\(\int \frac {(c+d x)^2 (A+B x+C x^2)}{x^3 \sqrt {a+b x^2}} \, dx\) [110]

Optimal result

Integrand size = 32, antiderivative size = 162 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=\frac {C d^2 \sqrt {a+b x^2}}{b}-\frac {A c^2 \sqrt {a+b x^2}}{2 a x^2}-\frac {c (B c+2 A d) \sqrt {a+b x^2}}{a x}+\frac {d (2 c C+B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {\left (2 a c (c C+2 B d)-A \left (b c^2-2 a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-A b c (c+4 d x)+2 x \left (-b B c^2+a C d^2 x\right )\right )}{2 a b x^2}-\frac {A b c^2 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 \left (c^2 C+2 B c d+A d^2\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {d (2 c C+B d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2338, 25, 2338, 25, 2340, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 538

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

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

rule 2340

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {c \sqrt {b \,x^{2}+a}\, \left (4 A d x +2 B c x +A c \right )}{2 a \,x^{2}}+\frac {-\frac {\left (2 A a \,d^{2}-b A \,c^{2}+4 B a c d +2 C a \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+\frac {2 a B \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {4 C a c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {2 a C \,d^{2} \sqrt {b \,x^{2}+a}}{b}}{2 a}\)	\(166\)
default	\(\frac {d^{2} B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+A \,c^{2} \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )+\frac {C \,d^{2} \sqrt {b \,x^{2}+a}}{b}-\frac {c \left (2 A d +B c \right ) \sqrt {b \,x^{2}+a}}{a x}+\frac {2 c d C \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\)	\(189\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.29 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=\left [\frac {2 \, {\left (2 \, C a^{2} c d + B a^{2} d^{2}\right )} \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + {\left (4 \, B a b c d + 2 \, A a b d^{2} + {\left (2 \, C a b - A b^{2}\right )} c^{2}\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 (2 \, C a^{2} d^{2} x^{2} - A a b c^{2} - 2 \, {\left (B a b c^{2} + 2 \, A a b c d\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, a^{2} b x^{2}}, -\frac {4 \, {\left (2 \, C a^{2} c d + B a^{2} d^{2}\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (4 \, B a b c d + 2 \, A a b d^{2} + {\left (2 \, C a b - A b^{2}\right )} c^{2}\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 (2 \, C a^{2} d^{2} x^{2} - A a b c^{2} - 2 \, {\left (B a b c^{2} + 2 \, A a b c d\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, a^{2} b x^{2}}, \frac {{\left (4 \, B a b c d + 2 \, A a b d^{2} + {\left (2 \, C a b - A b^{2}\right )} c^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (2 \, C a^{2} c d + B a^{2} d^{2}\right )} \sqrt {b} x^{2} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + {\left (2 \, C a^{2} d^{2} x^{2} - A a b c^{2} - 2 \, {\left (B a b c^{2} + 2 \, A a b c d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, a^{2} b x^{2}}, -\frac {2 \, {\left (2 \, C a^{2} c d + B a^{2} d^{2}\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (4 \, B a b c d + 2 \, A a b d^{2} + {\left (2 \, C a b - A b^{2}\right )} c^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - {\left (2 \, C a^{2} d^{2} x^{2} - A a b c^{2} - 2 \, {\left (B a b c^{2} + 2 \, A a b c d\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, a^{2} b x^{2}}\right ] \] Input:

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

Output:

[1/4*(2*(2*C*a^2*c*d + B*a^2*d^2)*sqrt(b)*x^2*log(-2*b*x^2 - 2*sqrt(b*x^2 
+ a)*sqrt(b)*x - a) + (4*B*a*b*c*d + 2*A*a*b*d^2 + (2*C*a*b - A*b^2)*c^2)* 
sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(2*C*a 
^2*d^2*x^2 - A*a*b*c^2 - 2*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/( 
a^2*b*x^2), -1/4*(4*(2*C*a^2*c*d + B*a^2*d^2)*sqrt(-b)*x^2*arctan(sqrt(-b) 
*x/sqrt(b*x^2 + a)) - (4*B*a*b*c*d + 2*A*a*b*d^2 + (2*C*a*b - A*b^2)*c^2)* 
sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2*C*a 
^2*d^2*x^2 - A*a*b*c^2 - 2*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/( 
a^2*b*x^2), 1/2*((4*B*a*b*c*d + 2*A*a*b*d^2 + (2*C*a*b - A*b^2)*c^2)*sqrt( 
-a)*x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (2*C*a^2*c*d + B*a^2*d^2)*sqr 
t(b)*x^2*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (2*C*a^2*d^2*x^ 
2 - A*a*b*c^2 - 2*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(a^2*b*x^2 
), -1/2*(2*(2*C*a^2*c*d + B*a^2*d^2)*sqrt(-b)*x^2*arctan(sqrt(-b)*x/sqrt(b 
*x^2 + a)) - (4*B*a*b*c*d + 2*A*a*b*d^2 + (2*C*a*b - A*b^2)*c^2)*sqrt(-a)* 
x^2*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (2*C*a^2*d^2*x^2 - A*a*b*c^2 - 2* 
(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(a^2*b*x^2)]
 

