\(\int \frac {(c+d x)^2}{x^2 (a+b x^2)^3} \, dx\) [237]

Optimal result

Integrand size = 20, antiderivative size = 151 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=-\frac {c^2}{a^3 x}+\frac {2 a c d-\left (b c^2-a d^2\right ) x}{4 a^2 \left (a+b x^2\right )^2}+\frac {8 a c d-\left (7 b c^2-3 a d^2\right ) x}{8 a^3 \left (a+b x^2\right )}-\frac {3 \left (5 b c^2-a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2} \sqrt {b}}+\frac {2 c d \log (x)}{a^3}-\frac {c d \log \left (a+b x^2\right )}{a^3} \] Output:

-c^2/a^3/x+1/4*(2*a*c*d-(-a*d^2+b*c^2)*x)/a^2/(b*x^2+a)^2+1/8*(8*a*c*d-(-3 
*a*d^2+7*b*c^2)*x)/a^3/(b*x^2+a)-3/8*(-a*d^2+5*b*c^2)*arctan(b^(1/2)*x/a^( 
1/2))/a^(7/2)/b^(1/2)+2*c*d*ln(x)/a^3-c*d*ln(b*x^2+a)/a^3
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=-\frac {c^2}{a^3 x}+\frac {2 a c d-b c^2 x+a d^2 x}{4 a^2 \left (a+b x^2\right )^2}+\frac {8 a c d-7 b c^2 x+3 a d^2 x}{8 a^3 \left (a+b x^2\right )}+\frac {3 \left (-5 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2} \sqrt {b}}+\frac {2 c d \log (x)}{a^3}-\frac {c d \log \left (a+b x^2\right )}{a^3} \] Input:

Integrate[(c + d*x)^2/(x^2*(a + b*x^2)^3),x]
 

Output:

-(c^2/(a^3*x)) + (2*a*c*d - b*c^2*x + a*d^2*x)/(4*a^2*(a + b*x^2)^2) + (8* 
a*c*d - 7*b*c^2*x + 3*a*d^2*x)/(8*a^3*(a + b*x^2)) + (3*(-5*b*c^2 + a*d^2) 
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*Sqrt[b]) + (2*c*d*Log[x])/a^3 - ( 
c*d*Log[a + b*x^2])/a^3
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {532, 25, 2336, 25, 2333, 2009}

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.

\(\displaystyle \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {2 c d-x \left (\frac {b c^2}{a}-d^2\right )}{4 a \left (a+b x^2\right )^2}-\frac {\int -\frac {4 c^2+8 d x c-3 \left (\frac {b c^2}{a}-d^2\right ) x^2}{x^2 \left (b x^2+a\right )^2}dx}{4 a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {4 c^2+8 d x c-3 \left (\frac {b c^2}{a}-d^2\right ) x^2}{x^2 \left (b x^2+a\right )^2}dx}{4 a}+\frac {2 c d-x \left (\frac {b c^2}{a}-d^2\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {\frac {8 c d-x \left (\frac {7 b c^2}{a}-3 d^2\right )}{2 a \left (a+b x^2\right )}-\frac {\int -\frac {8 c^2+16 d x c-\left (\frac {7 b c^2}{a}-3 d^2\right ) x^2}{x^2 \left (b x^2+a\right )}dx}{2 a}}{4 a}+\frac {2 c d-x \left (\frac {b c^2}{a}-d^2\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {8 c^2+16 d x c-\left (\frac {7 b c^2}{a}-3 d^2\right ) x^2}{x^2 \left (b x^2+a\right )}dx}{2 a}+\frac {8 c d-x \left (\frac {7 b c^2}{a}-3 d^2\right )}{2 a \left (a+b x^2\right )}}{4 a}+\frac {2 c d-x \left (\frac {b c^2}{a}-d^2\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 2333

