\(\int \frac {(c+d x) \sqrt {a+b x^2} (A+B x+C x^2)}{x^5} \, dx\) [11]

Optimal result

Integrand size = 30, antiderivative size = 176 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=\frac {(A b c-4 a (c C+B d)) \sqrt {a+b x^2}}{8 a x^2}-\frac {C d \sqrt {a+b x^2}}{x}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}-\frac {(B c+A d) \left (a+b x^2\right )^{3/2}}{3 a x^3}+\sqrt {b} C d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {b (A b c-4 a (c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{3/2}} \] Output:

1/8*(A*b*c-4*a*(B*d+C*c))*(b*x^2+a)^(1/2)/a/x^2-C*d*(b*x^2+a)^(1/2)/x-1/4* 
A*c*(b*x^2+a)^(3/2)/a/x^4-1/3*(A*d+B*c)*(b*x^2+a)^(3/2)/a/x^3+b^(1/2)*C*d* 
arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))+1/8*b*(A*b*c-4*a*(B*d+C*c))*arctanh((b* 
x^2+a)^(1/2)/a^(1/2))/a^(3/2)
 

Mathematica [A] (verified)

Time = 1.60 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=-\frac {\sqrt {a+b x^2} \left (2 a A (3 c+4 d x)+b x^2 (3 A c+8 B c x+8 A d x)+4 a x (3 C x (c+2 d x)+B (2 c+3 d x))\right )}{24 a x^4}-\frac {A b^2 c \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {b (c C+B d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\sqrt {b} C d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \] Input:

Integrate[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^5,x]
 

Output:

-1/24*(Sqrt[a + b*x^2]*(2*a*A*(3*c + 4*d*x) + b*x^2*(3*A*c + 8*B*c*x + 8*A 
*d*x) + 4*a*x*(3*C*x*(c + 2*d*x) + B*(2*c + 3*d*x))))/(a*x^4) - (A*b^2*c*A 
rcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/(4*a^(3/2)) - (b*(c*C + B*d 
)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/Sqrt[a] - Sqrt[b]*C*d 
*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2338, 25, 2338, 27, 537, 25, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} (c+d x) \left (A+B x+C x^2\right )}{x^5} \, dx\)

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

\(\displaystyle \frac {\frac {1}{2} b \int -\frac {A b c-4 a (c C+B d)-8 a C d x}{x \sqrt {b x^2+a}}dx+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {1}{2} b \int \frac {A b c-4 a (c C+B d)-8 a C d x}{x \sqrt {b x^2+a}}dx+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {-\frac {1}{2} b \left ((A b c-4 a (B d+c C)) \int \frac {1}{x \sqrt {b x^2+a}}dx-8 a C d \int \frac {1}{\sqrt {b x^2+a}}dx\right )+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {1}{2} b \left ((A b c-4 a (B d+c C)) \int \frac {1}{x \sqrt {b x^2+a}}dx-8 a C d \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}\right )+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {1}{2} b \left ((A b c-4 a (B d+c C)) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {8 a C d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-\frac {1}{2} b \left (\frac {1}{2} (A b c-4 a (B d+c C)) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {8 a C d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {1}{2} b \left (\frac {(A b c-4 a (B d+c C)) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {8 a C d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {1}{2} b \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (A b c-4 a (B d+c C))}{\sqrt {a}}-\frac {8 a C d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}\right )+\frac {\sqrt {a+b x^2} (-4 a (B d+c C)-8 a C d x+A b c)}{2 x^2}-\frac {4 \left (a+b x^2\right )^{3/2} (A d+B c)}{3 x^3}}{4 a}-\frac {A c \left (a+b x^2\right )^{3/2}}{4 a x^4}\)

Input:

Int[((c + d*x)*Sqrt[a + b*x^2]*(A + B*x + C*x^2))/x^5,x]
 

Output:

-1/4*(A*c*(a + b*x^2)^(3/2))/(a*x^4) + (((A*b*c - 4*a*(c*C + B*d) - 8*a*C* 
d*x)*Sqrt[a + b*x^2])/(2*x^2) - (4*(B*c + A*d)*(a + b*x^2)^(3/2))/(3*x^3) 
- (b*((-8*a*C*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b] - ((A*b*c - 
4*a*(c*C + B*d))*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2)/(4*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 537

