\(\int \frac {(c+d x) \sqrt {a+b x^2} (A+B x+C x^2)}{x^4} \, dx\) [10]

Optimal result

Integrand size = 30, antiderivative size = 180 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {(b B c+A b d+2 a C d) \sqrt {a+b x^2}}{2 a}-\frac {(c C+B d) \sqrt {a+b x^2}}{x}-\frac {A c \left (a+b x^2\right )^{3/2}}{3 a x^3}-\frac {(B c+A d) \left (a+b x^2\right )^{3/2}}{2 a x^2}+\sqrt {b} (c C+B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {(b B c+A b d+2 a C d) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {\sqrt {a+b x^2} \left (-2 A b c x^2-a (A (2 c+3 d x)+3 x (2 C x (c-d x)+B (c+2 d x)))\right )}{6 a x^3}+2 \sqrt {a} C d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {b (B c+A d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\sqrt {b} (c C+B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2338, 27, 2338, 25, 27, 536, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} (c+d x) \left (A+B x+C x^2\right )}{x^4} \, dx\)

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^4,x]
 

Output:

-1/3*(A*c*(a + b*x^2)^(3/2))/(a*x^3) + (-1/2*((B*c + A*d)*(a + b*x^2)^(3/2 
))/x^2 + (-(((2*a*(c*C + B*d) - (b*B*c + A*b*d + 2*a*C*d)*x)*Sqrt[a + b*x^ 
2])/x) + 2*a*Sqrt[b]*(c*C + B*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] - Sq 
rt[a]*(b*B*c + A*b*d + 2*a*C*d)*ArcTanh[Sqrt[a + b*x^2]/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 536

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S 
imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( 
a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer 
Q[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 = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.88

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (2 A b c \,x^{2}+6 B a d \,x^{2}+6 C a c \,x^{2}+3 A a d x +3 B a c x +2 A a c \right )}{6 x^{3} a}-\frac {\left (A b d +B b c +2 a C d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 \sqrt {a}}+B \sqrt {b}\, d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+C \sqrt {b}\, c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+d C \sqrt {b \,x^{2}+a}\)	\(159\)
default	\(\left (A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )-\frac {A c \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}}+d C \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\)	\(199\)

Input:

int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/6*(b*x^2+a)^(1/2)*(2*A*b*c*x^2+6*B*a*d*x^2+6*C*a*c*x^2+3*A*a*d*x+3*B*a* 
c*x+2*A*a*c)/x^3/a-1/2*(A*b*d+B*b*c+2*C*a*d)/a^(1/2)*ln((2*a+2*a^(1/2)*(b* 
x^2+a)^(1/2))/x)+B*b^(1/2)*d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))+C*b^(1/2)*c*ln( 
b^(1/2)*x+(b*x^2+a)^(1/2))+d*C*(b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.43 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\left [\frac {6 \, {\left (C a c + B a d\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 3 \, {\left (B b c + {\left (2 \, C a + A b\right )} d\right )} \sqrt {a} x^{3} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (6 \, C a d x^{3} - 2 \, A a c - 2 \, {\left (3 \, B a d + {\left (3 \, C a + A b\right )} c\right )} x^{2} - 3 \, {\left (B a c + A a d\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, a x^{3}}, -\frac {12 \, {\left (C a c + B a d\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 3 \, {\left (B b c + {\left (2 \, C a + A b\right )} d\right )} \sqrt {a} x^{3} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (6 \, C a d x^{3} - 2 \, A a c - 2 \, {\left (3 \, B a d + {\left (3 \, C a + A b\right )} c\right )} x^{2} - 3 \, {\left (B a c + A a d\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, a x^{3}}, \frac {3 \, {\left (B b c + {\left (2 \, C a + A b\right )} d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + 3 \, {\left (C a c + B a d\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + {\left (6 \, C a d x^{3} - 2 \, A a c - 2 \, {\left (3 \, B a d + {\left (3 \, C a + A b\right )} c\right )} x^{2} - 3 \, {\left (B a c + A a d\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, a x^{3}}, -\frac {6 \, {\left (C a c + B a d\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 3 \, {\left (B b c + {\left (2 \, C a + A b\right )} d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (6 \, C a d x^{3} - 2 \, A a c - 2 \, {\left (3 \, B a d + {\left (3 \, C a + A b\right )} c\right )} x^{2} - 3 \, {\left (B a c + A a d\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, a x^{3}}\right ] \] Input:

integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4,x, algorithm="fricas")
 