\(\displaystyle \frac {\frac {\int \left (\frac {8 c^2}{a x^2}+\frac {16 d c}{a x}+\frac {-15 b c^2-16 b d x c+3 a d^2}{a \left (b x^2+a\right )}\right )dx}{2 a}+\frac {8 c d-x \left (\frac {7 b c^2}{a}-3 d^2\right )}{2 a \left (a+b x^2\right )}}{4 a}+\frac {2 c d-x \left (\frac {b c^2}{a}-d^2\right )}{4 a \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {-\frac {3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (5 b c^2-a d^2\right )}{a^{3/2} \sqrt {b}}-\frac {8 c d \log \left (a+b x^2\right )}{a}-\frac {8 c^2}{a x}+\frac {16 c d \log (x)}{a}}{2 a}+\frac {8 c d-x \left (\frac {7 b c^2}{a}-3 d^2\right )}{2 a \left (a+b x^2\right )}}{4 a}+\frac {2 c d-x \left (\frac {b c^2}{a}-d^2\right )}{4 a \left (a+b x^2\right )^2}\)

Input:

Int[(c + d*x)^2/(x^2*(a + b*x^2)^3),x]
 

Output:

(2*c*d - ((b*c^2)/a - d^2)*x)/(4*a*(a + b*x^2)^2) + ((8*c*d - ((7*b*c^2)/a 
 - 3*d^2)*x)/(2*a*(a + b*x^2)) + ((-8*c^2)/(a*x) - (3*(5*b*c^2 - a*d^2)*Ar 
cTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) + (16*c*d*Log[x])/a - (8*c*d* 
Log[a + b*x^2])/a)/(2*a))/(4*a)
 

Defintions of rubi rules used

rule 25

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

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2333

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] 
&& PolyQ[Pq, x] && IGtQ[p, -2]
 

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 = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87

method	result	size
default	\(-\frac {c^{2}}{a^{3} x}+\frac {2 c d \ln \left (x \right )}{a^{3}}+\frac {\frac {\left (\frac {3}{8} a b \,d^{2}-\frac {7}{8} b^{2} c^{2}\right ) x^{3}+a b c d \,x^{2}+\frac {a \left (5 a \,d^{2}-9 b \,c^{2}\right ) x}{8}+\frac {3 a^{2} c d}{2}}{\left (b \,x^{2}+a \right )^{2}}-d c \ln \left (b \,x^{2}+a \right )+\frac {\left (3 a \,d^{2}-15 b \,c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}}{a^{3}}\)	\(131\)
risch	\(\frac {\frac {3 b \left (a \,d^{2}-5 b \,c^{2}\right ) x^{4}}{8 a^{3}}+\frac {b c d \,x^{3}}{a^{2}}+\frac {5 \left (a \,d^{2}-5 b \,c^{2}\right ) x^{2}}{8 a^{2}}+\frac {3 c d x}{2 a}-\frac {c^{2}}{a}}{x \left (b \,x^{2}+a \right )^{2}}+\frac {2 c d \ln \left (x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} b \,\textit {\_Z}^{2}+32 a^{4} b c d \textit {\_Z} +9 a^{2} d^{4}+166 b \,c^{2} d^{2} a +225 b^{2} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\textit {\_R}^{2} a^{7} b +16 \textit {\_R} \,a^{4} b c d +6 a^{2} d^{4}-60 b \,c^{2} d^{2} a +150 b^{2} c^{4}\right ) x +\left (-a^{5} d^{2}+5 a^{4} b \,c^{2}\right ) \textit {\_R} +32 d^{3} c \,a^{2}-160 a b \,c^{3} d \right )\right )}{16}\)	\(229\)

Input:

int((d*x+c)^2/x^2/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-c^2/a^3/x+2*c*d*ln(x)/a^3+1/a^3*(((3/8*a*b*d^2-7/8*b^2*c^2)*x^3+a*b*c*d*x 
^2+1/8*a*(5*a*d^2-9*b*c^2)*x+3/2*a^2*c*d)/(b*x^2+a)^2-d*c*ln(b*x^2+a)+1/8* 
(3*a*d^2-15*b*c^2)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (137) = 274\).

