\(\int \frac {A+B x+C x^2}{x (a+b x^2)^{9/2}} \, dx\) [9]

Optimal result

Integrand size = 25, antiderivative size = 138 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \] Output:

1/7*(B*b*x+A*b-C*a)/a/b/(b*x^2+a)^(7/2)+1/35*(6*B*x+7*A)/a^2/(b*x^2+a)^(5/ 
2)+1/105*(24*B*x+35*A)/a^3/(b*x^2+a)^(3/2)+1/35*(16*B*x+35*A)/a^4/(b*x^2+a 
)^(1/2)-A*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\frac {-15 a^4 C+14 a b^3 x^4 (25 A+12 B x)+14 a^2 b^2 x^2 (29 A+15 B x)+3 b^4 x^6 (35 A+16 B x)+a^3 b (176 A+105 B x)}{105 a^4 b \left (a+b x^2\right )^{7/2}}+\frac {2 A \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \] Input:

Integrate[(A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x]
 

Output:

(-15*a^4*C + 14*a*b^3*x^4*(25*A + 12*B*x) + 14*a^2*b^2*x^2*(29*A + 15*B*x) 
 + 3*b^4*x^6*(35*A + 16*B*x) + a^3*b*(176*A + 105*B*x))/(105*a^4*b*(a + b* 
x^2)^(7/2)) + (2*A*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/a^(9/2)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2336, 25, 532, 25, 532, 27, 532, 27, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int -\frac {7 A+6 B x}{x \left (b x^2+a\right )^{7/2}}dx}{7 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {7 A+6 B x}{x \left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}-\frac {\int -\frac {35 A+24 B x}{x \left (b x^2+a\right )^{5/2}}dx}{5 a}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {35 A+24 B x}{x \left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}-\frac {\int -\frac {3 (35 A+16 B x)}{x \left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {35 A+16 B x}{x \left (b x^2+a\right )^{3/2}}dx}{a}+\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {\frac {\frac {35 A+16 B x}{a \sqrt {a+b x^2}}-\frac {\int -\frac {35 A}{x \sqrt {b x^2+a}}dx}{a}}{a}+\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {35 A \int \frac {1}{x \sqrt {b x^2+a}}dx}{a}+\frac {35 A+16 B x}{a \sqrt {a+b x^2}}}{a}+\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {\frac {\frac {35 A \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2}{2 a}+\frac {35 A+16 B x}{a \sqrt {a+b x^2}}}{a}+\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {35 A \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{a b}+\frac {35 A+16 B x}{a \sqrt {a+b x^2}}}{a}+\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {35 A+16 B x}{a \sqrt {a+b x^2}}-\frac {35 A \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {35 A+24 B x}{3 a \left (a+b x^2\right )^{3/2}}}{5 a}+\frac {7 A+6 B x}{5 a \left (a+b x^2\right )^{5/2}}}{7 a}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}}\)

Input:

Int[(A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x]
 

Output:

(A*b - a*C + b*B*x)/(7*a*b*(a + b*x^2)^(7/2)) + ((7*A + 6*B*x)/(5*a*(a + b 
*x^2)^(5/2)) + ((35*A + 24*B*x)/(3*a*(a + b*x^2)^(3/2)) + ((35*A + 16*B*x) 
/(a*Sqrt[a + b*x^2]) - (35*A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(3/2))/a) 
/(5*a))/(7*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 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

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]
 

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.40

method	result	size
default	\(B \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+A \left (\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\right )-\frac {C}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\)	\(193\)

Input:

int((C*x^2+B*x+A)/x/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

B*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/(b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/( 
b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2))))+A*(1/7/a/(b*x^2+a)^(7/2)+1/a*( 
1/5/a/(b*x^2+a)^(5/2)+1/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)))))-1/7*C/b/(b*x^2+a)^(7/2 
)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.39 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (A b^{5} x^{8} + 4 \, A a b^{4} x^{6} + 6 \, A a^{2} b^{3} x^{4} + 4 \, A a^{3} b^{2} x^{2} + A a^{4} b\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (48 \, B a b^{4} x^{7} + 105 \, A a b^{4} x^{6} + 168 \, B a^{2} b^{3} x^{5} + 350 \, A a^{2} b^{3} x^{4} + 210 \, B a^{3} b^{2} x^{3} + 406 \, A a^{3} b^{2} x^{2} + 105 \, B a^{4} b x - 15 \, C a^{5} + 176 \, A a^{4} b\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{5} b^{5} x^{8} + 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} + 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}, \frac {105 \, {\left (A b^{5} x^{8} + 4 \, A a b^{4} x^{6} + 6 \, A a^{2} b^{3} x^{4} + 4 \, A a^{3} b^{2} x^{2} + A a^{4} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (48 \, B a b^{4} x^{7} + 105 \, A a b^{4} x^{6} + 168 \, B a^{2} b^{3} x^{5} + 350 \, A a^{2} b^{3} x^{4} + 210 \, B a^{3} b^{2} x^{3} + 406 \, A a^{3} b^{2} x^{2} + 105 \, B a^{4} b x - 15 \, C a^{5} + 176 \, A a^{4} b\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{5} x^{8} + 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} + 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}\right ] \] Input:

integrate((C*x^2+B*x+A)/x/(b*x^2+a)^(9/2),x, algorithm="fricas")
 