Output:

[1/12*(6*(C*a*c + B*a*d)*sqrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt 
(b)*x - a) + 3*(B*b*c + (2*C*a + A*b)*d)*sqrt(a)*x^3*log(-(b*x^2 - 2*sqrt( 
b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(6*C*a*d*x^3 - 2*A*a*c - 2*(3*B*a*d + ( 
3*C*a + A*b)*c)*x^2 - 3*(B*a*c + A*a*d)*x)*sqrt(b*x^2 + a))/(a*x^3), -1/12 
*(12*(C*a*c + B*a*d)*sqrt(-b)*x^3*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 3*( 
B*b*c + (2*C*a + A*b)*d)*sqrt(a)*x^3*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt( 
a) + 2*a)/x^2) - 2*(6*C*a*d*x^3 - 2*A*a*c - 2*(3*B*a*d + (3*C*a + A*b)*c)* 
x^2 - 3*(B*a*c + A*a*d)*x)*sqrt(b*x^2 + a))/(a*x^3), 1/6*(3*(B*b*c + (2*C* 
a + A*b)*d)*sqrt(-a)*x^3*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 3*(C*a*c + B 
*a*d)*sqrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (6*C*a 
*d*x^3 - 2*A*a*c - 2*(3*B*a*d + (3*C*a + A*b)*c)*x^2 - 3*(B*a*c + A*a*d)*x 
)*sqrt(b*x^2 + a))/(a*x^3), -1/6*(6*(C*a*c + B*a*d)*sqrt(-b)*x^3*arctan(sq 
rt(-b)*x/sqrt(b*x^2 + a)) - 3*(B*b*c + (2*C*a + A*b)*d)*sqrt(-a)*x^3*arcta 
n(sqrt(b*x^2 + a)*sqrt(-a)/a) - (6*C*a*d*x^3 - 2*A*a*c - 2*(3*B*a*d + (3*C 
*a + A*b)*c)*x^2 - 3*(B*a*c + A*a*d)*x)*sqrt(b*x^2 + a))/(a*x^3)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (165) = 330\).

Time = 4.36 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.94 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=- \frac {A \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {A b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} - \frac {A b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {B \sqrt {a} d}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + B \sqrt {b} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {B b c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {B b d x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {C \sqrt {a} c}{x \sqrt {1 + \frac {b x^{2}}{a}}} - C \sqrt {a} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {C a d}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + C \sqrt {b} c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + \frac {C \sqrt {b} d x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {C b c x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

integrate((d*x+c)*(b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**4,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.91 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=-C \sqrt {a} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \sqrt {b x^{2} + a} C d + {\left (C c + B d\right )} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (B c + A d\right )} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {\sqrt {b x^{2} + a} {\left (B c + A d\right )} b}{2 \, a} - \frac {\sqrt {b x^{2} + a} {\left (C c + B d\right )}}{x} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )}}{2 \, a x^{2}} \] Input:

integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (152) = 304\).

Time = 0.23 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.14 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\sqrt {b x^{2} + a} C d - {\left (C \sqrt {b} c + B \sqrt {b} d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {{\left (B b c + 2 \, C a d + A b d\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B b c + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A b d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a \sqrt {b} c + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} c + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{2} \sqrt {b} c - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b d + 6 \, C a^{3} \sqrt {b} c + 2 \, A a^{2} b^{\frac {3}{2}} c + 6 \, B a^{3} \sqrt {b} d}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \] Input:

integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4,x, algorithm="giac")
 

Output:

sqrt(b*x^2 + a)*C*d - (C*sqrt(b)*c + B*sqrt(b)*d)*log(abs(-sqrt(b)*x + sqr 
t(b*x^2 + a))) + (B*b*c + 2*C*a*d + A*b*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 
 + a))/sqrt(-a))/sqrt(-a) + 1/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B*b*c + 
 3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*b*d + 6*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^4*C*a*sqrt(b)*c + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2)*c + 6*(sqrt 
(b)*x - sqrt(b*x^2 + a))^4*B*a*sqrt(b)*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^2*C*a^2*sqrt(b)*c - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b)*d - 
 3*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^2*b*c - 3*(sqrt(b)*x - sqrt(b*x^2 + a 
))*A*a^2*b*d + 6*C*a^3*sqrt(b)*c + 2*A*a^2*b^(3/2)*c + 6*B*a^3*sqrt(b)*d)/ 
((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^4} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^4,x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 623, normalized size of antiderivative = 3.46 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {-4 \sqrt {b \,x^{2}+a}\, a^{2} c -6 \sqrt {b \,x^{2}+a}\, a^{2} d x -4 \sqrt {b \,x^{2}+a}\, a b c \,x^{2}-6 \sqrt {b \,x^{2}+a}\, a b c x -12 \sqrt {b \,x^{2}+a}\, a b d \,x^{2}-12 \sqrt {b \,x^{2}+a}\, a \,c^{2} x^{2}+12 \sqrt {b \,x^{2}+a}\, a c d \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\frac {-\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x -\sqrt {b}\, \sqrt {a}\, x +a +b \,x^{2}}{\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {a}\, x}\right ) a b d \,x^{3}+6 \sqrt {a}\, \mathrm {log}\left (\frac {-\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x -\sqrt {b}\, \sqrt {a}\, x +a +b \,x^{2}}{\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {a}\, x}\right ) a c d \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\frac {-\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x -\sqrt {b}\, \sqrt {a}\, x +a +b \,x^{2}}{\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {a}\, x}\right ) b^{2} c \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x +\sqrt {b}\, \sqrt {a}\, x +a +b \,x^{2}}{\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {a}\, x}\right ) a b d \,x^{3}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x +\sqrt {b}\, \sqrt {a}\, x +a +b \,x^{2}}{\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {a}\, x}\right ) a c d \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x +\sqrt {b}\, \sqrt {a}\, x +a +b \,x^{2}}{\sqrt {a}\, \sqrt {b \,x^{2}+a}+\sqrt {b}\, \sqrt {a}\, x}\right ) b^{2} c \,x^{3}+12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b d \,x^{3}+12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,c^{2} x^{3}-4 \sqrt {b}\, a b c \,x^{3}+4 \sqrt {b}\, a b d \,x^{3}+4 \sqrt {b}\, a \,c^{2} x^{3}}{12 a \,x^{3}} \] Input:

int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^4,x)
 

Output:

( - 4*sqrt(a + b*x**2)*a**2*c - 6*sqrt(a + b*x**2)*a**2*d*x - 4*sqrt(a + b 
*x**2)*a*b*c*x**2 - 6*sqrt(a + b*x**2)*a*b*c*x - 12*sqrt(a + b*x**2)*a*b*d 
*x**2 - 12*sqrt(a + b*x**2)*a*c**2*x**2 + 12*sqrt(a + b*x**2)*a*c*d*x**3 + 
 3*sqrt(a)*log(( - sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a + b*x**2)*x - 
 sqrt(b)*sqrt(a)*x + a + b*x**2)/(sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt( 
a)*x))*a*b*d*x**3 + 6*sqrt(a)*log(( - sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*s 
qrt(a + b*x**2)*x - sqrt(b)*sqrt(a)*x + a + b*x**2)/(sqrt(a)*sqrt(a + b*x* 
*2) + sqrt(b)*sqrt(a)*x))*a*c*d*x**3 + 3*sqrt(a)*log(( - sqrt(a)*sqrt(a + 
b*x**2) + sqrt(b)*sqrt(a + b*x**2)*x - sqrt(b)*sqrt(a)*x + a + b*x**2)/(sq 
rt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a)*x))*b**2*c*x**3 - 3*sqrt(a)*log(( 
sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a + b*x**2)*x + sqrt(b)*sqrt(a)*x 
+ a + b*x**2)/(sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a)*x))*a*b*d*x**3 - 
 6*sqrt(a)*log((sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a + b*x**2)*x + sq 
rt(b)*sqrt(a)*x + a + b*x**2)/(sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a)* 
x))*a*c*d*x**3 - 3*sqrt(a)*log((sqrt(a)*sqrt(a + b*x**2) + sqrt(b)*sqrt(a 
+ b*x**2)*x + sqrt(b)*sqrt(a)*x + a + b*x**2)/(sqrt(a)*sqrt(a + b*x**2) + 
sqrt(b)*sqrt(a)*x))*b**2*c*x**3 + 12*sqrt(b)*log((sqrt(a + b*x**2) + sqrt( 
b)*x)/sqrt(a))*a*b*d*x**3 + 12*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/ 
sqrt(a))*a*c**2*x**3 - 4*sqrt(b)*a*b*c*x**3 + 4*sqrt(b)*a*b*d*x**3 + 4*sqr 
t(b)*a*c**2*x**3)/(12*a*x**3)