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), 
 x] - Simp[2*b*(p/((m + 1)*(m + 2)))   Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) 
*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && 
 GtQ[p, 0] &&  !ILtQ[m + 2*p + 3, 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 = 0.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (8 A b d \,x^{3}+8 B b c \,x^{3}+24 C a d \,x^{3}+3 A b c \,x^{2}+12 B a d \,x^{2}+12 C a c \,x^{2}+8 A a d x +8 B a c x +6 A a c \right )}{24 x^{4} a}-\frac {b \left (-\frac {\left (A b c -4 B a d -4 C a c \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}-\frac {8 a C d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\right )}{8 a}\)	\(159\)
default	\(-\frac {\left (A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}}+\left (B d +C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )+A c \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}\right )+d C \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )\)	\(247\)

Input:

int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/24*(b*x^2+a)^(1/2)*(8*A*b*d*x^3+8*B*b*c*x^3+24*C*a*d*x^3+3*A*b*c*x^2+12 
*B*a*d*x^2+12*C*a*c*x^2+8*A*a*d*x+8*B*a*c*x+6*A*a*c)/x^4/a-1/8*b/a*(-(A*b* 
c-4*B*a*d-4*C*a*c)/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-8*a*C*d*l 
n(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 734, normalized size of antiderivative = 4.17 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5,x, algorithm="fricas")
 

Output:

[1/48*(24*C*a^2*sqrt(b)*d*x^4*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - 
 a) + 3*(4*B*a*b*d + (4*C*a*b - A*b^2)*c)*sqrt(a)*x^4*log(-(b*x^2 - 2*sqrt 
(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(6*A*a^2*c + 8*(B*a*b*c + (3*C*a^2 + A 
*a*b)*d)*x^3 + 3*(4*B*a^2*d + (4*C*a^2 + A*a*b)*c)*x^2 + 8*(B*a^2*c + A*a^ 
2*d)*x)*sqrt(b*x^2 + a))/(a^2*x^4), -1/48*(48*C*a^2*sqrt(-b)*d*x^4*arctan( 
sqrt(-b)*x/sqrt(b*x^2 + a)) - 3*(4*B*a*b*d + (4*C*a*b - A*b^2)*c)*sqrt(a)* 
x^4*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(6*A*a^2*c + 8 
*(B*a*b*c + (3*C*a^2 + A*a*b)*d)*x^3 + 3*(4*B*a^2*d + (4*C*a^2 + A*a*b)*c) 
*x^2 + 8*(B*a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + a))/(a^2*x^4), 1/24*(12*C*a^2 
*sqrt(b)*d*x^4*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(4*B*a* 
b*d + (4*C*a*b - A*b^2)*c)*sqrt(-a)*x^4*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) 
 - (6*A*a^2*c + 8*(B*a*b*c + (3*C*a^2 + A*a*b)*d)*x^3 + 3*(4*B*a^2*d + (4* 
C*a^2 + A*a*b)*c)*x^2 + 8*(B*a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + a))/(a^2*x^4 
), -1/24*(24*C*a^2*sqrt(-b)*d*x^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 3*( 
4*B*a*b*d + (4*C*a*b - A*b^2)*c)*sqrt(-a)*x^4*arctan(sqrt(b*x^2 + a)*sqrt( 
-a)/a) + (6*A*a^2*c + 8*(B*a*b*c + (3*C*a^2 + A*a*b)*d)*x^3 + 3*(4*B*a^2*d 
 + (4*C*a^2 + A*a*b)*c)*x^2 + 8*(B*a^2*c + A*a^2*d)*x)*sqrt(b*x^2 + a))/(a 
^2*x^4)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (160) = 320\).

