\(\int \frac {A+B x+C x^2+D x^3}{x^3 (a+b x^2)^{5/2}} \, dx\) [116]

Optimal result

Integrand size = 30, antiderivative size = 156 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=-\frac {A b-a C+(b B-a D) x}{3 a^2 \left (a+b x^2\right )^{3/2}}-\frac {3 (2 A b-a C)+(5 b B-2 a D) x}{3 a^3 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{2 a^3 x^2}-\frac {B \sqrt {a+b x^2}}{a^3 x}+\frac {(5 A b-2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{7/2}} \] Output:

-1/3*(A*b-C*a+(B*b-D*a)*x)/a^2/(b*x^2+a)^(3/2)-1/3*(6*A*b-3*C*a+(5*B*b-2*D 
*a)*x)/a^3/(b*x^2+a)^(1/2)-1/2*A*(b*x^2+a)^(1/2)/a^3/x^2-B*(b*x^2+a)^(1/2) 
/a^3/x+1/2*(5*A*b-2*C*a)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(7/2)
 

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {-b^2 x^4 (15 A+16 B x)+a^2 \left (-3 A-6 B x+8 C x^2+6 D x^3\right )+2 a b x^2 \left (-10 A+x \left (-12 B+3 C x+2 D x^2\right )\right )}{6 a^3 x^2 \left (a+b x^2\right )^{3/2}}+\frac {(-5 A b+2 a C) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{7/2}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^(5/2)),x]
 

Output:

(-(b^2*x^4*(15*A + 16*B*x)) + a^2*(-3*A - 6*B*x + 8*C*x^2 + 6*D*x^3) + 2*a 
*b*x^2*(-10*A + x*(-12*B + 3*C*x + 2*D*x^2)))/(6*a^3*x^2*(a + b*x^2)^(3/2) 
) + ((-5*A*b + 2*a*C)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(7 
/2)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2336, 25, 2336, 27, 2338, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2336

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2336

\(\displaystyle \frac {-\frac {\int -\frac {3 \left (-\left (\left (\frac {2 A b}{a}-C\right ) x^2\right )+B x+A\right )}{x^3 \sqrt {b x^2+a}}dx}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 \int \frac {-\left (\left (\frac {2 A b}{a}-C\right ) x^2\right )+B x+A}{x^3 \sqrt {b x^2+a}}dx}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {\frac {3 \left (-\frac {\int -\frac {2 a B-(5 A b-2 a C) x}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {2 a B-(5 A b-2 a C) x}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {3 \left (\frac {-(5 A b-2 a C) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {3 \left (\frac {-\frac {1}{2} (5 A b-2 a C) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {3 \left (\frac {-\frac {(5 A b-2 a C) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {(5 A b-2 a C) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 B \sqrt {a+b x^2}}{x}}{2 a}-\frac {A \sqrt {a+b x^2}}{2 a x^2}\right )}{a}-\frac {3 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {5 b B}{a}-2 D\right )}{a \sqrt {a+b x^2}}}{3 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{3 a \left (a+b x^2\right )^{3/2}}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)^(5/2)),x]
 

Output:

-1/3*((A*b)/a - C + ((b*B)/a - D)*x)/(a*(a + b*x^2)^(3/2)) + (-((3*((2*A*b 
)/a - C) + ((5*b*B)/a - 2*D)*x)/(a*Sqrt[a + b*x^2])) + (3*(-1/2*(A*Sqrt[a 
+ b*x^2])/(a*x^2) + ((-2*B*Sqrt[a + b*x^2])/x + ((5*A*b - 2*a*C)*ArcTanh[S 
qrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])/(2*a)))/a)/(3*a)
 

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 73

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]
 

rule 221

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]
 

rule 243

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]
 

rule 534

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]
 

rule 2336

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]
 

rule 2338

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])
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.55

method	result	size
default	\(D \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+A \left (-\frac {1}{2 a \,x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 b \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )}{2 a}\right )+B \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 b \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{a}\right )+C \left (\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )\)	\(242\)

Input:

int((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.82 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, C a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \, {\left (2 \, C a^{2} b - 5 \, A a b^{2}\right )} x^{4} + {\left (2 \, C a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (4 \, {\left (D a^{2} b - 4 \, B a b^{2}\right )} x^{5} - 6 \, B a^{3} x + 3 \, {\left (2 \, C a^{2} b - 5 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} + 6 \, {\left (D a^{3} - 4 \, B a^{2} b\right )} x^{3} + 4 \, {\left (2 \, C a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}, \frac {3 \, {\left ({\left (2 \, C a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \, {\left (2 \, C a^{2} b - 5 \, A a b^{2}\right )} x^{4} + {\left (2 \, C a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (4 \, {\left (D a^{2} b - 4 \, B a b^{2}\right )} x^{5} - 6 \, B a^{3} x + 3 \, {\left (2 \, C a^{2} b - 5 \, A a b^{2}\right )} x^{4} - 3 \, A a^{3} + 6 \, {\left (D a^{3} - 4 \, B a^{2} b\right )} x^{3} + 4 \, {\left (2 \, C a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}\right ] \] Input:

integrate((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/12*(3*((2*C*a*b^2 - 5*A*b^3)*x^6 + 2*(2*C*a^2*b - 5*A*a*b^2)*x^4 + (2* 
C*a^3 - 5*A*a^2*b)*x^2)*sqrt(a)*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 
2*a)/x^2) - 2*(4*(D*a^2*b - 4*B*a*b^2)*x^5 - 6*B*a^3*x + 3*(2*C*a^2*b - 5* 
A*a*b^2)*x^4 - 3*A*a^3 + 6*(D*a^3 - 4*B*a^2*b)*x^3 + 4*(2*C*a^3 - 5*A*a^2* 
b)*x^2)*sqrt(b*x^2 + a))/(a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2), 1/6*(3*((2 
*C*a*b^2 - 5*A*b^3)*x^6 + 2*(2*C*a^2*b - 5*A*a*b^2)*x^4 + (2*C*a^3 - 5*A*a 
^2*b)*x^2)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (4*(D*a^2*b - 4*B 
*a*b^2)*x^5 - 6*B*a^3*x + 3*(2*C*a^2*b - 5*A*a*b^2)*x^4 - 3*A*a^3 + 6*(D*a 
^3 - 4*B*a^2*b)*x^3 + 4*(2*C*a^3 - 5*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a^4*b 
^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (138) = 276\).

Time = 28.33 (sec) , antiderivative size = 1875, normalized size of antiderivative = 12.02 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((D*x**3+C*x**2+B*x+A)/x**3/(b*x**2+a)**(5/2),x)
 

Output:

A*(-6*a**17*sqrt(1 + b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 
36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 46*a**16*b*x**2*sqrt(1 
+ b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x 
**6 + 12*a**(33/2)*b**3*x**8) - 15*a**16*b*x**2*log(b*x**2/a)/(12*a**(39/2 
)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3* 
x**8) + 30*a**16*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 3 
6*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 70 
*a**15*b**2*x**4*sqrt(1 + b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x* 
*4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 45*a**15*b**2*x**4 
*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b** 
2*x**6 + 12*a**(33/2)*b**3*x**8) + 90*a**15*b**2*x**4*log(sqrt(1 + b*x**2/ 
a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 
+ 12*a**(33/2)*b**3*x**8) - 30*a**14*b**3*x**6*sqrt(1 + b*x**2/a)/(12*a**( 
39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b 
**3*x**8) - 45*a**14*b**3*x**6*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(3 
7/2)*b*x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) + 90*a**14* 
b**3*x**6*log(sqrt(1 + b*x**2/a) + 1)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b* 
x**4 + 36*a**(35/2)*b**2*x**6 + 12*a**(33/2)*b**3*x**8) - 15*a**13*b**4*x* 
*8*log(b*x**2/a)/(12*a**(39/2)*x**2 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b 
**2*x**6 + 12*a**(33/2)*b**3*x**8) + 30*a**13*b**4*x**8*log(sqrt(1 + b*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {2 \, D x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {D x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {8 \, B b x}{3 \, \sqrt {b x^{2} + a} a^{3}} - \frac {4 \, B b x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {C \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {5}{2}}} + \frac {5 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} + \frac {C}{\sqrt {b x^{2} + a} a^{2}} + \frac {C}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {5 \, A b}{2 \, \sqrt {b x^{2} + a} a^{3}} - \frac {5 \, A b}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {B}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} a x} - \frac {A}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

2/3*D*x/(sqrt(b*x^2 + a)*a^2) + 1/3*D*x/((b*x^2 + a)^(3/2)*a) - 8/3*B*b*x/ 
(sqrt(b*x^2 + a)*a^3) - 4/3*B*b*x/((b*x^2 + a)^(3/2)*a^2) - C*arcsinh(a/(s 
qrt(a*b)*abs(x)))/a^(5/2) + 5/2*A*b*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) 
+ C/(sqrt(b*x^2 + a)*a^2) + 1/3*C/((b*x^2 + a)^(3/2)*a) - 5/2*A*b/(sqrt(b* 
x^2 + a)*a^3) - 5/6*A*b/((b*x^2 + a)^(3/2)*a^2) - B/((b*x^2 + a)^(3/2)*a*x 
) - 1/2*A/((b*x^2 + a)^(3/2)*a*x^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (131) = 262\).