Output:

[1/210*(105*(A*b^5*x^8 + 4*A*a*b^4*x^6 + 6*A*a^2*b^3*x^4 + 4*A*a^3*b^2*x^2 
 + A*a^4*b)*sqrt(a)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 
2*(48*B*a*b^4*x^7 + 105*A*a*b^4*x^6 + 168*B*a^2*b^3*x^5 + 350*A*a^2*b^3*x^ 
4 + 210*B*a^3*b^2*x^3 + 406*A*a^3*b^2*x^2 + 105*B*a^4*b*x - 15*C*a^5 + 176 
*A*a^4*b)*sqrt(b*x^2 + a))/(a^5*b^5*x^8 + 4*a^6*b^4*x^6 + 6*a^7*b^3*x^4 + 
4*a^8*b^2*x^2 + a^9*b), 1/105*(105*(A*b^5*x^8 + 4*A*a*b^4*x^6 + 6*A*a^2*b^ 
3*x^4 + 4*A*a^3*b^2*x^2 + A*a^4*b)*sqrt(-a)*arctan(sqrt(b*x^2 + a)*sqrt(-a 
)/a) + (48*B*a*b^4*x^7 + 105*A*a*b^4*x^6 + 168*B*a^2*b^3*x^5 + 350*A*a^2*b 
^3*x^4 + 210*B*a^3*b^2*x^3 + 406*A*a^3*b^2*x^2 + 105*B*a^4*b*x - 15*C*a^5 
+ 176*A*a^4*b)*sqrt(b*x^2 + a))/(a^5*b^5*x^8 + 4*a^6*b^4*x^6 + 6*a^7*b^3*x 
^4 + 4*a^8*b^2*x^2 + a^9*b)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5251 vs. \(2 (122) = 244\).

Time = 39.15 (sec) , antiderivative size = 6613, normalized size of antiderivative = 47.92 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x**2+B*x+A)/x/(b*x**2+a)**(9/2),x)
 

Output:

A*(352*a**32*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9 
450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4 
*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a* 
*(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 
 + 210*a**(53/2)*b**10*x**20) + 105*a**32*log(b*x**2/a)/(210*a**(73/2) + 2 
100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x** 
6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/ 
2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2 
100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 210*a**32*log(sqrt 
(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2 
)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 5292 
0*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7 
*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(5 
3/2)*b**10*x**20) + 2924*a**31*b*x**2*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 
2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x* 
*6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61 
/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 
2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 1050*a**31*b*x**2 
*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b** 
2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\frac {16 \, B x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {A \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {A}{\sqrt {b x^{2} + a} a^{4}} + \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {C}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \] Input:

integrate((C*x^2+B*x+A)/x/(b*x^2+a)^(9/2),x, algorithm="maxima")
 

Output:

16/35*B*x/(sqrt(b*x^2 + a)*a^4) + 8/35*B*x/((b*x^2 + a)^(3/2)*a^3) + 6/35* 
B*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*B*x/((b*x^2 + a)^(7/2)*a) - A*arcsinh(a/ 
(sqrt(a*b)*abs(x)))/a^(9/2) + A/(sqrt(b*x^2 + a)*a^4) + 1/3*A/((b*x^2 + a) 
^(3/2)*a^3) + 1/5*A/((b*x^2 + a)^(5/2)*a^2) + 1/7*A/((b*x^2 + a)^(7/2)*a) 
- 1/7*C/((b*x^2 + a)^(7/2)*b)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left (3 \, {\left ({\left (\frac {16 \, B b^{3} x}{a^{4}} + \frac {35 \, A b^{3}}{a^{4}}\right )} x + \frac {56 \, B b^{2}}{a^{3}}\right )} x + \frac {350 \, A b^{2}}{a^{3}}\right )} x + \frac {210 \, B b}{a^{2}}\right )} x + \frac {406 \, A b}{a^{2}}\right )} x + \frac {105 \, B}{a}\right )} x - \frac {15 \, C a^{14} b^{2} - 176 \, A a^{13} b^{3}}{a^{14} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, A \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} \] Input:

integrate((C*x^2+B*x+A)/x/(b*x^2+a)^(9/2),x, algorithm="giac")
 