Sympy [A] (verification not implemented)

Time = 4.31 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=- \frac {A \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} - \frac {2 A \sqrt {b} c d \sqrt {\frac {a}{b x^{2}} + 1}}{a} - \frac {A d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} + \frac {A b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} + B d^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \wedge b \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a}} & \text {otherwise} \end {cases}\right ) - \frac {B \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{a} - \frac {2 B c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} + 2 C c d \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \wedge b \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a}} & \text {otherwise} \end {cases}\right ) + C d^{2} \left (\begin {cases} \frac {\sqrt {a + b x^{2}}}{b} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 \sqrt {a}} & \text {otherwise} \end {cases}\right ) - \frac {C c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} \] Input:

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

Output:

-A*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/(2*a*x) - 2*A*sqrt(b)*c*d*sqrt(a/(b*x 
**2) + 1)/a - A*d**2*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a) + A*b*c**2*asinh(s 
qrt(a)/(sqrt(b)*x))/(2*a**(3/2)) + B*d**2*Piecewise((log(2*sqrt(b)*sqrt(a 
+ b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0) & Ne(b, 0)), (x*log(x)/sqrt(b*x**2), 
Ne(b, 0)), (x/sqrt(a), True)) - B*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/a - 2* 
B*c*d*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a) + 2*C*c*d*Piecewise((log(2*sqrt(b 
)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0) & Ne(b, 0)), (x*log(x)/sqrt( 
b*x**2), Ne(b, 0)), (x/sqrt(a), True)) + C*d**2*Piecewise((sqrt(a + b*x**2 
)/b, Ne(b, 0)), (x**2/(2*sqrt(a)), True)) - C*c**2*asinh(sqrt(a)/(sqrt(b)* 
x))/sqrt(a)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.22 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=\frac {2 \, C c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} + \frac {B d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {C c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} + \frac {A b c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {2 \, B c d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} - \frac {A d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} + \frac {\sqrt {b x^{2} + a} C d^{2}}{b} - \frac {\sqrt {b x^{2} + a} B c^{2}}{a x} - \frac {2 \, \sqrt {b x^{2} + a} A c d}{a x} - \frac {\sqrt {b x^{2} + a} A c^{2}}{2 \, a x^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.69 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} C d^{2}}{b} - \frac {{\left (2 \, C c d + B d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{\sqrt {b}} + \frac {{\left (2 \, C a c^{2} - A b c^{2} + 4 \, B a c d + 2 \, A a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b c^{2} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} c^{2} + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a \sqrt {b} c d + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b c^{2} - 2 \, B a^{2} \sqrt {b} c^{2} - 4 \, A a^{2} \sqrt {b} c d}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.53 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^3 \sqrt {a+b x^2}} \, dx=\frac {-\sqrt {b \,x^{2}+a}\, a b \,c^{2}-4 \sqrt {b \,x^{2}+a}\, a b c d x +2 \sqrt {b \,x^{2}+a}\, a c \,d^{2} x^{2}-2 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x +2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}-\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{2}+4 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{2}+2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{3} x^{2}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}+\sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c^{2} x^{2}-4 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{2} c d \,x^{2}-2 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{3} x^{2}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,d^{2} x^{2}+4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,c^{2} d \,x^{2}}{2 a b \,x^{2}} \] Input:

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

Output:

( - sqrt(a + b*x**2)*a*b*c**2 - 4*sqrt(a + b*x**2)*a*b*c*d*x + 2*sqrt(a + 
b*x**2)*a*c*d**2*x**2 - 2*sqrt(a + b*x**2)*b**2*c**2*x + 2*sqrt(a)*log((sq 
rt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*d**2*x**2 - sqrt(a)*log 
((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c**2*x**2 + 4*sqrt 
(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**2*c*d*x**2 + 
2*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b*c**3*x** 
2 - 2*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b*d* 
*2*x**2 + sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*b* 
*2*c**2*x**2 - 4*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt 
(a))*b**2*c*d*x**2 - 2*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x 
)/sqrt(a))*b*c**3*x**2 + 2*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt 
(a))*a*b*d**2*x**2 + 4*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a)) 
*a*c**2*d*x**2)/(2*a*b*x**2)