Time = 4.96 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.11 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=- \frac {A a c}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A \sqrt {b} c}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A b^{\frac {3}{2}} c}{8 a x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} + \frac {A b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {3}{2}}} - \frac {B \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {B \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {B b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} - \frac {B b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {C \sqrt {a} d}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {C \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + C \sqrt {b} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {C b c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {C b d x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

integrate((d*x+c)*(b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**5,x)
 

Output:

-A*a*c/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*A*sqrt(b)*c/(8*x**3*sqrt( 
a/(b*x**2) + 1)) - A*sqrt(b)*d*sqrt(a/(b*x**2) + 1)/(3*x**2) - A*b**(3/2)* 
c/(8*a*x*sqrt(a/(b*x**2) + 1)) - A*b**(3/2)*d*sqrt(a/(b*x**2) + 1)/(3*a) + 
 A*b**2*c*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(3/2)) - B*sqrt(b)*c*sqrt(a/(b* 
x**2) + 1)/(3*x**2) - B*sqrt(b)*d*sqrt(a/(b*x**2) + 1)/(2*x) - B*b**(3/2)* 
c*sqrt(a/(b*x**2) + 1)/(3*a) - B*b*d*asinh(sqrt(a)/(sqrt(b)*x))/(2*sqrt(a) 
) - C*sqrt(a)*d/(x*sqrt(1 + b*x**2/a)) - C*sqrt(b)*c*sqrt(a/(b*x**2) + 1)/ 
(2*x) + C*sqrt(b)*d*asinh(sqrt(b)*x/sqrt(a)) - C*b*c*asinh(sqrt(a)/(sqrt(b 
)*x))/(2*sqrt(a)) - C*b*d*x/(sqrt(a)*sqrt(1 + b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=C \sqrt {b} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + \frac {A b^{2} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} A b^{2} c}{8 \, a^{2}} - \frac {{\left (C c + B d\right )} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + \frac {\sqrt {b x^{2} + a} {\left (C c + B d\right )} b}{2 \, a} - \frac {\sqrt {b x^{2} + a} C d}{x} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b c}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B c}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A d}{3 \, a x^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (C c + B d\right )}}{2 \, a x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c}{4 \, a x^{4}} \] Input:

integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5,x, algorithm="maxima")
 

Output:

C*sqrt(b)*d*arcsinh(b*x/sqrt(a*b)) + 1/8*A*b^2*c*arcsinh(a/(sqrt(a*b)*abs( 
x)))/a^(3/2) - 1/8*sqrt(b*x^2 + a)*A*b^2*c/a^2 - 1/2*(C*c + B*d)*b*arcsinh 
(a/(sqrt(a*b)*abs(x)))/sqrt(a) + 1/2*sqrt(b*x^2 + a)*(C*c + B*d)*b/a - sqr 
t(b*x^2 + a)*C*d/x + 1/8*(b*x^2 + a)^(3/2)*A*b*c/(a^2*x^2) - 1/3*(b*x^2 + 
a)^(3/2)*B*c/(a*x^3) - 1/3*(b*x^2 + a)^(3/2)*A*d/(a*x^3) - 1/2*(b*x^2 + a) 
^(3/2)*(C*c + B*d)/(a*x^2) - 1/4*(b*x^2 + a)^(3/2)*A*c/(a*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (148) = 296\).

Time = 0.23 (sec) , antiderivative size = 716, normalized size of antiderivative = 4.07 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=-C \sqrt {b} d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {{\left (4 \, C a b c - A b^{2} c + 4 \, B a b d\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a} + \frac {12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} C a b c + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} A b^{2} c + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} B a b d + 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a b^{\frac {3}{2}} c + 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{2} \sqrt {b} d + 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a b^{\frac {3}{2}} d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} C a^{2} b c + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A a b^{2} c - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B a^{2} b d - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{2} b^{\frac {3}{2}} c - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{3} \sqrt {b} d - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {3}{2}} d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} C a^{3} b c + 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a^{2} b^{2} c - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B a^{3} b d + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{3} b^{\frac {3}{2}} c + 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{4} \sqrt {b} d + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {3}{2}} d + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} C a^{4} b c + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{3} b^{2} c + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{4} b d - 8 \, B a^{4} b^{\frac {3}{2}} c - 24 \, C a^{5} \sqrt {b} d - 8 \, A a^{4} b^{\frac {3}{2}} d}{12 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{4} a} \] Input:

integrate((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5,x, algorithm="giac")
 

