\(\int \frac {A+B x^2+C x^4}{(a+b x^2)^{5/2}} \, dx\) [119]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (\frac {A}{a}-\frac {b B-a C}{b^2}\right ) x}{3 \left (a+b x^2\right )^{3/2}}+\frac {\left (2 A b^2+a (b B-4 a C)\right ) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Output:

1/3*(A/a-(B*b-C*a)/b^2)*x/(b*x^2+a)^(3/2)+1/3*(2*A*b^2+a*(B*b-4*C*a))*x/a^ 
2/b^2/(b*x^2+a)^(1/2)+C*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {3 a A b^2 x-3 a^3 C x+2 A b^3 x^3+a b^2 B x^3-4 a^2 b C x^3}{3 a^2 b^2 \left (a+b x^2\right )^{3/2}}-\frac {C \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(a + b*x^2)^(5/2),x]
 

Output:

(3*a*A*b^2*x - 3*a^3*C*x + 2*A*b^3*x^3 + a*b^2*B*x^3 - 4*a^2*b*C*x^3)/(3*a 
^2*b^2*(a + b*x^2)^(3/2)) - (C*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(5/2 
)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 108, 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, 298, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {x \left (A b^2-a (b B-a C)\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {3 a C x^2+b \left (2 A+\frac {a (b B-a C)}{b^2}\right )}{b \left (b x^2+a\right )^{3/2}}dx}{3 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 a C x^2+2 A b+a \left (B-\frac {a C}{b}\right )}{b \left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x \left (A b^2-a (b B-a C)\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 a C x^2+2 A b+a \left (B-\frac {a C}{b}\right )}{\left (b x^2+a\right )^{3/2}}dx}{3 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {3 a C \int \frac {1}{\sqrt {b x^2+a}}dx}{b}+\frac {x \left (\frac {2 A b}{a}-\frac {4 a C}{b}+B\right )}{\sqrt {a+b x^2}}}{3 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {3 a C \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b}+\frac {x \left (\frac {2 A b}{a}-\frac {4 a C}{b}+B\right )}{\sqrt {a+b x^2}}}{3 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {x \left (\frac {2 A b}{a}-\frac {4 a C}{b}+B\right )}{\sqrt {a+b x^2}}+\frac {3 a C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}}{3 a b}+\frac {x \left (A b^2-a (b B-a C)\right )}{3 a b^2 \left (a+b x^2\right )^{3/2}}\)

Input:

Int[(A + B*x^2 + C*x^4)/(a + b*x^2)^(5/2),x]
 

Output:

((A*b^2 - a*(b*B - a*C))*x)/(3*a*b^2*(a + b*x^2)^(3/2)) + ((((2*A*b)/a + B 
 - (4*a*C)/b)*x)/Sqrt[a + b*x^2] + (3*a*C*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x 
^2]])/b^(3/2))/(3*a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 224

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]
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 1471

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]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(\frac {a^{2} C \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}+\frac {2 \left (A \,x^{2} b^{3}+\frac {3 a \left (\frac {x^{2} B}{3}+A \right ) b^{2}}{2}-2 C \,a^{2} b \,x^{2}-\frac {3 C \,a^{3}}{2}\right ) b^{\frac {5}{2}} x}{3}}{\left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{\frac {9}{2}} a^{2}}\)	\(97\)
default	\(A \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+C \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+B \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\)	\(150\)

Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/(b*x^2+a)^(3/2)*(a^2*C*arctanh((b*x^2+a)^(1/2)/x/b^(1/2))*b^2*(b*x^2+a)^ 
(3/2)+2/3*(A*x^2*b^3+3/2*a*(1/3*x^2*B+A)*b^2-2*C*a^2*b*x^2-3/2*C*a^3)*b^(5 
/2)*x)/b^(9/2)/a^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.86 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (C a^{2} b^{2} x^{4} + 2 \, C a^{3} b x^{2} + C a^{4}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left ({\left (4 \, C a^{2} b^{2} - B a b^{3} - 2 \, A b^{4}\right )} x^{3} + 3 \, {\left (C a^{3} b - A a b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {3 \, {\left (C a^{2} b^{2} x^{4} + 2 \, C a^{3} b x^{2} + C a^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left ({\left (4 \, C a^{2} b^{2} - B a b^{3} - 2 \, A b^{4}\right )} x^{3} + 3 \, {\left (C a^{3} b - A a b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(C*a^2*b^2*x^4 + 2*C*a^3*b*x^2 + C*a^4)*sqrt(b)*log(-2*b*x^2 - 2*s 
qrt(b*x^2 + a)*sqrt(b)*x - a) - 2*((4*C*a^2*b^2 - B*a*b^3 - 2*A*b^4)*x^3 + 
 3*(C*a^3*b - A*a*b^3)*x)*sqrt(b*x^2 + a))/(a^2*b^5*x^4 + 2*a^3*b^4*x^2 + 
a^4*b^3), -1/3*(3*(C*a^2*b^2*x^4 + 2*C*a^3*b*x^2 + C*a^4)*sqrt(-b)*arctan( 
sqrt(-b)*x/sqrt(b*x^2 + a)) + ((4*C*a^2*b^2 - B*a*b^3 - 2*A*b^4)*x^3 + 3*( 
C*a^3*b - A*a*b^3)*x)*sqrt(b*x^2 + a))/(a^2*b^5*x^4 + 2*a^3*b^4*x^2 + a^4* 
b^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (92) = 184\).

