\(\int \frac {(c+d x)^2 \sqrt {a+b x^2} (A+B x+C x^2)}{x^4} \, dx\) [23]

Optimal result

Integrand size = 32, antiderivative size = 283 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {(b c (B c+2 A d)+2 a d (2 c C+B d)) \sqrt {a+b x^2}}{2 a}+\frac {\left (a C d^2+2 b \left (c^2 C+2 B c d+A d^2\right )\right ) x \sqrt {a+b x^2}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{3/2}}{3 a x^3}-\frac {c (B c+2 A d) \left (a+b x^2\right )^{3/2}}{2 a x^2}-\frac {\left (c^2 C+2 B c d+A d^2\right ) \left (a+b x^2\right )^{3/2}}{a x}+\frac {\left (a C d^2+2 b \left (c^2 C+2 B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}-\frac {(b c (B c+2 A d)+2 a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.75 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {1}{6} \left (\frac {\sqrt {a+b x^2} \left (-2 A b c^2 x^2-2 a A \left (c^2+3 c d x+3 d^2 x^2\right )-3 a x \left (B \left (c^2+4 c d x-2 d^2 x^2\right )-C x \left (-2 c^2+4 c d x+d^2 x^2\right )\right )\right )}{a x^3}-\frac {6 (b c (B c+2 A d)+2 a d (2 c C+B d)) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {3 \left (a C d^2+2 b \left (c^2 C+2 B c d+A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x^2]*(-2*A*b*c^2*x^2 - 2*a*A*(c^2 + 3*c*d*x + 3*d^2*x^2) - 3* 
a*x*(B*(c^2 + 4*c*d*x - 2*d^2*x^2) - C*x*(-2*c^2 + 4*c*d*x + d^2*x^2))))/( 
a*x^3) - (6*(b*c*(B*c + 2*A*d) + 2*a*d*(2*c*C + B*d))*ArcTanh[(-(Sqrt[b]*x 
) + Sqrt[a + b*x^2])/Sqrt[a]])/Sqrt[a] - (3*(a*C*d^2 + 2*b*(c^2*C + 2*B*c* 
d + A*d^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/6
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.95, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2338, 27, 2338, 25, 2338, 25, 27, 535, 27, 538, 224, 219, 243, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 73

