\(\int \frac {(c+d x) \sqrt {a+b x^2} (A+B x+C x^2)}{x^3} \, dx\) [9]

Optimal result

Integrand size = 30, antiderivative size = 202 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {(A b c+2 a (c C+B d)) \sqrt {a+b x^2}}{2 a}+\frac {(a C d+2 b (B c+A d)) x \sqrt {a+b x^2}}{2 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{2 a x^2}-\frac {(B c+A d) \left (a+b x^2\right )^{3/2}}{a x}+\frac {(a C d+2 b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}-\frac {(A b c+2 a (c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {1}{2} \left (\frac {\sqrt {a+b x^2} (-A (c+2 d x)+x (-2 B (c-d x)+C x (2 c+d x)))}{x^2}+\frac {2 A b c \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-4 \sqrt {a} (c C+B d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {(a C d+2 b (B c+A d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.90, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2338, 25, 2338, 25, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (2 A d x +2 B c x +A c \right )}{2 x^{2}}-\frac {\left (A b c +2 B a d +2 C a c \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 \sqrt {a}}+A \sqrt {b}\, d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+B \sqrt {b}\, c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {a C d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+B d \sqrt {b \,x^{2}+a}+C c \sqrt {b \,x^{2}+a}+C b d \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )\)	\(206\)
default	\(d C \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )+\left (B d +C c \right ) \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )+A c \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )\)	\(218\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.38 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\left [\frac {{\left (2 \, B a b c + {\left (C a^{2} + 2 \, A a b\right )} d\right )} \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + {\left (2 \, B a b d + {\left (2 \, C a b + A b^{2}\right )} c\right )} \sqrt {a} x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (C a b d x^{3} - A a b c + 2 \, {\left (C a b c + B a b d\right )} x^{2} - 2 \, {\left (B a b c + A a b d\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, a b x^{2}}, -\frac {2 \, {\left (2 \, B a b c + {\left (C a^{2} + 2 \, A a b\right )} d\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B a b d + {\left (2 \, C a b + A b^{2}\right )} c\right )} \sqrt {a} x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (C a b d x^{3} - A a b c + 2 \, {\left (C a b c + B a b d\right )} x^{2} - 2 \, {\left (B a b c + A a b d\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, a b x^{2}}, \frac {2 \, {\left (2 \, B a b d + {\left (2 \, C a b + A b^{2}\right )} c\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (2 \, B a b c + {\left (C a^{2} + 2 \, A a b\right )} d\right )} \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (C a b d x^{3} - A a b c + 2 \, {\left (C a b c + B a b d\right )} x^{2} - 2 \, {\left (B a b c + A a b d\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, a b x^{2}}, -\frac {{\left (2 \, B a b c + {\left (C a^{2} + 2 \, A a b\right )} d\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B a b d + {\left (2 \, C a b + A b^{2}\right )} c\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (C a b d x^{3} - A a b c + 2 \, {\left (C a b c + B a b d\right )} x^{2} - 2 \, {\left (B a b c + A a b d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, a b x^{2}}\right ] \] Input:

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

Output:

[1/4*((2*B*a*b*c + (C*a^2 + 2*A*a*b)*d)*sqrt(b)*x^2*log(-2*b*x^2 - 2*sqrt( 
b*x^2 + a)*sqrt(b)*x - a) + (2*B*a*b*d + (2*C*a*b + A*b^2)*c)*sqrt(a)*x^2* 
log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(C*a*b*d*x^3 - A*a 
*b*c + 2*(C*a*b*c + B*a*b*d)*x^2 - 2*(B*a*b*c + A*a*b*d)*x)*sqrt(b*x^2 + a 
))/(a*b*x^2), -1/4*(2*(2*B*a*b*c + (C*a^2 + 2*A*a*b)*d)*sqrt(-b)*x^2*arcta 
n(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2*B*a*b*d + (2*C*a*b + A*b^2)*c)*sqrt(a)* 
x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(C*a*b*d*x^3 - 
 A*a*b*c + 2*(C*a*b*c + B*a*b*d)*x^2 - 2*(B*a*b*c + A*a*b*d)*x)*sqrt(b*x^2 
 + a))/(a*b*x^2), 1/4*(2*(2*B*a*b*d + (2*C*a*b + A*b^2)*c)*sqrt(-a)*x^2*ar 
ctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (2*B*a*b*c + (C*a^2 + 2*A*a*b)*d)*sqrt( 
b)*x^2*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(C*a*b*d*x^3 - 
A*a*b*c + 2*(C*a*b*c + B*a*b*d)*x^2 - 2*(B*a*b*c + A*a*b*d)*x)*sqrt(b*x^2 
+ a))/(a*b*x^2), -1/2*((2*B*a*b*c + (C*a^2 + 2*A*a*b)*d)*sqrt(-b)*x^2*arct 
an(sqrt(-b)*x/sqrt(b*x^2 + a)) - (2*B*a*b*d + (2*C*a*b + A*b^2)*c)*sqrt(-a 
)*x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (C*a*b*d*x^3 - A*a*b*c + 2*(C*a 
*b*c + B*a*b*d)*x^2 - 2*(B*a*b*c + A*a*b*d)*x)*sqrt(b*x^2 + a))/(a*b*x^2)]
 

