Integrand size = 30, antiderivative size = 110 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=-\frac {A \sqrt {a+b x^2}}{3 a x^3}-\frac {B \sqrt {a+b x^2}}{2 a x^2}+\frac {(2 A b-3 a C) \sqrt {a+b x^2}}{3 a^2 x}+\frac {(b B-2 a D) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Output:
-1/3*A*(b*x^2+a)^(1/2)/a/x^3-1/2*B*(b*x^2+a)^(1/2)/a/x^2+1/3*(2*A*b-3*C*a) *(b*x^2+a)^(1/2)/a^2/x+1/2*(B*b-2*D*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^ (3/2)
Time = 0.61 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-2 a A-3 a B x+4 A b x^2-6 a C x^2\right )}{6 a^2 x^3}+\frac {(-b B+2 a D) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(x^4*Sqrt[a + b*x^2]),x]
Output:
(Sqrt[a + b*x^2]*(-2*a*A - 3*a*B*x + 4*A*b*x^2 - 6*a*C*x^2))/(6*a^2*x^3) + ((-(b*B) + 2*a*D)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(3/2)
Time = 0.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2338, 25, 2338, 27, 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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle -\frac {\int -\frac {3 a D x^2-(2 A b-3 a C) x+3 a B}{x^3 \sqrt {b x^2+a}}dx}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 a D x^2-(2 A b-3 a C) x+3 a B}{x^3 \sqrt {b x^2+a}}dx}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \frac {-\frac {\int \frac {a (2 (2 A b-3 a C)+3 (b B-2 a D) x)}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {3 B \sqrt {a+b x^2}}{2 x^2}}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {1}{2} \int \frac {2 (2 A b-3 a C)+3 (b B-2 a D) x}{x^2 \sqrt {b x^2+a}}dx-\frac {3 B \sqrt {a+b x^2}}{2 x^2}}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} (2 A b-3 a C)}{a x}-3 (b B-2 a D) \int \frac {1}{x \sqrt {b x^2+a}}dx\right )-\frac {3 B \sqrt {a+b x^2}}{2 x^2}}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} (2 A b-3 a C)}{a x}-\frac {3}{2} (b B-2 a D) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2\right )-\frac {3 B \sqrt {a+b x^2}}{2 x^2}}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} (2 A b-3 a C)}{a x}-\frac {3 (b B-2 a D) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}\right )-\frac {3 B \sqrt {a+b x^2}}{2 x^2}}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {2 \sqrt {a+b x^2} (2 A b-3 a C)}{a x}+\frac {3 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (b B-2 a D)}{\sqrt {a}}\right )-\frac {3 B \sqrt {a+b x^2}}{2 x^2}}{3 a}-\frac {A \sqrt {a+b x^2}}{3 a x^3}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(x^4*Sqrt[a + b*x^2]),x]
Output:
-1/3*(A*Sqrt[a + b*x^2])/(a*x^3) + ((-3*B*Sqrt[a + b*x^2])/(2*x^2) + ((2*( 2*A*b - 3*a*C)*Sqrt[a + b*x^2])/(a*x) + (3*(b*B - 2*a*D)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/2)/(3*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.54 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.24
method | result | size |
default | \(A \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )+B \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )-\frac {C \sqrt {b \,x^{2}+a}}{a x}-\frac {D \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}\) | \(136\) |
Input:
int((D*x^3+C*x^2+B*x+A)/x^4/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
A*(-1/3*(b*x^2+a)^(1/2)/a/x^3+2/3*b/a^2*(b*x^2+a)^(1/2)/x)+B*(-1/2*(b*x^2+ a)^(1/2)/a/x^2+1/2*b/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x))-C*(b*x ^2+a)^(1/2)/a/x-D/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.57 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=\left [-\frac {3 \, {\left (2 \, D a - B b\right )} \sqrt {a} x^{3} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, B a x + 2 \, {\left (3 \, C a - 2 \, A b\right )} x^{2} + 2 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, a^{2} x^{3}}, \frac {3 \, {\left (2 \, D a - B b\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (3 \, B a x + 2 \, {\left (3 \, C a - 2 \, A b\right )} x^{2} + 2 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, a^{2} x^{3}}\right ] \] Input:
integrate((D*x^3+C*x^2+B*x+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
[-1/12*(3*(2*D*a - B*b)*sqrt(a)*x^3*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a ) + 2*a)/x^2) + 2*(3*B*a*x + 2*(3*C*a - 2*A*b)*x^2 + 2*A*a)*sqrt(b*x^2 + a ))/(a^2*x^3), 1/6*(3*(2*D*a - B*b)*sqrt(-a)*x^3*arctan(sqrt(b*x^2 + a)*sqr t(-a)/a) - (3*B*a*x + 2*(3*C*a - 2*A*b)*x^2 + 2*A*a)*sqrt(b*x^2 + a))/(a^2 *x^3)]