Time = 0.14 (sec) , antiderivative size = 578, normalized size of antiderivative = 3.83 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=\left [\frac {16 \, a^{2} b^{2} c d x^{3} + 24 \, a^{3} b c d x - 16 \, a^{3} b c^{2} - 6 \, {\left (5 \, a b^{3} c^{2} - a^{2} b^{2} d^{2}\right )} x^{4} - 10 \, {\left (5 \, a^{2} b^{2} c^{2} - a^{3} b d^{2}\right )} x^{2} + 3 \, {\left ({\left (5 \, b^{3} c^{2} - a b^{2} d^{2}\right )} x^{5} + 2 \, {\left (5 \, a b^{2} c^{2} - a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a^{2} b c^{2} - a^{3} d^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 16 \, {\left (a b^{3} c d x^{5} + 2 \, a^{2} b^{2} c d x^{3} + a^{3} b c d x\right )} \log \left (b x^{2} + a\right ) + 32 \, {\left (a b^{3} c d x^{5} + 2 \, a^{2} b^{2} c d x^{3} + a^{3} b c d x\right )} \log \left (x\right )}{16 \, {\left (a^{4} b^{3} x^{5} + 2 \, a^{5} b^{2} x^{3} + a^{6} b x\right )}}, \frac {8 \, a^{2} b^{2} c d x^{3} + 12 \, a^{3} b c d x - 8 \, a^{3} b c^{2} - 3 \, {\left (5 \, a b^{3} c^{2} - a^{2} b^{2} d^{2}\right )} x^{4} - 5 \, {\left (5 \, a^{2} b^{2} c^{2} - a^{3} b d^{2}\right )} x^{2} - 3 \, {\left ({\left (5 \, b^{3} c^{2} - a b^{2} d^{2}\right )} x^{5} + 2 \, {\left (5 \, a b^{2} c^{2} - a^{2} b d^{2}\right )} x^{3} + {\left (5 \, a^{2} b c^{2} - a^{3} d^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 8 \, {\left (a b^{3} c d x^{5} + 2 \, a^{2} b^{2} c d x^{3} + a^{3} b c d x\right )} \log \left (b x^{2} + a\right ) + 16 \, {\left (a b^{3} c d x^{5} + 2 \, a^{2} b^{2} c d x^{3} + a^{3} b c d x\right )} \log \left (x\right )}{8 \, {\left (a^{4} b^{3} x^{5} + 2 \, a^{5} b^{2} x^{3} + a^{6} b x\right )}}\right ] \] Input:

integrate((d*x+c)^2/x^2/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

[1/16*(16*a^2*b^2*c*d*x^3 + 24*a^3*b*c*d*x - 16*a^3*b*c^2 - 6*(5*a*b^3*c^2 
 - a^2*b^2*d^2)*x^4 - 10*(5*a^2*b^2*c^2 - a^3*b*d^2)*x^2 + 3*((5*b^3*c^2 - 
 a*b^2*d^2)*x^5 + 2*(5*a*b^2*c^2 - a^2*b*d^2)*x^3 + (5*a^2*b*c^2 - a^3*d^2 
)*x)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 16*(a*b^3* 
c*d*x^5 + 2*a^2*b^2*c*d*x^3 + a^3*b*c*d*x)*log(b*x^2 + a) + 32*(a*b^3*c*d* 
x^5 + 2*a^2*b^2*c*d*x^3 + a^3*b*c*d*x)*log(x))/(a^4*b^3*x^5 + 2*a^5*b^2*x^ 
3 + a^6*b*x), 1/8*(8*a^2*b^2*c*d*x^3 + 12*a^3*b*c*d*x - 8*a^3*b*c^2 - 3*(5 
*a*b^3*c^2 - a^2*b^2*d^2)*x^4 - 5*(5*a^2*b^2*c^2 - a^3*b*d^2)*x^2 - 3*((5* 
b^3*c^2 - a*b^2*d^2)*x^5 + 2*(5*a*b^2*c^2 - a^2*b*d^2)*x^3 + (5*a^2*b*c^2 
- a^3*d^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - 8*(a*b^3*c*d*x^5 + 2*a^2*b 
^2*c*d*x^3 + a^3*b*c*d*x)*log(b*x^2 + a) + 16*(a*b^3*c*d*x^5 + 2*a^2*b^2*c 
*d*x^3 + a^3*b*c*d*x)*log(x))/(a^4*b^3*x^5 + 2*a^5*b^2*x^3 + a^6*b*x)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**2/x**2/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {8 \, a b c d x^{3} + 12 \, a^{2} c d x - 3 \, {\left (5 \, b^{2} c^{2} - a b d^{2}\right )} x^{4} - 8 \, a^{2} c^{2} - 5 \, {\left (5 \, a b c^{2} - a^{2} d^{2}\right )} x^{2}}{8 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} - \frac {c d \log \left (b x^{2} + a\right )}{a^{3}} + \frac {2 \, c d \log \left (x\right )}{a^{3}} - \frac {3 \, {\left (5 \, b c^{2} - a d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} \] Input:

integrate((d*x+c)^2/x^2/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

1/8*(8*a*b*c*d*x^3 + 12*a^2*c*d*x - 3*(5*b^2*c^2 - a*b*d^2)*x^4 - 8*a^2*c^ 
2 - 5*(5*a*b*c^2 - a^2*d^2)*x^2)/(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x) - c*d 
*log(b*x^2 + a)/a^3 + 2*c*d*log(x)/a^3 - 3/8*(5*b*c^2 - a*d^2)*arctan(b*x/ 
sqrt(a*b))/(sqrt(a*b)*a^3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=-\frac {c d \log \left (b x^{2} + a\right )}{a^{3}} + \frac {2 \, c d \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {3 \, {\left (5 \, b c^{2} - a d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} + \frac {8 \, a b c d x^{3} + 12 \, a^{2} c d x - 3 \, {\left (5 \, b^{2} c^{2} - a b d^{2}\right )} x^{4} - 8 \, a^{2} c^{2} - 5 \, {\left (5 \, a b c^{2} - a^{2} d^{2}\right )} x^{2}}{8 \, {\left (b x^{2} + a\right )}^{2} a^{3} x} \] Input:

integrate((d*x+c)^2/x^2/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

-c*d*log(b*x^2 + a)/a^3 + 2*c*d*log(abs(x))/a^3 - 3/8*(5*b*c^2 - a*d^2)*ar 
ctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) + 1/8*(8*a*b*c*d*x^3 + 12*a^2*c*d*x - 
3*(5*b^2*c^2 - a*b*d^2)*x^4 - 8*a^2*c^2 - 5*(5*a*b*c^2 - a^2*d^2)*x^2)/((b 
*x^2 + a)^2*a^3*x)
 

Mupad [B] (verification not implemented)

Time = 7.41 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.24 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=\ln \left (a\,d^2\,\sqrt {-a^7\,b}-5\,b\,c^2\,\sqrt {-a^7\,b}+5\,a^3\,b^2\,c^2\,x+16\,a^4\,b\,c\,d-a^4\,b\,d^2\,x+16\,b\,c\,d\,x\,\sqrt {-a^7\,b}\right )\,\left (\frac {\frac {15\,c^2\,\sqrt {-a^7\,b}}{16}-a^4\,c\,d}{a^7}-\frac {3\,d^2\,\sqrt {-a^7\,b}}{16\,a^6\,b}\right )-\ln \left (a\,d^2\,\sqrt {-a^7\,b}-5\,b\,c^2\,\sqrt {-a^7\,b}-5\,a^3\,b^2\,c^2\,x-16\,a^4\,b\,c\,d+a^4\,b\,d^2\,x+16\,b\,c\,d\,x\,\sqrt {-a^7\,b}\right )\,\left (\frac {\frac {15\,c^2\,\sqrt {-a^7\,b}}{16}+a^4\,c\,d}{a^7}-\frac {3\,d^2\,\sqrt {-a^7\,b}}{16\,a^6\,b}\right )+\frac {\frac {5\,x^2\,\left (a\,d^2-5\,b\,c^2\right )}{8\,a^2}-\frac {c^2}{a}+\frac {3\,c\,d\,x}{2\,a}+\frac {3\,b\,x^4\,\left (a\,d^2-5\,b\,c^2\right )}{8\,a^3}+\frac {b\,c\,d\,x^3}{a^2}}{a^2\,x+2\,a\,b\,x^3+b^2\,x^5}+\frac {2\,c\,d\,\ln \left (x\right )}{a^3} \] Input:

int((c + d*x)^2/(x^2*(a + b*x^2)^3),x)
 