Output:

-C*sqrt(b)*d*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a))) + 1/4*(4*C*a*b*c - A*b 
^2*c + 4*B*a*b*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a 
)*a) + 1/12*(12*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a*b*c + 3*(sqrt(b)*x - s 
qrt(b*x^2 + a))^7*A*b^2*c + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^7*B*a*b*d + 2 
4*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a*b^(3/2)*c + 24*(sqrt(b)*x - sqrt(b*x 
^2 + a))^6*C*a^2*sqrt(b)*d + 24*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a*b^(3/2 
)*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^2*b*c + 21*(sqrt(b)*x - sqrt( 
b*x^2 + a))^5*A*a*b^2*c - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B*a^2*b*d - 2 
4*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*b^(3/2)*c - 72*(sqrt(b)*x - sqrt(b 
*x^2 + a))^4*C*a^3*sqrt(b)*d - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^ 
(3/2)*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*a^3*b*c + 21*(sqrt(b)*x - s 
qrt(b*x^2 + a))^3*A*a^2*b^2*c - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a^3*b 
*d + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^3*b^(3/2)*c + 72*(sqrt(b)*x - s 
qrt(b*x^2 + a))^2*C*a^4*sqrt(b)*d + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^ 
3*b^(3/2)*d + 12*(sqrt(b)*x - sqrt(b*x^2 + a))*C*a^4*b*c + 3*(sqrt(b)*x - 
sqrt(b*x^2 + a))*A*a^3*b^2*c + 12*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^4*b*d 
- 8*B*a^4*b^(3/2)*c - 24*C*a^5*sqrt(b)*d - 8*A*a^4*b^(3/2)*d)/(((sqrt(b)*x 
 - sqrt(b*x^2 + a))^2 - a)^4*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^5} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^5,x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x)*(A + B*x + C*x^2))/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.32 \[ \int \frac {(c+d x) \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^5} \, dx=\frac {-6 \sqrt {b \,x^{2}+a}\, a^{2} c -8 \sqrt {b \,x^{2}+a}\, a^{2} d x -3 \sqrt {b \,x^{2}+a}\, a b c \,x^{2}-8 \sqrt {b \,x^{2}+a}\, a b c x -8 \sqrt {b \,x^{2}+a}\, a b d \,x^{3}-12 \sqrt {b \,x^{2}+a}\, a b d \,x^{2}-12 \sqrt {b \,x^{2}+a}\, a \,c^{2} x^{2}-24 \sqrt {b \,x^{2}+a}\, a c d \,x^{3}-8 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{4}+12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d \,x^{4}+12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} x^{4}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c \,x^{4}-12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} d \,x^{4}-12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2} x^{4}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a c d \,x^{4}-4 \sqrt {b}\, a b d \,x^{4}+12 \sqrt {b}\, a c d \,x^{4}-4 \sqrt {b}\, b^{2} c \,x^{4}}{24 a \,x^{4}} \] Input:

int((d*x+c)*(b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^5,x)
 

Output:

( - 6*sqrt(a + b*x**2)*a**2*c - 8*sqrt(a + b*x**2)*a**2*d*x - 3*sqrt(a + b 
*x**2)*a*b*c*x**2 - 8*sqrt(a + b*x**2)*a*b*c*x - 8*sqrt(a + b*x**2)*a*b*d* 
x**3 - 12*sqrt(a + b*x**2)*a*b*d*x**2 - 12*sqrt(a + b*x**2)*a*c**2*x**2 - 
24*sqrt(a + b*x**2)*a*c*d*x**3 - 8*sqrt(a + b*x**2)*b**2*c*x**3 - 3*sqrt(a 
)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*x**4 + 12*s 
qrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*d*x**4 + 
 12*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**2*x 
**4 + 3*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2 
*c*x**4 - 12*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a)) 
*b**2*d*x**4 - 12*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqr 
t(a))*b*c**2*x**4 + 24*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a)) 
*a*c*d*x**4 - 4*sqrt(b)*a*b*d*x**4 + 12*sqrt(b)*a*c*d*x**4 - 4*sqrt(b)*b** 
2*c*x**4)/(24*a*x**4)