\(\int \frac {(c+d x) (A+B x+C x^2)}{x^6 \sqrt {a+b x^2}} \, dx\) [103]

Optimal result

Integrand size = 30, antiderivative size = 208 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=-\frac {A c \sqrt {a+b x^2}}{5 a x^5}-\frac {(B c+A d) \sqrt {a+b x^2}}{4 a x^4}+\frac {(4 A b c-5 a (c C+B d)) \sqrt {a+b x^2}}{15 a^2 x^3}-\frac {(4 a C d-3 b (B c+A d)) \sqrt {a+b x^2}}{8 a^2 x^2}-\frac {2 b (4 A b c-5 a (c C+B d)) \sqrt {a+b x^2}}{15 a^3 x}+\frac {b (4 a C d-3 b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \] Output:

-1/5*A*c*(b*x^2+a)^(1/2)/a/x^5-1/4*(A*d+B*c)*(b*x^2+a)^(1/2)/a/x^4+1/15*(4 
*A*b*c-5*a*(B*d+C*c))*(b*x^2+a)^(1/2)/a^2/x^3-1/8*(4*a*C*d-3*b*(A*d+B*c))* 
(b*x^2+a)^(1/2)/a^2/x^2-2/15*b*(4*A*b*c-5*a*(B*d+C*c))*(b*x^2+a)^(1/2)/a^3 
/x+1/8*b*(4*a*C*d-3*b*(A*d+B*c))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-64 A b^2 c x^4+a b x^2 (A (32 c+45 d x)+5 x (9 B c+16 c C x+16 B d x))-2 a^2 \left (3 A (4 c+5 d x)+5 x \left (3 B c+4 c C x+4 B d x+6 C d x^2\right )\right )\right )}{120 a^3 x^5}-\frac {b C d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {3 b^2 (B c+A d) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Input:

Integrate[((c + d*x)*(A + B*x + C*x^2))/(x^6*Sqrt[a + b*x^2]),x]
 

Output:

(Sqrt[a + b*x^2]*(-64*A*b^2*c*x^4 + a*b*x^2*(A*(32*c + 45*d*x) + 5*x*(9*B* 
c + 16*c*C*x + 16*B*d*x)) - 2*a^2*(3*A*(4*c + 5*d*x) + 5*x*(3*B*c + 4*c*C* 
x + 4*B*d*x + 6*C*d*x^2))))/(120*a^3*x^5) - (b*C*d*ArcTanh[(Sqrt[b]*x - Sq 
rt[a + b*x^2])/Sqrt[a]])/a^(3/2) - (3*b^2*(B*c + A*d)*ArcTanh[(-(Sqrt[b]*x 
) + Sqrt[a + b*x^2])/Sqrt[a]])/(4*a^(5/2))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07, 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, 539, 539, 25, 27, 534, 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.

\(\displaystyle \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx\)

\(\Big \downarrow \) 2338

\(\displaystyle -\frac {\int -\frac {5 a C d x^2-(4 A b c-5 a (c C+B d)) x+5 a (B c+A d)}{x^5 \sqrt {b x^2+a}}dx}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 a C d x^2-(4 A b c-5 a (c C+B d)) x+5 a (B c+A d)}{x^5 \sqrt {b x^2+a}}dx}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int \frac {a (4 (4 A b c-5 a (c C+B d))-5 (4 a C d-3 b (B c+A d)) x)}{x^4 \sqrt {b x^2+a}}dx}{4 a}-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{4} \int \frac {4 (4 A b c-5 a (c C+B d))-5 (4 a C d-3 b (B c+A d)) x}{x^4 \sqrt {b x^2+a}}dx-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\int \frac {15 a (4 a C d-3 b (B c+A d))+8 b (4 A b c-5 a (c C+B d)) x}{x^3 \sqrt {b x^2+a}}dx}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 539