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

rule 219

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

rule 221

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

rule 224

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

rule 243

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

rule 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 538

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

rule 2338

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.92

method	result	size
default	\(C \,d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}\right )-\frac {A \,c^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}}+c \left (2 A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {b \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{2 a}\right )+d \left (B d +2 C c \right ) \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\)	\(259\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (6 A a \,d^{2} x^{2}+2 A b \,c^{2} x^{2}+12 B a c d \,x^{2}+6 C a \,c^{2} x^{2}+6 A a c d x +3 B a \,c^{2} x +2 A \,c^{2} a \right )}{6 x^{3} a}-\frac {\left (2 A b c d +2 a B \,d^{2}+b B \,c^{2}+4 C a c d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 \sqrt {a}}+A \sqrt {b}\, d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+C \sqrt {b}\, c^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {a C \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+B \,d^{2} \sqrt {b \,x^{2}+a}+2 B \sqrt {b}\, c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+C b \,d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )+2 C c d \sqrt {b \,x^{2}+a}\)	\(304\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.45 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/12*(3*(2*C*a*b*c^2 + 4*B*a*b*c*d + (C*a^2 + 2*A*a*b)*d^2)*sqrt(b)*x^3*l 
og(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(B*b^2*c^2 + 2*B*a*b*d^ 
2 + 2*(2*C*a*b + A*b^2)*c*d)*sqrt(a)*x^3*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*s 
qrt(a) + 2*a)/x^2) + 2*(3*C*a*b*d^2*x^4 - 2*A*a*b*c^2 + 6*(2*C*a*b*c*d + B 
*a*b*d^2)*x^3 - 2*(6*B*a*b*c*d + 3*A*a*b*d^2 + (3*C*a*b + A*b^2)*c^2)*x^2 
- 3*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(a*b*x^3), -1/12*(6*(2*C 
*a*b*c^2 + 4*B*a*b*c*d + (C*a^2 + 2*A*a*b)*d^2)*sqrt(-b)*x^3*arctan(sqrt(- 
b)*x/sqrt(b*x^2 + a)) - 3*(B*b^2*c^2 + 2*B*a*b*d^2 + 2*(2*C*a*b + A*b^2)*c 
*d)*sqrt(a)*x^3*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(3 
*C*a*b*d^2*x^4 - 2*A*a*b*c^2 + 6*(2*C*a*b*c*d + B*a*b*d^2)*x^3 - 2*(6*B*a* 
b*c*d + 3*A*a*b*d^2 + (3*C*a*b + A*b^2)*c^2)*x^2 - 3*(B*a*b*c^2 + 2*A*a*b* 
c*d)*x)*sqrt(b*x^2 + a))/(a*b*x^3), 1/12*(6*(B*b^2*c^2 + 2*B*a*b*d^2 + 2*( 
2*C*a*b + A*b^2)*c*d)*sqrt(-a)*x^3*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + 3* 
(2*C*a*b*c^2 + 4*B*a*b*c*d + (C*a^2 + 2*A*a*b)*d^2)*sqrt(b)*x^3*log(-2*b*x 
^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(3*C*a*b*d^2*x^4 - 2*A*a*b*c^2 + 
 6*(2*C*a*b*c*d + B*a*b*d^2)*x^3 - 2*(6*B*a*b*c*d + 3*A*a*b*d^2 + (3*C*a*b 
 + A*b^2)*c^2)*x^2 - 3*(B*a*b*c^2 + 2*A*a*b*c*d)*x)*sqrt(b*x^2 + a))/(a*b* 
x^3), -1/6*(3*(2*C*a*b*c^2 + 4*B*a*b*c*d + (C*a^2 + 2*A*a*b)*d^2)*sqrt(-b) 
*x^3*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - 3*(B*b^2*c^2 + 2*B*a*b*d^2 + 2*( 
2*C*a*b + A*b^2)*c*d)*sqrt(-a)*x^3*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) -...
 

Sympy [A] (verification not implemented)

Time = 6.00 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.12 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=- \frac {A \sqrt {a} d^{2}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A \sqrt {b} c d \sqrt {\frac {a}{b x^{2}} + 1}}{x} + A \sqrt {b} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} - \frac {A b^{\frac {3}{2}} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} - \frac {A b c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} - \frac {A b d^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {2 B \sqrt {a} c d}{x \sqrt {1 + \frac {b x^{2}}{a}}} - B \sqrt {a} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a d^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + 2 B \sqrt {b} c d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + \frac {B \sqrt {b} d^{2} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 \sqrt {a}} - \frac {2 B b c d x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {C \sqrt {a} c^{2}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - 2 C \sqrt {a} c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {2 C a c d}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + C \sqrt {b} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + \frac {2 C \sqrt {b} c d x}{\sqrt {\frac {a}{b x^{2}} + 1}} + C d^{2} \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) - \frac {C b c^{2} x}{\sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:

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

Output:

-A*sqrt(a)*d**2/(x*sqrt(1 + b*x**2/a)) - A*sqrt(b)*c**2*sqrt(a/(b*x**2) + 
1)/(3*x**2) - A*sqrt(b)*c*d*sqrt(a/(b*x**2) + 1)/x + A*sqrt(b)*d**2*asinh( 
sqrt(b)*x/sqrt(a)) - A*b**(3/2)*c**2*sqrt(a/(b*x**2) + 1)/(3*a) - A*b*c*d* 
asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a) - A*b*d**2*x/(sqrt(a)*sqrt(1 + b*x**2/a 
)) - 2*B*sqrt(a)*c*d/(x*sqrt(1 + b*x**2/a)) - B*sqrt(a)*d**2*asinh(sqrt(a) 
/(sqrt(b)*x)) + B*a*d**2/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) - B*sqrt(b)*c**2 
*sqrt(a/(b*x**2) + 1)/(2*x) + 2*B*sqrt(b)*c*d*asinh(sqrt(b)*x/sqrt(a)) + B 
*sqrt(b)*d**2*x/sqrt(a/(b*x**2) + 1) - B*b*c**2*asinh(sqrt(a)/(sqrt(b)*x)) 
/(2*sqrt(a)) - 2*B*b*c*d*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - C*sqrt(a)*c**2/( 
x*sqrt(1 + b*x**2/a)) - 2*C*sqrt(a)*c*d*asinh(sqrt(a)/(sqrt(b)*x)) + 2*C*a 
*c*d/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)) + C*sqrt(b)*c**2*asinh(sqrt(b)*x/sqr 
t(a)) + 2*C*sqrt(b)*c*d*x/sqrt(a/(b*x**2) + 1) + C*d**2*Piecewise((a*Piece 
wise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x 
)/sqrt(b*x**2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqrt(a)*x, Tr 
ue)) - C*b*c**2*x/(sqrt(a)*sqrt(1 + b*x**2/a))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {1}{2} \, \sqrt {b x^{2} + a} C d^{2} x + \frac {C a d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} + {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - {\left (2 \, C c d + B d^{2}\right )} \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {{\left (B c^{2} + 2 \, A c d\right )} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, \sqrt {a}} + {\left (2 \, C c d + B d^{2}\right )} \sqrt {b x^{2} + a} + \frac {{\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a} b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A c^{2}}{3 \, a x^{3}} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a}}{x} - \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{2 \, a x^{2}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (249) = 498\).

Time = 0.25 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.84 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx=\frac {1}{2} \, {\left (C d^{2} x + 4 \, C c d + 2 \, B d^{2}\right )} \sqrt {b x^{2} + a} + \frac {{\left (B b c^{2} + 4 \, C a c d + 2 \, A b c d + 2 \, B a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {{\left (2 \, C b c^{2} + 4 \, B b c d + C a d^{2} + 2 \, A b d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, \sqrt {b}} + \frac {3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B b c^{2} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A b c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a \sqrt {b} c^{2} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A b^{\frac {3}{2}} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a \sqrt {b} d^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{2} \sqrt {b} c^{2} - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} c d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} \sqrt {b} d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b c d + 6 \, C a^{3} \sqrt {b} c^{2} + 2 \, A a^{2} b^{\frac {3}{2}} c^{2} + 12 \, B a^{3} \sqrt {b} c d + 6 \, A a^{3} \sqrt {b} d^{2}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \] Input:

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

Output:

1/2*(C*d^2*x + 4*C*c*d + 2*B*d^2)*sqrt(b*x^2 + a) + (B*b*c^2 + 4*C*a*c*d + 
 2*A*b*c*d + 2*B*a*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/sq 
rt(-a) - 1/2*(2*C*b*c^2 + 4*B*b*c*d + C*a*d^2 + 2*A*b*d^2)*log(abs(-sqrt(b 
)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B 
*b*c^2 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*b*c*d + 6*(sqrt(b)*x - sqrt(b 
*x^2 + a))^4*C*a*sqrt(b)*c^2 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(3/2) 
*c^2 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a*sqrt(b)*c*d + 6*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^4*A*a*sqrt(b)*d^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C 
*a^2*sqrt(b)*c^2 - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b)*c*d - 
12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^2*sqrt(b)*d^2 - 3*(sqrt(b)*x - sqrt 
(b*x^2 + a))*B*a^2*b*c^2 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^2*b*c*d + 6 
*C*a^3*sqrt(b)*c^2 + 2*A*a^2*b^(3/2)*c^2 + 12*B*a^3*sqrt(b)*c*d + 6*A*a^3* 
sqrt(b)*d^2)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 22.79 (sec) , antiderivative size = 676, normalized size of antiderivative = 2.39 \[ \int \frac {(c+d x)^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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