Output:

1/105*(((((3*((16*B*b^3*x/a^4 + 35*A*b^3/a^4)*x + 56*B*b^2/a^3)*x + 350*A* 
b^2/a^3)*x + 210*B*b/a^2)*x + 406*A*b/a^2)*x + 105*B/a)*x - (15*C*a^14*b^2 
 - 176*A*a^13*b^3)/(a^14*b^3))/(b*x^2 + a)^(7/2) + 2*A*arctan(-(sqrt(b)*x 
- sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^4)
 

Mupad [B] (verification not implemented)

Time = 2.83 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\frac {\frac {A}{7\,a}+\frac {A\,{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {A\,{\left (b\,x^2+a\right )}^3}{a^4}+\frac {A\,\left (b\,x^2+a\right )}{5\,a^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {C}{7\,b\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {A\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {16\,B\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,B\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,B\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {B\,x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}} \] Input:

int((A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x)
 

Output:

(A/(7*a) + (A*(a + b*x^2)^2)/(3*a^3) + (A*(a + b*x^2)^3)/a^4 + (A*(a + b*x 
^2))/(5*a^2))/(a + b*x^2)^(7/2) - C/(7*b*(a + b*x^2)^(7/2)) - (A*atanh((a 
+ b*x^2)^(1/2)/a^(1/2)))/a^(9/2) + (16*B*x)/(35*a^4*(a + b*x^2)^(1/2)) + ( 
8*B*x)/(35*a^3*(a + b*x^2)^(3/2)) + (6*B*x)/(35*a^2*(a + b*x^2)^(5/2)) + ( 
B*x)/(7*a*(a + b*x^2)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 588, normalized size of antiderivative = 4.26 \[ \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx=\frac {176 \sqrt {b \,x^{2}+a}\, a^{4} b -15 \sqrt {b \,x^{2}+a}\, a^{4} c +406 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{2}+105 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x +350 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{4}+210 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{3}+105 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{6}+168 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{5}+48 \sqrt {b \,x^{2}+a}\, b^{5} x^{7}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b +420 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{2}+630 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{4}+420 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{6}+105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{8}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b -420 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{2}-630 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{4}-420 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{6}-105 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{8}-48 \sqrt {b}\, a^{4} b -192 \sqrt {b}\, a^{3} b^{2} x^{2}-288 \sqrt {b}\, a^{2} b^{3} x^{4}-192 \sqrt {b}\, a \,b^{4} x^{6}-48 \sqrt {b}\, b^{5} x^{8}}{105 a^{4} b \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

int((C*x^2+B*x+A)/x/(b*x^2+a)^(9/2),x)
 

Output:

(176*sqrt(a + b*x**2)*a**4*b - 15*sqrt(a + b*x**2)*a**4*c + 406*sqrt(a + b 
*x**2)*a**3*b**2*x**2 + 105*sqrt(a + b*x**2)*a**3*b**2*x + 350*sqrt(a + b* 
x**2)*a**2*b**3*x**4 + 210*sqrt(a + b*x**2)*a**2*b**3*x**3 + 105*sqrt(a + 
b*x**2)*a*b**4*x**6 + 168*sqrt(a + b*x**2)*a*b**4*x**5 + 48*sqrt(a + b*x** 
2)*b**5*x**7 + 105*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sq 
rt(a))*a**4*b + 420*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/s 
qrt(a))*a**3*b**2*x**2 + 630*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqr 
t(b)*x)/sqrt(a))*a**2*b**3*x**4 + 420*sqrt(a)*log((sqrt(a + b*x**2) - sqrt 
(a) + sqrt(b)*x)/sqrt(a))*a*b**4*x**6 + 105*sqrt(a)*log((sqrt(a + b*x**2) 
- sqrt(a) + sqrt(b)*x)/sqrt(a))*b**5*x**8 - 105*sqrt(a)*log((sqrt(a + b*x* 
*2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**4*b - 420*sqrt(a)*log((sqrt(a + b*x 
**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b**2*x**2 - 630*sqrt(a)*log((sqr 
t(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**3*x**4 - 420*sqrt(a) 
*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**4*x**6 - 105*s 
qrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**5*x**8 - 4 
8*sqrt(b)*a**4*b - 192*sqrt(b)*a**3*b**2*x**2 - 288*sqrt(b)*a**2*b**3*x**4 
 - 192*sqrt(b)*a*b**4*x**6 - 48*sqrt(b)*b**5*x**8)/(105*a**4*b*(a**4 + 4*a 
**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))