\(\displaystyle \frac {\frac {1}{4} \left (\frac {-\frac {\int -\frac {a b (16 (4 A b c-5 a (c C+B d))-15 (4 a C d-3 b (B c+A d)) x)}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {\int \frac {a b (16 (4 A b c-5 a (c C+B d))-15 (4 a C d-3 b (B c+A d)) x)}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {1}{2} b \int \frac {16 (4 A b c-5 a (c C+B d))-15 (4 a C d-3 b (B c+A d)) x}{x^2 \sqrt {b x^2+a}}dx-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {1}{2} b \left (-15 (4 a C d-3 b (A d+B c)) \int \frac {1}{x \sqrt {b x^2+a}}dx-\frac {16 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{a x}\right )-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {1}{2} b \left (-\frac {15}{2} (4 a C d-3 b (A d+B c)) \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {16 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{a x}\right )-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {1}{2} b \left (-\frac {15 (4 a C d-3 b (A d+B c)) \int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}}{b}-\frac {16 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{a x}\right )-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {1}{4} \left (\frac {\frac {1}{2} b \left (\frac {15 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (4 a C d-3 b (A d+B c))}{\sqrt {a}}-\frac {16 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{a x}\right )-\frac {15 \sqrt {a+b x^2} (4 a C d-3 b (A d+B c))}{2 x^2}}{3 a}+\frac {4 \sqrt {a+b x^2} (4 A b c-5 a (B d+c C))}{3 a x^3}\right )-\frac {5 \sqrt {a+b x^2} (A d+B c)}{4 x^4}}{5 a}-\frac {A c \sqrt {a+b x^2}}{5 a x^5}\)

Input:

Int[((c + d*x)*(A + B*x + C*x^2))/(x^6*Sqrt[a + b*x^2]),x]
 

Output:

-1/5*(A*c*Sqrt[a + b*x^2])/(a*x^5) + ((-5*(B*c + A*d)*Sqrt[a + b*x^2])/(4* 
x^4) + ((4*(4*A*b*c - 5*a*(c*C + B*d))*Sqrt[a + b*x^2])/(3*a*x^3) + ((-15* 
(4*a*C*d - 3*b*(B*c + A*d))*Sqrt[a + b*x^2])/(2*x^2) + (b*((-16*(4*A*b*c - 
 5*a*(c*C + B*d))*Sqrt[a + b*x^2])/(a*x) + (15*(4*a*C*d - 3*b*(B*c + A*d)) 
*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/Sqrt[a]))/2)/(3*a))/4)/(5*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 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 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 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 539

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) 
   Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] 
/; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

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 = 0.36 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (64 A \,b^{2} c \,x^{4}-80 B a b d \,x^{4}-80 C a b c \,x^{4}-45 A b d \,x^{3} a -45 B b c \,x^{3} a +60 C \,a^{2} d \,x^{3}-32 A a b c \,x^{2}+40 B \,a^{2} d \,x^{2}+40 C \,a^{2} c \,x^{2}+30 A \,a^{2} d x +30 B \,a^{2} c x +24 A \,a^{2} c \right )}{120 a^{3} x^{5}}-\frac {\left (3 A b d +3 B b c -4 a C d \right ) b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{8 a^{\frac {5}{2}}}\)	\(173\)
default	\(\left (A d +B c \right ) \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+\left (B d +C c \right ) \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )+A c \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )+d C \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )\)	\(238\)

Input:

int((d*x+c)*(C*x^2+B*x+A)/x^6/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(b*x^2+a)^(1/2)*(64*A*b^2*c*x^4-80*B*a*b*d*x^4-80*C*a*b*c*x^4-45*A* 
a*b*d*x^3-45*B*a*b*c*x^3+60*C*a^2*d*x^3-32*A*a*b*c*x^2+40*B*a^2*d*x^2+40*C 
*a^2*c*x^2+30*A*a^2*d*x+30*B*a^2*c*x+24*A*a^2*c)/a^3/x^5-1/8*(3*A*b*d+3*B* 
b*c-4*C*a*d)/a^(5/2)*b*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.73 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=\left [\frac {15 \, {\left (3 \, B b^{2} c - {\left (4 \, C a b - 3 \, A b^{2}\right )} d\right )} \sqrt {a} x^{5} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (16 \, {\left (5 \, B a b d + {\left (5 \, C a b - 4 \, A b^{2}\right )} c\right )} x^{4} - 24 \, A a^{2} c + 15 \, {\left (3 \, B a b c - {\left (4 \, C a^{2} - 3 \, A a b\right )} d\right )} x^{3} - 8 \, {\left (5 \, B a^{2} d + {\left (5 \, C a^{2} - 4 \, A a b\right )} c\right )} x^{2} - 30 \, {\left (B a^{2} c + A a^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{240 \, a^{3} x^{5}}, \frac {15 \, {\left (3 \, B b^{2} c - {\left (4 \, C a b - 3 \, A b^{2}\right )} d\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (16 \, {\left (5 \, B a b d + {\left (5 \, C a b - 4 \, A b^{2}\right )} c\right )} x^{4} - 24 \, A a^{2} c + 15 \, {\left (3 \, B a b c - {\left (4 \, C a^{2} - 3 \, A a b\right )} d\right )} x^{3} - 8 \, {\left (5 \, B a^{2} d + {\left (5 \, C a^{2} - 4 \, A a b\right )} c\right )} x^{2} - 30 \, {\left (B a^{2} c + A a^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{120 \, a^{3} x^{5}}\right ] \] Input:

integrate((d*x+c)*(C*x^2+B*x+A)/x^6/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/240*(15*(3*B*b^2*c - (4*C*a*b - 3*A*b^2)*d)*sqrt(a)*x^5*log(-(b*x^2 - 2 
*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(16*(5*B*a*b*d + (5*C*a*b - 4*A*b 
^2)*c)*x^4 - 24*A*a^2*c + 15*(3*B*a*b*c - (4*C*a^2 - 3*A*a*b)*d)*x^3 - 8*( 
5*B*a^2*d + (5*C*a^2 - 4*A*a*b)*c)*x^2 - 30*(B*a^2*c + A*a^2*d)*x)*sqrt(b* 
x^2 + a))/(a^3*x^5), 1/120*(15*(3*B*b^2*c - (4*C*a*b - 3*A*b^2)*d)*sqrt(-a 
)*x^5*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (16*(5*B*a*b*d + (5*C*a*b - 4*A 
*b^2)*c)*x^4 - 24*A*a^2*c + 15*(3*B*a*b*c - (4*C*a^2 - 3*A*a*b)*d)*x^3 - 8 
*(5*B*a^2*d + (5*C*a^2 - 4*A*a*b)*c)*x^2 - 30*(B*a^2*c + A*a^2*d)*x)*sqrt( 
b*x^2 + a))/(a^3*x^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (196) = 392\).

Time = 7.20 (sec) , antiderivative size = 694, normalized size of antiderivative = 3.34 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=- \frac {3 A a^{4} b^{\frac {9}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {2 A a^{3} b^{\frac {11}{2}} c x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {3 A a^{2} b^{\frac {13}{2}} c x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {12 A a b^{\frac {15}{2}} c x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {8 A b^{\frac {17}{2}} c x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {A d}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A \sqrt {b} d}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 A b^{\frac {3}{2}} d}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} - \frac {B c}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B \sqrt {b} c}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} + \frac {3 B b^{\frac {3}{2}} c}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {2 B b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} - \frac {3 B b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} - \frac {C \sqrt {b} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} - \frac {C \sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} + \frac {2 C b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} + \frac {C b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} \] Input:

integrate((d*x+c)*(C*x**2+B*x+A)/x**6/(b*x**2+a)**(1/2),x)
 

Output:

-3*A*a**4*b**(9/2)*c*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b** 
5*x**6 + 15*a**3*b**6*x**8) - 2*A*a**3*b**(11/2)*c*x**2*sqrt(a/(b*x**2) + 
1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 3*A*a**2* 
b**(13/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x* 
*6 + 15*a**3*b**6*x**8) - 12*A*a*b**(15/2)*c*x**6*sqrt(a/(b*x**2) + 1)/(15 
*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 8*A*b**(17/2)*c 
*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a** 
3*b**6*x**8) - A*d/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + A*sqrt(b)*d/(8* 
a*x**3*sqrt(a/(b*x**2) + 1)) + 3*A*b**(3/2)*d/(8*a**2*x*sqrt(a/(b*x**2) + 
1)) - 3*A*b**2*d*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(5/2)) - B*c/(4*sqrt(b)* 
x**5*sqrt(a/(b*x**2) + 1)) + B*sqrt(b)*c/(8*a*x**3*sqrt(a/(b*x**2) + 1)) - 
 B*sqrt(b)*d*sqrt(a/(b*x**2) + 1)/(3*a*x**2) + 3*B*b**(3/2)*c/(8*a**2*x*sq 
rt(a/(b*x**2) + 1)) + 2*B*b**(3/2)*d*sqrt(a/(b*x**2) + 1)/(3*a**2) - 3*B*b 
**2*c*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(5/2)) - C*sqrt(b)*c*sqrt(a/(b*x**2 
) + 1)/(3*a*x**2) - C*sqrt(b)*d*sqrt(a/(b*x**2) + 1)/(2*a*x) + 2*C*b**(3/2 
)*c*sqrt(a/(b*x**2) + 1)/(3*a**2) + C*b*d*asinh(sqrt(a)/(sqrt(b)*x))/(2*a* 
*(3/2))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=\frac {C b d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {3 \, {\left (B c + A d\right )} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} - \frac {8 \, \sqrt {b x^{2} + a} A b^{2} c}{15 \, a^{3} x} - \frac {\sqrt {b x^{2} + a} C d}{2 \, a x^{2}} + \frac {2 \, \sqrt {b x^{2} + a} {\left (C c + B d\right )} b}{3 \, a^{2} x} + \frac {4 \, \sqrt {b x^{2} + a} A b c}{15 \, a^{2} x^{3}} + \frac {3 \, \sqrt {b x^{2} + a} {\left (B c + A d\right )} b}{8 \, a^{2} x^{2}} - \frac {\sqrt {b x^{2} + a} {\left (C c + B d\right )}}{3 \, a x^{3}} - \frac {\sqrt {b x^{2} + a} A c}{5 \, a x^{5}} - \frac {\sqrt {b x^{2} + a} {\left (B c + A d\right )}}{4 \, a x^{4}} \] Input:

integrate((d*x+c)*(C*x^2+B*x+A)/x^6/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

1/2*C*b*d*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 3/8*(B*c + A*d)*b^2*arcs 
inh(a/(sqrt(a*b)*abs(x)))/a^(5/2) - 8/15*sqrt(b*x^2 + a)*A*b^2*c/(a^3*x) - 
 1/2*sqrt(b*x^2 + a)*C*d/(a*x^2) + 2/3*sqrt(b*x^2 + a)*(C*c + B*d)*b/(a^2* 
x) + 4/15*sqrt(b*x^2 + a)*A*b*c/(a^2*x^3) + 3/8*sqrt(b*x^2 + a)*(B*c + A*d 
)*b/(a^2*x^2) - 1/3*sqrt(b*x^2 + a)*(C*c + B*d)/(a*x^3) - 1/5*sqrt(b*x^2 + 
 a)*A*c/(a*x^5) - 1/4*sqrt(b*x^2 + a)*(B*c + A*d)/(a*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (180) = 360\).

Time = 0.15 (sec) , antiderivative size = 672, normalized size of antiderivative = 3.23 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=\frac {{\left (3 \, B b^{2} c - 4 \, C a b d + 3 \, A b^{2} d\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} - \frac {45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} B b^{2} c - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} C a b d + 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{9} A b^{2} d - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} B a b^{2} c + 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} C a^{2} b d - 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{7} A a b^{2} d - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} C a^{2} b^{\frac {3}{2}} c - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{2} b^{\frac {3}{2}} d + 560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a^{3} b^{\frac {3}{2}} c - 640 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {5}{2}} c + 560 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} b^{\frac {3}{2}} d + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B a^{3} b^{2} c - 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} C a^{4} b d + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a^{3} b^{2} d - 400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{4} b^{\frac {3}{2}} c + 320 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {5}{2}} c - 400 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} b^{\frac {3}{2}} d - 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{4} b^{2} c + 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} C a^{5} b d - 45 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{4} b^{2} d + 80 \, C a^{5} b^{\frac {3}{2}} c - 64 \, A a^{4} b^{\frac {5}{2}} c + 80 \, B a^{5} b^{\frac {3}{2}} d}{60 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{2}} \] Input:

integrate((d*x+c)*(C*x^2+B*x+A)/x^6/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