Sympy [A] (verification not implemented)

Time = 4.23 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=- \frac {A \sqrt {a} d}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + A \sqrt {b} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {A b c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {A b d x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B \sqrt {a} c}{x \sqrt {1 + \frac {b x^{2}}{a}}} - B \sqrt {a} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a d}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + B \sqrt {b} c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + \frac {B \sqrt {b} d x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b c x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - C \sqrt {a} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {C a c}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {C \sqrt {b} c x}{\sqrt {\frac {a}{b x^{2}} + 1}} + C d \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-A*sqrt(a)*d/(x*sqrt(1 + b*x**2/a)) - A*sqrt(b)*c*sqrt(a/(b*x**2) + 1)/(2* 
x) + A*sqrt(b)*d*asinh(sqrt(b)*x/sqrt(a)) - A*b*c*asinh(sqrt(a)/(sqrt(b)*x 
))/(2*sqrt(a)) - A*b*d*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - B*sqrt(a)*c/(x*sqr 
t(1 + b*x**2/a)) - B*sqrt(a)*d*asinh(sqrt(a)/(sqrt(b)*x)) + B*a*d/(sqrt(b) 
*x*sqrt(a/(b*x**2) + 1)) + B*sqrt(b)*c*asinh(sqrt(b)*x/sqrt(a)) + B*sqrt(b 
)*d*x/sqrt(a/(b*x**2) + 1) - B*b*c*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - C*sqrt 
(a)*c*asinh(sqrt(a)/(sqrt(b)*x)) + C*a*c/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) 
+ C*sqrt(b)*c*x/sqrt(a/(b*x**2) + 1) + C*d*Piecewise((a*Piecewise((log(2*s 
qrt(b)*sqrt(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))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} C d x + \frac {C a d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} - \frac {A b c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {\sqrt {b x^{2} + a} A b c}{2 \, a} + {\left (B c + A d\right )} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - {\left (C c + B d\right )} \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \sqrt {b x^{2} + a} {\left (C c + B d\right )} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c}{2 \, a x^{2}} - \frac {\sqrt {b x^{2} + a} {\left (B c + A d\right )}}{x} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {1}{2} \, {\left (C d x + 2 \, C c + 2 \, B d\right )} \sqrt {b x^{2} + a} + \frac {{\left (2 \, C a c + A b c + 2 \, B a d\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {{\left (2 \, B b c + C a d + 2 \, A b d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, \sqrt {b}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b c + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} c + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b c - 2 \, B a^{2} \sqrt {b} c - 2 \, A a^{2} \sqrt {b} d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.90 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^3} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a b c -2 \sqrt {b \,x^{2}+a}\, a b d x -2 \sqrt {b \,x^{2}+a}\, b^{2} c x +2 \sqrt {b \,x^{2}+a}\, b^{2} d \,x^{2}+2 \sqrt {b \,x^{2}+a}\, b \,c^{2} x^{2}+\sqrt {b \,x^{2}+a}\, b c d \,x^{3}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{2}+2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d \,x^{2}+2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{2}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d \,x^{2}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} x^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b d \,x^{2}+\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a c d \,x^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{2}}{2 b \,x^{2}} \] Input:

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

Output:

( - sqrt(a + b*x**2)*a*b*c - 2*sqrt(a + b*x**2)*a*b*d*x - 2*sqrt(a + b*x** 
2)*b**2*c*x + 2*sqrt(a + b*x**2)*b**2*d*x**2 + 2*sqrt(a + b*x**2)*b*c**2*x 
**2 + sqrt(a + b*x**2)*b*c*d*x**3 + sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a 
) + sqrt(b)*x)/sqrt(a))*b**2*c*x**2 + 2*sqrt(a)*log((sqrt(a + b*x**2) - sq 
rt(a) + sqrt(b)*x)/sqrt(a))*b**2*d*x**2 + 2*sqrt(a)*log((sqrt(a + b*x**2) 
- sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**2*x**2 - sqrt(a)*log((sqrt(a + b*x**2 
) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*x**2 - 2*sqrt(a)*log((sqrt(a + b* 
x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*d*x**2 - 2*sqrt(a)*log((sqrt(a 
+ b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**2*x**2 + 2*sqrt(b)*log((sqr 
t(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*d*x**2 + sqrt(b)*log((sqrt(a + b*x 
**2) + sqrt(b)*x)/sqrt(a))*a*c*d*x**2 + 2*sqrt(b)*log((sqrt(a + b*x**2) + 
sqrt(b)*x)/sqrt(a))*b**2*c*x**2)/(2*b*x**2)