Time = 6.19 (sec) , antiderivative size = 450, normalized size of antiderivative = 4.46 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=A \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {B x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + C \left (\frac {3 a^{\frac {39}{2}} b^{11} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{\frac {37}{2}} b^{12} x^{2} \sqrt {1 + \frac {b x^{2}}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{19} b^{\frac {23}{2}} x}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {4 a^{18} b^{\frac {25}{2}} x^{3}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) \] Input:

integrate((C*x**4+B*x**2+A)/(b*x**2+a)**(5/2),x)
 

Output:

A*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x** 
2/a)) + 2*b*x**3/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 
 + b*x**2/a))) + B*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b*x**2 
*sqrt(1 + b*x**2/a)) + C*(3*a**(39/2)*b**11*sqrt(1 + b*x**2/a)*asinh(sqrt( 
b)*x/sqrt(a))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**( 
29/2)*x**2*sqrt(1 + b*x**2/a)) + 3*a**(37/2)*b**12*x**2*sqrt(1 + b*x**2/a) 
*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a* 
*(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) - 3*a**19*b**(23/2)*x/(3*a**(39 
/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x 
**2/a)) - 4*a**18*b**(25/2)*x**3/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) 
 + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, C x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {2 \, A x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {C x}{3 \, \sqrt {b x^{2} + a} b^{2}} - \frac {B x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {B x}{3 \, \sqrt {b x^{2} + a} a b} + \frac {C \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-1/3*C*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2)) + 2/3 
*A*x/(sqrt(b*x^2 + a)*a^2) + 1/3*A*x/((b*x^2 + a)^(3/2)*a) - 1/3*C*x/(sqrt 
(b*x^2 + a)*b^2) - 1/3*B*x/((b*x^2 + a)^(3/2)*b) + 1/3*B*x/(sqrt(b*x^2 + a 
)*a*b) + C*arcsinh(b*x/sqrt(a*b))/b^(5/2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {x {\left (\frac {{\left (4 \, C a^{2} b^{2} - B a b^{3} - 2 \, A b^{4}\right )} x^{2}}{a^{2} b^{3}} + \frac {3 \, {\left (C a^{3} b - A a b^{3}\right )}}{a^{2} b^{3}}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {C \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {5}{2}}} \] Input:

integrate((C*x^4+B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

-1/3*x*((4*C*a^2*b^2 - B*a*b^3 - 2*A*b^4)*x^2/(a^2*b^3) + 3*(C*a^3*b - A*a 
*b^3)/(a^2*b^3))/(b*x^2 + a)^(3/2) - C*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a 
)))/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int((A + B*x^2 + C*x^4)/(a + b*x^2)^(5/2),x)
 

Output:

int((A + B*x^2 + C*x^4)/(a + b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.12 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a^{2} b c x +3 \sqrt {b \,x^{2}+a}\, a \,b^{3} x -4 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{3}+3 \sqrt {b \,x^{2}+a}\, b^{4} x^{3}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} c +6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b c \,x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c \,x^{4}-\sqrt {b}\, a^{2} b^{2}-2 \sqrt {b}\, a \,b^{3} x^{2}-\sqrt {b}\, b^{4} x^{4}}{3 a \,b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int((C*x^4+B*x^2+A)/(b*x^2+a)^(5/2),x)
 

Output:

( - 3*sqrt(a + b*x**2)*a**2*b*c*x + 3*sqrt(a + b*x**2)*a*b**3*x - 4*sqrt(a 
 + b*x**2)*a*b**2*c*x**3 + 3*sqrt(a + b*x**2)*b**4*x**3 + 3*sqrt(b)*log((s 
qrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*c + 6*sqrt(b)*log((sqrt(a + b*x 
**2) + sqrt(b)*x)/sqrt(a))*a**2*b*c*x**2 + 3*sqrt(b)*log((sqrt(a + b*x**2) 
 + sqrt(b)*x)/sqrt(a))*a*b**2*c*x**4 - sqrt(b)*a**2*b**2 - 2*sqrt(b)*a*b** 
3*x**2 - sqrt(b)*b**4*x**4)/(3*a*b**3*(a**2 + 2*a*b*x**2 + b**2*x**4))