1/4*(3*B*b^2*c - 4*C*a*b*d + 3*A*b^2*d)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + 
a))/sqrt(-a))/(sqrt(-a)*a^2) - 1/60*(45*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B* 
b^2*c - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9*C*a*b*d + 45*(sqrt(b)*x - sqrt( 
b*x^2 + a))^9*A*b^2*d - 210*(sqrt(b)*x - sqrt(b*x^2 + a))^7*B*a*b^2*c + 12 
0*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^2*b*d - 210*(sqrt(b)*x - sqrt(b*x^2 
+ a))^7*A*a*b^2*d - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^2*b^(3/2)*c - 
240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2)*d + 560*(sqrt(b)*x - sqr 
t(b*x^2 + a))^4*C*a^3*b^(3/2)*c - 640*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^ 
2*b^(5/2)*c + 560*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2)*d + 210*(s 
qrt(b)*x - sqrt(b*x^2 + a))^3*B*a^3*b^2*c - 120*(sqrt(b)*x - sqrt(b*x^2 + 
a))^3*C*a^4*b*d + 210*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^3*b^2*d - 400*(s 
qrt(b)*x - sqrt(b*x^2 + a))^2*C*a^4*b^(3/2)*c + 320*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^2*A*a^3*b^(5/2)*c - 400*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3 
/2)*d - 45*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^4*b^2*c + 60*(sqrt(b)*x - sqr 
t(b*x^2 + a))*C*a^5*b*d - 45*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^4*b^2*d + 8 
0*C*a^5*b^(3/2)*c - 64*A*a^4*b^(5/2)*c + 80*B*a^5*b^(3/2)*d)/(((sqrt(b)*x 
- sqrt(b*x^2 + a))^2 - a)^5*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=\int \frac {\left (c+d\,x\right )\,\left (C\,x^2+B\,x+A\right )}{x^6\,\sqrt {b\,x^2+a}} \,d x \] Input:

int(((c + d*x)*(A + B*x + C*x^2))/(x^6*(a + b*x^2)^(1/2)),x)
 

Output:

int(((c + d*x)*(A + B*x + C*x^2))/(x^6*(a + b*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.18 \[ \int \frac {(c+d x) \left (A+B x+C x^2\right )}{x^6 \sqrt {a+b x^2}} \, dx=\frac {-24 \sqrt {b \,x^{2}+a}\, a^{3} c -30 \sqrt {b \,x^{2}+a}\, a^{3} d x +32 \sqrt {b \,x^{2}+a}\, a^{2} b c \,x^{2}-30 \sqrt {b \,x^{2}+a}\, a^{2} b c x +45 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{3}-40 \sqrt {b \,x^{2}+a}\, a^{2} b d \,x^{2}-40 \sqrt {b \,x^{2}+a}\, a^{2} c^{2} x^{2}-60 \sqrt {b \,x^{2}+a}\, a^{2} c d \,x^{3}-64 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{4}+45 \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,x^{3}+80 \sqrt {b \,x^{2}+a}\, a \,b^{2} d \,x^{4}+80 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x^{4}+45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{5}-60 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{5}+45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{5}+60 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c d \,x^{5}-45 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c \,x^{5}+64 \sqrt {b}\, a \,b^{2} c \,x^{5}-80 \sqrt {b}\, a \,b^{2} d \,x^{5}-80 \sqrt {b}\, a b \,c^{2} x^{5}}{120 a^{3} x^{5}} \] Input:

int((d*x+c)*(C*x^2+B*x+A)/x^6/(b*x^2+a)^(1/2),x)
 

Output:

( - 24*sqrt(a + b*x**2)*a**3*c - 30*sqrt(a + b*x**2)*a**3*d*x + 32*sqrt(a 
+ b*x**2)*a**2*b*c*x**2 - 30*sqrt(a + b*x**2)*a**2*b*c*x + 45*sqrt(a + b*x 
**2)*a**2*b*d*x**3 - 40*sqrt(a + b*x**2)*a**2*b*d*x**2 - 40*sqrt(a + b*x** 
2)*a**2*c**2*x**2 - 60*sqrt(a + b*x**2)*a**2*c*d*x**3 - 64*sqrt(a + b*x**2 
)*a*b**2*c*x**4 + 45*sqrt(a + b*x**2)*a*b**2*c*x**3 + 80*sqrt(a + b*x**2)* 
a*b**2*d*x**4 + 80*sqrt(a + b*x**2)*a*b*c**2*x**4 + 45*sqrt(a)*log((sqrt(a 
 + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*d*x**5 - 60*sqrt(a)*log( 
(sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c*d*x**5 + 45*sqrt(a 
)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**3*c*x**5 - 45*s 
qrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**2*d*x**5 
 + 60*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*c* 
d*x**5 - 45*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))* 
b**3*c*x**5 + 64*sqrt(b)*a*b**2*c*x**5 - 80*sqrt(b)*a*b**2*d*x**5 - 80*sqr 
t(b)*a*b*c**2*x**5)/(120*a**3*x**5)