Output:

log(a*d^2*(-a^7*b)^(1/2) - 5*b*c^2*(-a^7*b)^(1/2) + 5*a^3*b^2*c^2*x + 16*a 
^4*b*c*d - a^4*b*d^2*x + 16*b*c*d*x*(-a^7*b)^(1/2))*(((15*c^2*(-a^7*b)^(1/ 
2))/16 - a^4*c*d)/a^7 - (3*d^2*(-a^7*b)^(1/2))/(16*a^6*b)) - log(a*d^2*(-a 
^7*b)^(1/2) - 5*b*c^2*(-a^7*b)^(1/2) - 5*a^3*b^2*c^2*x - 16*a^4*b*c*d + a^ 
4*b*d^2*x + 16*b*c*d*x*(-a^7*b)^(1/2))*(((15*c^2*(-a^7*b)^(1/2))/16 + a^4* 
c*d)/a^7 - (3*d^2*(-a^7*b)^(1/2))/(16*a^6*b)) + ((5*x^2*(a*d^2 - 5*b*c^2)) 
/(8*a^2) - c^2/a + (3*c*d*x)/(2*a) + (3*b*x^4*(a*d^2 - 5*b*c^2))/(8*a^3) + 
 (b*c*d*x^3)/a^2)/(a^2*x + b^2*x^5 + 2*a*b*x^3) + (2*c*d*log(x))/a^3
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.46 \[ \int \frac {(c+d x)^2}{x^2 \left (a+b x^2\right )^3} \, dx=\frac {3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} d^{2} x -15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,c^{2} x +6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b \,d^{2} x^{3}-30 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} c^{2} x^{3}+3 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{2} d^{2} x^{5}-15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b^{3} c^{2} x^{5}-8 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b c d x -16 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{2} c d \,x^{3}-8 \,\mathrm {log}\left (b \,x^{2}+a \right ) a \,b^{3} c d \,x^{5}+16 \,\mathrm {log}\left (x \right ) a^{3} b c d x +32 \,\mathrm {log}\left (x \right ) a^{2} b^{2} c d \,x^{3}+16 \,\mathrm {log}\left (x \right ) a \,b^{3} c d \,x^{5}-8 a^{3} b \,c^{2}+8 a^{3} b c d x +5 a^{3} b \,d^{2} x^{2}-25 a^{2} b^{2} c^{2} x^{2}+3 a^{2} b^{2} d^{2} x^{4}-15 a \,b^{3} c^{2} x^{4}-4 a \,b^{3} c d \,x^{5}}{8 a^{4} b x \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:

int((d*x+c)^2/x^2/(b*x^2+a)^3,x)
 

Output:

(3*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**3*d**2*x - 15*sqrt(b)* 
sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*c**2*x + 6*sqrt(b)*sqrt(a)*at 
an((b*x)/(sqrt(b)*sqrt(a)))*a**2*b*d**2*x**3 - 30*sqrt(b)*sqrt(a)*atan((b* 
x)/(sqrt(b)*sqrt(a)))*a*b**2*c**2*x**3 + 3*sqrt(b)*sqrt(a)*atan((b*x)/(sqr 
t(b)*sqrt(a)))*a*b**2*d**2*x**5 - 15*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*s 
qrt(a)))*b**3*c**2*x**5 - 8*log(a + b*x**2)*a**3*b*c*d*x - 16*log(a + b*x* 
*2)*a**2*b**2*c*d*x**3 - 8*log(a + b*x**2)*a*b**3*c*d*x**5 + 16*log(x)*a** 
3*b*c*d*x + 32*log(x)*a**2*b**2*c*d*x**3 + 16*log(x)*a*b**3*c*d*x**5 - 8*a 
**3*b*c**2 + 8*a**3*b*c*d*x + 5*a**3*b*d**2*x**2 - 25*a**2*b**2*c**2*x**2 
+ 3*a**2*b**2*d**2*x**4 - 15*a*b**3*c**2*x**4 - 4*a*b**3*c*d*x**5)/(8*a**4 
*b*x*(a**2 + 2*a*b*x**2 + b**2*x**4))