a \right ) b^{2} c \,x^{2}+3 a b d x -b^{2} c \,x^{2}+2 b^{2} d \,x^{3}}{2 b^{3} \left (b \,x^{2}+a \right )} \] Input:
int(x^3*(d*x+c)/(b*x^2+a)^2,x)
Output:
( - 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*d - 3*sqrt(b)*sqrt(a )*atan((b*x)/(sqrt(b)*sqrt(a)))*b*d*x**2 + log(a + b*x**2)*a*b*c + log(a + b*x**2)*b**2*c*x**2 + 3*a*b*d*x - b**2*c*x**2 + 2*b**2*d*x**3)/(2*b**3*(a + b*x**2))