\(\int \frac {(c+d x)^2 \sqrt {a+b x^2} (A+B x+C x^2)}{x^3} \, dx\) [22]

Optimal result

Integrand size = 32, antiderivative size = 250 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\left (c^2 C+2 B c d+A d^2\right ) \sqrt {a+b x^2}-\frac {A c^2 \sqrt {a+b x^2}}{2 x^2}+\frac {(2 b c (B c+2 A d)+a d (2 c C+B d)) x \sqrt {a+b x^2}}{2 a}+\frac {C d^2 \left (a+b x^2\right )^{3/2}}{3 b}-\frac {c (B c+2 A d) \left (a+b x^2\right )^{3/2}}{a x}+\frac {(2 b c (B c+2 A d)+a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}-\frac {\left (2 a c (c C+2 B d)+A \left (b c^2+2 a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {\sqrt {a+b x^2} \left (-3 A b \left (c^2+4 c d x-2 d^2 x^2\right )+x \left (2 a C d^2 x+2 b C x \left (3 c^2+3 c d x+d^2 x^2\right )+b B \left (-6 c^2+12 c d x+3 d^2 x^2\right )\right )\right )}{6 b x^2}+\frac {A b c^2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-2 \sqrt {a} \left (c^2 C+2 B c d+A d^2\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {(2 b c (B c+2 A d)+a d (2 c C+B d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 \sqrt {b}} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(-3*A*b*(c^2 + 4*c*d*x - 2*d^2*x^2) + x*(2*a*C*d^2*x + 2* 
b*C*x*(3*c^2 + 3*c*d*x + d^2*x^2) + b*B*(-6*c^2 + 12*c*d*x + 3*d^2*x^2)))) 
/(6*b*x^2) + (A*b*c^2*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/Sqrt 
[a] - 2*Sqrt[a]*(c^2*C + 2*B*c*d + A*d^2)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + 
 b*x^2])/Sqrt[a]] - ((2*b*c*(B*c + 2*A*d) + a*d*(2*c*C + B*d))*Log[-(Sqrt[ 
b]*x) + Sqrt[a + b*x^2]])/(2*Sqrt[b])
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2338, 25, 2338, 25, 2340, 27, 535, 27, 538, 224, 219, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(A*c^2*(a + b*x^2)^(3/2))/(a*x^2) + ((-2*c*(B*c + 2*A*d)*(a + b*x^2)^ 
(3/2))/x + ((2*a^2*C*d^2*(a + b*x^2)^(3/2))/(3*b) + a*((2*a*c*(c*C + 2*B*d 
) + A*(b*c^2 + 2*a*d^2) + (2*b*c*(B*c + 2*A*d) + a*d*(2*c*C + B*d))*x)*Sqr 
t[a + b*x^2] + a*(((2*b*c*(B*c + 2*A*d) + a*d*(2*c*C + B*d))*ArcTanh[(Sqrt 
[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - ((2*a*c*(c*C + 2*B*d) + A*(b*c^2 + 2*a* 
d^2))*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a])))/a)/(2*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 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 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 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 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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

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])
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.16

method	result	size
default	\(B \,d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )+A \,c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )+\frac {C \,d^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}+c \left (2 A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )+2 C c d \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )\)	\(291\)
risch	\(-\frac {c \sqrt {b \,x^{2}+a}\, \left (4 A d x +2 B c x +A c \right )}{2 x^{2}}+b d \left (B d +2 C c \right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+\frac {\left (2 A b \,d^{2}+4 B b c d +2 a C \,d^{2}+2 C b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{2 b}-\frac {\left (2 A a \,d^{2}+b A \,c^{2}+4 B a c d +2 C a \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 \sqrt {a}}+\frac {a B \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\sqrt {b}\, B \,c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+2 A \sqrt {b}\, c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {2 C a c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+C \,d^{2} b \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )\)	\(314\)

Input:

int((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3,x,method=_RETURNVERBOSE)
 

Output:

B*d^2*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+ 
(A*d^2+2*B*c*d+C*c^2)*((b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a) 
^(1/2))/x))+A*c^2*(-1/2/a/x^2*(b*x^2+a)^(3/2)+1/2*b/a*((b*x^2+a)^(1/2)-a^( 
1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)))+1/3*C*d^2*(b*x^2+a)^(3/2)/b+c 
*(2*A*d+B*c)*(-1/a/x*(b*x^2+a)^(3/2)+2*b/a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^ 
(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))+2*C*c*d*(1/2*x*(b*x^2+a)^(1/2)+1/2*a 
/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 1.15 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.90 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3,x, algorithm="fricas 
")
 

Output:

[1/12*(3*(2*B*a*b*c^2 + B*a^2*d^2 + 2*(C*a^2 + 2*A*a*b)*c*d)*sqrt(b)*x^2*l 
og(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(4*B*a*b*c*d + 2*A*a*b* 
d^2 + (2*C*a*b + A*b^2)*c^2)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*s 
qrt(a) + 2*a)/x^2) + 2*(2*C*a*b*d^2*x^4 - 3*A*a*b*c^2 + 3*(2*C*a*b*c*d + B 
*a*b*d^2)*x^3 + 2*(3*C*a*b*c^2 + 6*B*a*b*c*d + (C*a^2 + 3*A*a*b)*d^2)*x^2 
- 6*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(a*b*x^2), -1/12*(6*(2*B 
*a*b*c^2 + B*a^2*d^2 + 2*(C*a^2 + 2*A*a*b)*c*d)*sqrt(-b)*x^2*arctan(sqrt(- 
b)*x/sqrt(b*x^2 + a)) - 3*(4*B*a*b*c*d + 2*A*a*b*d^2 + (2*C*a*b + A*b^2)*c 
^2)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2 
*C*a*b*d^2*x^4 - 3*A*a*b*c^2 + 3*(2*C*a*b*c*d + B*a*b*d^2)*x^3 + 2*(3*C*a* 
b*c^2 + 6*B*a*b*c*d + (C*a^2 + 3*A*a*b)*d^2)*x^2 - 6*(B*a*b*c^2 + 2*A*a*b* 
c*d)*x)*sqrt(b*x^2 + a))/(a*b*x^2), 1/12*(6*(4*B*a*b*c*d + 2*A*a*b*d^2 + ( 
2*C*a*b + A*b^2)*c^2)*sqrt(-a)*x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 3* 
(2*B*a*b*c^2 + B*a^2*d^2 + 2*(C*a^2 + 2*A*a*b)*c*d)*sqrt(b)*x^2*log(-2*b*x 
^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(2*C*a*b*d^2*x^4 - 3*A*a*b*c^2 + 
 3*(2*C*a*b*c*d + B*a*b*d^2)*x^3 + 2*(3*C*a*b*c^2 + 6*B*a*b*c*d + (C*a^2 + 
 3*A*a*b)*d^2)*x^2 - 6*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(a*b* 
x^2), -1/6*(3*(2*B*a*b*c^2 + B*a^2*d^2 + 2*(C*a^2 + 2*A*a*b)*c*d)*sqrt(-b) 
*x^2*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 3*(4*B*a*b*c*d + 2*A*a*b*d^2 + ( 
2*C*a*b + A*b^2)*c^2)*sqrt(-a)*x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) -...
 

Sympy [A] (verification not implemented)

Time = 5.31 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.47 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**2*(b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**3,x)
 

Output:

-2*A*sqrt(a)*c*d/(x*sqrt(1 + b*x**2/a)) - A*sqrt(a)*d**2*asinh(sqrt(a)/(sq 
rt(b)*x)) + A*a*d**2/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) - A*sqrt(b)*c**2*sqr 
t(a/(b*x**2) + 1)/(2*x) + 2*A*sqrt(b)*c*d*asinh(sqrt(b)*x/sqrt(a)) + A*sqr 
t(b)*d**2*x/sqrt(a/(b*x**2) + 1) - A*b*c**2*asinh(sqrt(a)/(sqrt(b)*x))/(2* 
sqrt(a)) - 2*A*b*c*d*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - B*sqrt(a)*c**2/(x*sq 
rt(1 + b*x**2/a)) - 2*B*sqrt(a)*c*d*asinh(sqrt(a)/(sqrt(b)*x)) + 2*B*a*c*d 
/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) + B*sqrt(b)*c**2*asinh(sqrt(b)*x/sqrt(a) 
) + 2*B*sqrt(b)*c*d*x/sqrt(a/(b*x**2) + 1) + B*d**2*Piecewise((a*Piecewise 
((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sq 
rt(b*x**2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True)) 
 - B*b*c**2*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - C*sqrt(a)*c**2*asinh(sqrt(a)/ 
(sqrt(b)*x)) + C*a*c**2/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) + C*sqrt(b)*c**2* 
x/sqrt(a/(b*x**2) + 1) + 2*C*c*d*Piecewise((a*Piecewise((log(2*sqrt(b)*sqr 
t(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/ 
2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, True)) + C*d**2*Piecewise 
((a*sqrt(a + b*x**2)/(3*b) + x**2*sqrt(a + b*x**2)/3, Ne(b, 0)), (sqrt(a)* 
x**2/2, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=-\frac {A b c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {\sqrt {b x^{2} + a} A b c^{2}}{2 \, a} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C d^{2}}{3 \, b} + \frac {1}{2} \, {\left (2 \, C c d + B d^{2}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (2 \, C c d + B d^{2}\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} + {\left (B c^{2} + 2 \, A c d\right )} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c^{2}}{2 \, a x^{2}} - \frac {{\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a}}{x} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3,x, algorithm="maxima 
")
 

Output:

-1/2*A*b*c^2*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 1/2*sqrt(b*x^2 + a)*A 
*b*c^2/a + 1/3*(b*x^2 + a)^(3/2)*C*d^2/b + 1/2*(2*C*c*d + B*d^2)*sqrt(b*x^ 
2 + a)*x + 1/2*(2*C*c*d + B*d^2)*a*arcsinh(b*x/sqrt(a*b))/sqrt(b) + (B*c^2 
 + 2*A*c*d)*sqrt(b)*arcsinh(b*x/sqrt(a*b)) - (C*c^2 + 2*B*c*d + A*d^2)*sqr 
t(a)*arcsinh(a/(sqrt(a*b)*abs(x))) + (C*c^2 + 2*B*c*d + A*d^2)*sqrt(b*x^2 
+ a) - 1/2*(b*x^2 + a)^(3/2)*A*c^2/(a*x^2) - (B*c^2 + 2*A*c*d)*sqrt(b*x^2 
+ a)/x
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.34 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {1}{6} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, C d^{2} x + \frac {3 \, {\left (2 \, C b c d + B b d^{2}\right )}}{b}\right )} x + \frac {2 \, {\left (3 \, C b c^{2} + 6 \, B b c d + C a d^{2} + 3 \, A b d^{2}\right )}}{b}\right )} + \frac {{\left (2 \, C a c^{2} + A b c^{2} + 4 \, B a c d + 2 \, A a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {{\left (2 \, B b c^{2} + 2 \, C a c d + 4 \, A b c d + B a d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, \sqrt {b}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b c^{2} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} c^{2} + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} c d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b c^{2} - 2 \, B a^{2} \sqrt {b} c^{2} - 4 \, A a^{2} \sqrt {b} c d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \] Input:

integrate((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/6*sqrt(b*x^2 + a)*((2*C*d^2*x + 3*(2*C*b*c*d + B*b*d^2)/b)*x + 2*(3*C*b* 
c^2 + 6*B*b*c*d + C*a*d^2 + 3*A*b*d^2)/b) + (2*C*a*c^2 + A*b*c^2 + 4*B*a*c 
*d + 2*A*a*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sqrt(-a) - 
 1/2*(2*B*b*c^2 + 2*C*a*c*d + 4*A*b*c*d + B*a*d^2)*log(abs(-sqrt(b)*x + sq 
rt(b*x^2 + a)))/sqrt(b) + ((sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b*c^2 + 2*(sq 
rt(b)*x - sqrt(b*x^2 + a))^2*B*a*sqrt(b)*c^2 + 4*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^2*A*a*sqrt(b)*c*d + (sqrt(b)*x - sqrt(b*x^2 + a))*A*a*b*c^2 - 2*B*a^2 
*sqrt(b)*c^2 - 4*A*a^2*sqrt(b)*c*d)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^ 
2
 

Mupad [F(-1)]

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

int(((a + b*x^2)^(1/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^3,x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.31 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, a b \,c^{2}-12 \sqrt {b \,x^{2}+a}\, a b c d x +6 \sqrt {b \,x^{2}+a}\, a b \,d^{2} x^{2}+2 \sqrt {b \,x^{2}+a}\, a c \,d^{2} x^{2}-6 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +12 \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{2}+3 \sqrt {b \,x^{2}+a}\, b^{2} d^{2} x^{3}+6 \sqrt {b \,x^{2}+a}\, b \,c^{3} x^{2}+6 \sqrt {b \,x^{2}+a}\, b \,c^{2} d \,x^{3}+2 \sqrt {b \,x^{2}+a}\, b c \,d^{2} x^{4}+6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{2}+12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{2}+6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{3} x^{2}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{2}-12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{2}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{3} x^{2}+12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}+6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,c^{2} d \,x^{2}+6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{2}}{6 b \,x^{2}} \] Input:

int((d*x+c)^2*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3,x)
 

Output:

( - 3*sqrt(a + b*x**2)*a*b*c**2 - 12*sqrt(a + b*x**2)*a*b*c*d*x + 6*sqrt(a 
 + b*x**2)*a*b*d**2*x**2 + 2*sqrt(a + b*x**2)*a*c*d**2*x**2 - 6*sqrt(a + b 
*x**2)*b**2*c**2*x + 12*sqrt(a + b*x**2)*b**2*c*d*x**2 + 3*sqrt(a + b*x**2 
)*b**2*d**2*x**3 + 6*sqrt(a + b*x**2)*b*c**3*x**2 + 6*sqrt(a + b*x**2)*b*c 
**2*d*x**3 + 2*sqrt(a + b*x**2)*b*c*d**2*x**4 + 6*sqrt(a)*log((sqrt(a + b* 
x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*d**2*x**2 + 3*sqrt(a)*log((sqrt( 
a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c**2*x**2 + 12*sqrt(a)*lo 
g((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*d*x**2 + 6*sqrt 
(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**3*x**2 - 6* 
sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*d**2*x** 
2 - 3*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c 
**2*x**2 - 12*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a) 
)*b**2*c*d*x**2 - 6*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/s 
qrt(a))*b*c**3*x**2 + 12*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a 
))*a*b*c*d*x**2 + 3*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a* 
b*d**2*x**2 + 6*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*c**2 
*d*x**2 + 6*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**2*c**2* 
x**2)/(6*b*x**2)