Integrand size = 24, antiderivative size = 90 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\left (\frac {A}{a}-\frac {b B-a C}{b^2}\right ) x}{\sqrt {a+b x^2}}+\frac {C x \sqrt {a+b x^2}}{2 b^2}+\frac {(2 b B-3 a C) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \] Output:
(A/a-(B*b-C*a)/b^2)*x/(b*x^2+a)^(1/2)+1/2*C*x*(b*x^2+a)^(1/2)/b^2+1/2*(2*B *b-3*C*a)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (2 A b^2-2 a b B+3 a^2 C+a b C x^2\right )}{2 a b^2 \sqrt {a+b x^2}}+\frac {(-2 b B+3 a C) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{5/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(a + b*x^2)^(3/2),x]
Output:
(x*(2*A*b^2 - 2*a*b*B + 3*a^2*C + a*b*C*x^2))/(2*a*b^2*Sqrt[a + b*x^2]) + ((-2*b*B + 3*a*C)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(2*b^(5/2))
Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1471, 25, 27, 299, 224, 219}
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.
\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle \frac {x \left (A b^2-a (b B-a C)\right )}{a b^2 \sqrt {a+b x^2}}-\frac {\int -\frac {a \left (b C x^2+b B-a C\right )}{b^2 \sqrt {b x^2+a}}dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a \left (b C x^2+b B-a C\right )}{b^2 \sqrt {b x^2+a}}dx}{a}+\frac {x \left (A b^2-a (b B-a C)\right )}{a b^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {b C x^2+b B-a C}{\sqrt {b x^2+a}}dx}{b^2}+\frac {x \left (A b^2-a (b B-a C)\right )}{a b^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\frac {1}{2} (2 b B-3 a C) \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} C x \sqrt {a+b x^2}}{b^2}+\frac {x \left (A b^2-a (b B-a C)\right )}{a b^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {1}{2} (2 b B-3 a C) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} C x \sqrt {a+b x^2}}{b^2}+\frac {x \left (A b^2-a (b B-a C)\right )}{a b^2 \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x \left (A b^2-a (b B-a C)\right )}{a b^2 \sqrt {a+b x^2}}+\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 b B-3 a C)}{2 \sqrt {b}}+\frac {1}{2} C x \sqrt {a+b x^2}}{b^2}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(a + b*x^2)^(3/2),x]
Output:
((A*b^2 - a*(b*B - a*C))*x)/(a*b^2*Sqrt[a + b*x^2]) + ((C*x*Sqrt[a + b*x^2 ])/2 + ((2*b*B - 3*a*C)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])) /b^2
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Time = 0.57 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {a \sqrt {b \,x^{2}+a}\, \left (B b -\frac {3 C a}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+\left (\frac {3 a^{2} C}{2}-\left (-\frac {C \,x^{2}}{2}+B \right ) b a +b^{2} A \right ) x \sqrt {b}}{\sqrt {b \,x^{2}+a}\, b^{\frac {5}{2}} a}\) | \(83\) |
risch | \(\frac {C x \sqrt {b \,x^{2}+a}}{2 b^{2}}+\frac {b \left (2 B b -3 C a \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )-\frac {a C x}{\sqrt {b \,x^{2}+a}}+\frac {2 b^{2} A x}{a \sqrt {b \,x^{2}+a}}}{2 b^{2}}\) | \(104\) |
default | \(\frac {A x}{a \sqrt {b \,x^{2}+a}}+C \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+B \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(117\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/(b*x^2+a)^(1/2)*(a*(b*x^2+a)^(1/2)*(B*b-3/2*C*a)*arctanh((b*x^2+a)^(1/2) /x/b^(1/2))+(3/2*a^2*C-(-1/2*C*x^2+B)*b*a+b^2*A)*x*b^(1/2))/b^(5/2)/a
Time = 0.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.77 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (3 \, C a^{3} - 2 \, B a^{2} b + {\left (3 \, C a^{2} b - 2 \, B a b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (C a b^{2} x^{3} + {\left (3 \, C a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac {{\left (3 \, C a^{3} - 2 \, B a^{2} b + {\left (3 \, C a^{2} b - 2 \, B a b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (C a b^{2} x^{3} + {\left (3 \, C a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
[-1/4*((3*C*a^3 - 2*B*a^2*b + (3*C*a^2*b - 2*B*a*b^2)*x^2)*sqrt(b)*log(-2* b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(C*a*b^2*x^3 + (3*C*a^2*b - 2 *B*a*b^2 + 2*A*b^3)*x)*sqrt(b*x^2 + a))/(a*b^4*x^2 + a^2*b^3), 1/2*((3*C*a ^3 - 2*B*a^2*b + (3*C*a^2*b - 2*B*a*b^2)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/s qrt(b*x^2 + a)) + (C*a*b^2*x^3 + (3*C*a^2*b - 2*B*a*b^2 + 2*A*b^3)*x)*sqrt (b*x^2 + a))/(a*b^4*x^2 + a^2*b^3)]
Time = 3.78 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} + B \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(3/2),x)
Output:
A*x/(a**(3/2)*sqrt(1 + b*x**2/a)) + B*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) + C*(3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x** 2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(5/2)) + x**3/(2*sqrt(a)*b*sqrt (1 + b*x**2/a)))
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {C x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {A x}{\sqrt {b x^{2} + a} a} + \frac {3 \, C a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {B x}{\sqrt {b x^{2} + a} b} - \frac {3 \, C a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
1/2*C*x^3/(sqrt(b*x^2 + a)*b) + A*x/(sqrt(b*x^2 + a)*a) + 3/2*C*a*x/(sqrt( b*x^2 + a)*b^2) - B*x/(sqrt(b*x^2 + a)*b) - 3/2*C*a*arcsinh(b*x/sqrt(a*b)) /b^(5/2) + B*arcsinh(b*x/sqrt(a*b))/b^(3/2)
Time = 0.13 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {C x^{2}}{b} + \frac {3 \, C a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3}}{a b^{3}}\right )} x}{2 \, \sqrt {b x^{2} + a}} + \frac {{\left (3 \, C a - 2 \, B b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
1/2*(C*x^2/b + (3*C*a^2*b - 2*B*a*b^2 + 2*A*b^3)/(a*b^3))*x/sqrt(b*x^2 + a ) + 1/2*(3*C*a - 2*B*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(a + b*x^2)^(3/2),x)
Output:
int((A + B*x^2 + C*x^4)/(a + b*x^2)^(3/2), x)
Time = 0.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.96 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {12 \sqrt {b \,x^{2}+a}\, a b c x +4 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{3}-12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} c +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2}-12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,x^{2}+8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} x^{2}+9 \sqrt {b}\, a^{2} c +9 \sqrt {b}\, a b c \,x^{2}}{8 b^{3} \left (b \,x^{2}+a \right )} \] Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)^(3/2),x)
Output:
(12*sqrt(a + b*x**2)*a*b*c*x + 4*sqrt(a + b*x**2)*b**2*c*x**3 - 12*sqrt(b) *log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2*c + 8*sqrt(b)*log((sqrt( a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2 - 12*sqrt(b)*log((sqrt(a + b*x**2 ) + sqrt(b)*x)/sqrt(a))*a*b*c*x**2 + 8*sqrt(b)*log((sqrt(a + b*x**2) + sqr t(b)*x)/sqrt(a))*b**3*x**2 + 9*sqrt(b)*a**2*c + 9*sqrt(b)*a*b*c*x**2)/(8*b **3*(a + b*x**2))