Time = 0.14 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.74 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (x {\left (\frac {{\left (2 \, D a^{8} b^{2} - 5 \, B a^{7} b^{3}\right )} x}{a^{10} b} + \frac {3 \, {\left (C a^{8} b^{2} - 2 \, A a^{7} b^{3}\right )}}{a^{10} b}\right )} + \frac {3 \, {\left (D a^{9} b - 2 \, B a^{8} b^{2}\right )}}{a^{10} b}\right )} x + \frac {4 \, C a^{9} b - 7 \, A a^{8} b^{2}}{a^{10} b}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} + \frac {{\left (2 \, C a - 5 \, A b\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{3}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

1/3*((x*((2*D*a^8*b^2 - 5*B*a^7*b^3)*x/(a^10*b) + 3*(C*a^8*b^2 - 2*A*a^7*b 
^3)/(a^10*b)) + 3*(D*a^9*b - 2*B*a^8*b^2)/(a^10*b))*x + (4*C*a^9*b - 7*A*a 
^8*b^2)/(a^10*b))/(b*x^2 + a)^(3/2) + (2*C*a - 5*A*b)*arctan(-(sqrt(b)*x - 
 sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^3) + ((sqrt(b)*x - sqrt(b*x^2 + a) 
)^3*A*b + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b) + (sqrt(b)*x - sqr 
t(b*x^2 + a))*A*a*b - 2*B*a^2*sqrt(b))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - 
 a)^2*a^3)
 

Mupad [B] (verification not implemented)

Time = 3.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {C}{3\,a}+\frac {C\,\left (b\,x^2+a\right )}{a^2}}{{\left (b\,x^2+a\right )}^{3/2}}+\frac {x\,D}{{\left (b\,x^2+a\right )}^{5/2}}-\frac {C\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {B\,a^2-8\,B\,{\left (b\,x^2+a\right )}^2+4\,B\,a\,\left (b\,x^2+a\right )}{3\,a^3\,x\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {10\,A\,b}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {A}{2\,a\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {5\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}+\frac {2\,b^2\,x^5\,D}{3\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {5\,A\,b^2\,x^2}{2\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {5\,b\,x^3\,D}{3\,a\,{\left (b\,x^2+a\right )}^{5/2}} \] Input:

int((A + B*x + C*x^2 + x^3*D)/(x^3*(a + b*x^2)^(5/2)),x)
 

Output:

(C/(3*a) + (C*(a + b*x^2))/a^2)/(a + b*x^2)^(3/2) + (x*D)/(a + b*x^2)^(5/2 
) - (C*atanh((a + b*x^2)^(1/2)/a^(1/2)))/a^(5/2) + (B*a^2 - 8*B*(a + b*x^2 
)^2 + 4*B*a*(a + b*x^2))/(3*a^3*x*(a + b*x^2)^(3/2)) - (10*A*b)/(3*a^2*(a 
+ b*x^2)^(3/2)) - A/(2*a*x^2*(a + b*x^2)^(3/2)) + (5*A*b*atanh((a + b*x^2) 
^(1/2)/a^(1/2)))/(2*a^(7/2)) + (2*b^2*x^5*D)/(3*a^2*(a + b*x^2)^(5/2)) - ( 
5*A*b^2*x^2)/(2*a^3*(a + b*x^2)^(3/2)) + (5*b*x^3*D)/(3*a*(a + b*x^2)^(5/2 
))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 675, normalized size of antiderivative = 4.33 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a)^(5/2),x)
 

Output:

( - 3*sqrt(a + b*x**2)*a**3*b - 20*sqrt(a + b*x**2)*a**2*b**2*x**2 - 6*sqr 
t(a + b*x**2)*a**2*b**2*x + 8*sqrt(a + b*x**2)*a**2*b*c*x**2 + 6*sqrt(a + 
b*x**2)*a**2*b*d*x**3 - 15*sqrt(a + b*x**2)*a*b**3*x**4 - 24*sqrt(a + b*x* 
*2)*a*b**3*x**3 + 6*sqrt(a + b*x**2)*a*b**2*c*x**4 + 4*sqrt(a + b*x**2)*a* 
b**2*d*x**5 - 16*sqrt(a + b*x**2)*b**4*x**5 - 15*sqrt(a)*log((sqrt(a + b*x 
**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**2*x**2 + 6*sqrt(a)*log((sqrt( 
a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b*c*x**2 - 30*sqrt(a)*log 
((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**3*x**4 + 12*sqrt(a 
)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*c*x**4 - 15 
*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**4*x**6 + 
 6*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c*x* 
*6 + 15*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2 
*b**2*x**2 - 6*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a 
))*a**2*b*c*x**2 + 30*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x) 
/sqrt(a))*a*b**3*x**4 - 12*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt( 
b)*x)/sqrt(a))*a*b**2*c*x**4 + 15*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) 
+ sqrt(b)*x)/sqrt(a))*b**4*x**6 - 6*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a 
) + sqrt(b)*x)/sqrt(a))*b**3*c*x**6 - 4*sqrt(b)*a**3*d*x**2 + 16*sqrt(b)*a 
**2*b**2*x**2 - 8*sqrt(b)*a**2*b*d*x**4 + 32*sqrt(b)*a*b**3*x**4 - 4*sqrt( 
b)*a*b**2*d*x**6 + 16*sqrt(b)*b**4*x**6)/(6*a**3*b*x**2*(a**2 + 2*a*b*x...