Integrand size = 30, antiderivative size = 88 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {a (b B-a D)-b (A b-a C) x}{a^2 b \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^2 x}-\frac {B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \] Output:
(a*(B*b-D*a)-b*(A*b-C*a)*x)/a^2/b/(b*x^2+a)^(1/2)-A*(b*x^2+a)^(1/2)/a^2/x- B*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a A b+a b B x-a^2 D x-2 A b^2 x^2+a b C x^2}{a^2 b x \sqrt {a+b x^2}}-\frac {B \log (x)}{a^{3/2}}+\frac {B \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{a^{3/2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(x^2*(a + b*x^2)^(3/2)),x]
Output:
(-(a*A*b) + a*b*B*x - a^2*D*x - 2*A*b^2*x^2 + a*b*C*x^2)/(a^2*b*x*Sqrt[a + b*x^2]) - (B*Log[x])/a^(3/2) + (B*Log[-Sqrt[a] + Sqrt[a + b*x^2]])/a^(3/2 )
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2336, 25, 534, 243, 73, 221}
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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle \frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{a b \sqrt {a+b x^2}}-\frac {\int -\frac {A+B x}{x^2 \sqrt {b x^2+a}}dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {A+B x}{x^2 \sqrt {b x^2+a}}dx}{a}+\frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {B \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {A \sqrt {a+b x^2}}{a x}}{a}+\frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} B \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {A \sqrt {a+b x^2}}{a x}}{a}+\frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {B \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {A \sqrt {a+b x^2}}{a x}}{a}+\frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{a b \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {A \sqrt {a+b x^2}}{a x}-\frac {B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{a}+\frac {-b x \left (\frac {A b}{a}-C\right )-a D+b B}{a b \sqrt {a+b x^2}}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(x^2*(a + b*x^2)^(3/2)),x]
Output:
(b*B - a*D - b*((A*b)/a - C)*x)/(a*b*Sqrt[a + b*x^2]) + (-((A*Sqrt[a + b*x ^2])/(a*x)) - (B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/a
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
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 = 0.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.27
method | result | size |
default | \(\frac {C x}{a \sqrt {b \,x^{2}+a}}+A \left (-\frac {1}{a x \sqrt {b \,x^{2}+a}}-\frac {2 b x}{a^{2} \sqrt {b \,x^{2}+a}}\right )+B \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )-\frac {D}{b \sqrt {b \,x^{2}+a}}\) | \(112\) |
Input:
int((D*x^3+C*x^2+B*x+A)/x^2/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
C*x/a/(b*x^2+a)^(1/2)+A*(-1/a/x/(b*x^2+a)^(1/2)-2*b/a^2*x/(b*x^2+a)^(1/2)) +B*(1/a/(b*x^2+a)^(1/2)-1/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))-D /b/(b*x^2+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.50 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (B b^{2} x^{3} + B a b x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (A a b - {\left (C a b - 2 \, A b^{2}\right )} x^{2} + {\left (D a^{2} - B a b\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{2} x^{3} + a^{3} b x\right )}}, \frac {{\left (B b^{2} x^{3} + B a b x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (A a b - {\left (C a b - 2 \, A b^{2}\right )} x^{2} + {\left (D a^{2} - B a b\right )} x\right )} \sqrt {b x^{2} + a}}{a^{2} b^{2} x^{3} + a^{3} b x}\right ] \] Input:
integrate((D*x^3+C*x^2+B*x+A)/x^2/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((B*b^2*x^3 + B*a*b*x)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a ) + 2*a)/x^2) - 2*(A*a*b - (C*a*b - 2*A*b^2)*x^2 + (D*a^2 - B*a*b)*x)*sqrt (b*x^2 + a))/(a^2*b^2*x^3 + a^3*b*x), ((B*b^2*x^3 + B*a*b*x)*sqrt(-a)*arct an(sqrt(b*x^2 + a)*sqrt(-a)/a) - (A*a*b - (C*a*b - 2*A*b^2)*x^2 + (D*a^2 - B*a*b)*x)*sqrt(b*x^2 + a))/(a^2*b^2*x^3 + a^3*b*x)]