Time = 2.75 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=- \frac {A \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} + \frac {B b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {C \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{a} - \frac {D \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/x**4/(b*x**2+a)**(1/2),x)
Output:
-A*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*a*x**2) + 2*A*b**(3/2)*sqrt(a/(b*x**2) + 1)/(3*a**2) - B*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*a*x) + B*b*asinh(sqrt(a) /(sqrt(b)*x))/(2*a**(3/2)) - C*sqrt(b)*sqrt(a/(b*x**2) + 1)/a - D*asinh(sq rt(a)/(sqrt(b)*x))/sqrt(a)
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=-\frac {D \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} + \frac {B b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} C}{a x} + \frac {2 \, \sqrt {b x^{2} + a} A b}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a} B}{2 \, a x^{2}} - \frac {\sqrt {b x^{2} + a} A}{3 \, a x^{3}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
-D*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 1/2*B*b*arcsinh(a/(sqrt(a*b)*ab s(x)))/a^(3/2) - sqrt(b*x^2 + a)*C/(a*x) + 2/3*sqrt(b*x^2 + a)*A*b/(a^2*x) - 1/2*sqrt(b*x^2 + a)*B/(a*x^2) - 1/3*sqrt(b*x^2 + a)*A/(a*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (91) = 182\).
Time = 0.14 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.00 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=\frac {{\left (2 \, D a - B b\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B b + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a \sqrt {b} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{2} \sqrt {b} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b + 6 \, C a^{3} \sqrt {b} - 4 \, A a^{2} b^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/x^4/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
(2*D*a - B*b)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B*b + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a*sqrt(b) - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^2*sqrt(b) + 1 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2) - 3*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^2*b + 6*C*a^3*sqrt(b) - 4*A*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x ^2 + a))^2 - a)^3*a)
Time = 3.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {-a}}\right )\,D}{\sqrt {-a}}-\frac {B\,\sqrt {b\,x^2+a}}{2\,a\,x^2}-\frac {C\,\sqrt {b\,x^2+a}}{a\,x}+\frac {B\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {A\,\sqrt {b\,x^2+a}\,\left (a-2\,b\,x^2\right )}{3\,a^2\,x^3} \] Input:
int((A + B*x + C*x^2 + x^3*D)/(x^4*(a + b*x^2)^(1/2)),x)
Output:
(atan((a + b*x^2)^(1/2)/(-a)^(1/2))*D)/(-a)^(1/2) - (B*(a + b*x^2)^(1/2))/ (2*a*x^2) - (C*(a + b*x^2)^(1/2))/(a*x) + (B*b*atanh((a + b*x^2)^(1/2)/a^( 1/2)))/(2*a^(3/2)) - (A*(a + b*x^2)^(1/2)*(a - 2*b*x^2))/(3*a^2*x^3)
Time = 0.16 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.90 \[ \int \frac {A+B x+C x^2+D x^3}{x^4 \sqrt {a+b x^2}} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a^{2}+4 \sqrt {b \,x^{2}+a}\, a b \,x^{2}-3 \sqrt {b \,x^{2}+a}\, a b x -6 \sqrt {b \,x^{2}+a}\, a c \,x^{2}+6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a d \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{3}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a d \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} x^{3}-4 \sqrt {b}\, a b \,x^{3}+2 \sqrt {b}\, a c \,x^{3}}{6 a^{2} x^{3}} \] Input:
int((D*x^3+C*x^2+B*x+A)/x^4/(b*x^2+a)^(1/2),x)
Output:
( - 2*sqrt(a + b*x**2)*a**2 + 4*sqrt(a + b*x**2)*a*b*x**2 - 3*sqrt(a + b*x **2)*a*b*x - 6*sqrt(a + b*x**2)*a*c*x**2 + 6*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*d*x**3 - 3*sqrt(a)*log((sqrt(a + b*x**2 ) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**3 - 6*sqrt(a)*log((sqrt(a + b*x* *2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*d*x**3 + 3*sqrt(a)*log((sqrt(a + b*x **2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*x**3 - 4*sqrt(b)*a*b*x**3 + 2*sq rt(b)*a*c*x**3)/(6*a**2*x**3)