Time = 8.50 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.20 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) + B \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{2}}{a}}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{3} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} + \frac {a^{2} b x^{2} \log {\left (\frac {b x^{2}}{a} \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}} - \frac {2 a^{2} b x^{2} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )}}{2 a^{\frac {9}{2}} + 2 a^{\frac {7}{2}} b x^{2}}\right ) + \frac {C x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + D \left (\begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((D*x**3+C*x**2+B*x+A)/x**2/(b*x**2+a)**(3/2),x)
Output:
A*(-1/(a*sqrt(b)*x**2*sqrt(a/(b*x**2) + 1)) - 2*sqrt(b)/(a**2*sqrt(a/(b*x* *2) + 1))) + B*(2*a**3*sqrt(1 + b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**3*log(b*x**2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**3*log(sqrt(1 + b*x**2/a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) + a**2*b*x**2*log(b*x** 2/a)/(2*a**(9/2) + 2*a**(7/2)*b*x**2) - 2*a**2*b*x**2*log(sqrt(1 + b*x**2/ a) + 1)/(2*a**(9/2) + 2*a**(7/2)*b*x**2)) + C*x/(a**(3/2)*sqrt(1 + b*x**2/ a)) + D*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True))
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {C x}{\sqrt {b x^{2} + a} a} - \frac {2 \, A b x}{\sqrt {b x^{2} + a} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {3}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a} - \frac {D}{\sqrt {b x^{2} + a} b} - \frac {A}{\sqrt {b x^{2} + a} a x} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/x^2/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
C*x/(sqrt(b*x^2 + a)*a) - 2*A*b*x/(sqrt(b*x^2 + a)*a^2) - B*arcsinh(a/(sqr t(a*b)*abs(x)))/a^(3/2) + B/(sqrt(b*x^2 + a)*a) - D/(sqrt(b*x^2 + a)*b) - A/(sqrt(b*x^2 + a)*a*x)
Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.42 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (C a^{2} b - A a b^{2}\right )} x}{a^{3} b} - \frac {D a^{3} - B a^{2} b}{a^{3} b}}{\sqrt {b x^{2} + a}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/x^2/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
((C*a^2*b - A*a*b^2)*x/(a^3*b) - (D*a^3 - B*a^2*b)/(a^3*b))/sqrt(b*x^2 + a ) + 2*B*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 2*A *sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a)
Time = 2.60 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {B}{a\,\sqrt {b\,x^2+a}}-\frac {\sqrt {b\,x^2+a}\,\left (\frac {A}{a}+\frac {2\,A\,b\,x^2}{a^2}\right )}{b\,x^3+a\,x}-\frac {D}{b\,\sqrt {b\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {C\,x}{a\,\sqrt {b\,x^2+a}} \] Input:
int((A + B*x + C*x^2 + x^3*D)/(x^2*(a + b*x^2)^(3/2)),x)
Output:
B/(a*(a + b*x^2)^(1/2)) - ((a + b*x^2)^(1/2)*(A/a + (2*A*b*x^2)/a^2))/(a*x + b*x^3) - D/(b*(a + b*x^2)^(1/2)) - (B*atanh((a + b*x^2)^(1/2)/a^(1/2))) /a^(3/2) + (C*x)/(a*(a + b*x^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.91 \[ \int \frac {A+B x+C x^2+D x^3}{x^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a^{2} b -\sqrt {b \,x^{2}+a}\, a^{2} d x -2 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{2}+\sqrt {b \,x^{2}+a}\, a \,b^{2} x +\sqrt {b \,x^{2}+a}\, a b c \,x^{2}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x +\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{3}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x -\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{3}-2 \sqrt {b}\, a^{2} b x +\sqrt {b}\, a^{2} c x -2 \sqrt {b}\, a \,b^{2} x^{3}+\sqrt {b}\, a b c \,x^{3}}{a^{2} b x \left (b \,x^{2}+a \right )} \] Input:
int((D*x^3+C*x^2+B*x+A)/x^2/(b*x^2+a)^(3/2),x)
Output:
( - sqrt(a + b*x**2)*a**2*b - sqrt(a + b*x**2)*a**2*d*x - 2*sqrt(a + b*x** 2)*a*b**2*x**2 + sqrt(a + b*x**2)*a*b**2*x + sqrt(a + b*x**2)*a*b*c*x**2 + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*x + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*x**3 - sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*x - s qrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*x**3 - 2 *sqrt(b)*a**2*b*x + sqrt(b)*a**2*c*x - 2*sqrt(b)*a*b**2*x**3 + sqrt(b)*a*b *c*x**3)/(a**2*b*x*(a + b*x**2))