\(\int \frac {(c+d x)^2 (A+B x+C x^2)}{x^4 \sqrt {a+b x^2}} \, dx\) [111]

Optimal result

Integrand size = 32, antiderivative size = 186 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx=-\frac {A c^2 \sqrt {a+b x^2}}{3 a x^3}-\frac {c (B c+2 A d) \sqrt {a+b x^2}}{2 a x^2}-\frac {\left (3 a c (c C+2 B d)-A \left (2 b c^2-3 a d^2\right )\right ) \sqrt {a+b x^2}}{3 a^2 x}+\frac {C d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}+\frac {(b c (B c+2 A d)-2 a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (4 A b c^2 x^2-2 a A \left (c^2+3 c d x+3 d^2 x^2\right )-3 a c x (2 c C x+B (c+4 d x))\right )}{6 a^2 x^3}+\frac {(b c (B c+2 A d)-2 a d (2 c C+B d)) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {C d^2 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.06, 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, 2338, 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.

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2338

\(\displaystyle \frac {-\frac {\int -\frac {6 a^2 C d^2 x^2-3 a (b c (B c+2 A d)-2 a d (2 c C+B d)) x+2 a \left (3 a c (c C+2 B d)-A \left (2 b c^2-3 a d^2\right )\right )}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {3 c \sqrt {a+b x^2} (2 A d+B c)}{2 x^2}}{3 a}-\frac {A c^2 \sqrt {a+b x^2}}{3 a x^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {6 a^2 C d^2 x^2-3 a (b c (B c+2 A d)-2 a d (2 c C+B d)) x+2 a \left (3 a c (c C+2 B d)-A \left (2 b c^2-3 a d^2\right )\right )}{x^2 \sqrt {b x^2+a}}dx}{2 a}-\frac {3 c \sqrt {a+b x^2} (2 A d+B c)}{2 x^2}}{3 a}-\frac {A c^2 \sqrt {a+b x^2}}{3 a x^3}\)

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (6 A a \,d^{2} x^{2}-4 A b \,c^{2} x^{2}+12 B a c d \,x^{2}+6 C a \,c^{2} x^{2}+6 A a c d x +3 B a \,c^{2} x +2 A \,c^{2} a \right )}{6 a^{2} x^{3}}-\frac {-\frac {\left (2 A b c d -2 a B \,d^{2}+b B \,c^{2}-4 C a c d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}-\frac {2 a C \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}}{2 a}\)	\(167\)
default	\(\frac {C \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{a x}+A \,c^{2} \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )+c \left (2 A d +B c \right ) \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )-\frac {d \left (B d +2 C c \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}\)	\(194\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 762, normalized size of antiderivative = 4.10 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 4.80 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.82 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx=- \frac {A \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} - \frac {A \sqrt {b} c d \sqrt {\frac {a}{b x^{2}} + 1}}{a x} - \frac {A \sqrt {b} d^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{a} + \frac {2 A b^{\frac {3}{2}} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} + \frac {A b c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{a^{\frac {3}{2}}} - \frac {B \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} - \frac {2 B \sqrt {b} c d \sqrt {\frac {a}{b x^{2}} + 1}}{a} - \frac {B d^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} + \frac {B b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} + C d^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \wedge b \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a}} & \text {otherwise} \end {cases}\right ) - \frac {C \sqrt {b} c^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{a} - \frac {2 C c d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} \] Input:

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

Output:

-A*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/(3*a*x**2) - A*sqrt(b)*c*d*sqrt(a/(b* 
x**2) + 1)/(a*x) - A*sqrt(b)*d**2*sqrt(a/(b*x**2) + 1)/a + 2*A*b**(3/2)*c* 
*2*sqrt(a/(b*x**2) + 1)/(3*a**2) + A*b*c*d*asinh(sqrt(a)/(sqrt(b)*x))/a**( 
3/2) - B*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/(2*a*x) - 2*B*sqrt(b)*c*d*sqrt( 
a/(b*x**2) + 1)/a - B*d**2*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a) + B*b*c**2*a 
sinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/2)) + C*d**2*Piecewise((log(2*sqrt(b)*s 
qrt(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*sqrt(b)*c**2*sqrt(a/(b*x**2) + 1)/ 
a - 2*C*c*d*asinh(sqrt(a)/(sqrt(b)*x))/sqrt(a)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.19 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx=\frac {C d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {2 \, C c d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} - \frac {B d^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} - \frac {\sqrt {b x^{2} + a} C c^{2}}{a x} + \frac {2 \, \sqrt {b x^{2} + a} A b c^{2}}{3 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a} B c d}{a x} - \frac {\sqrt {b x^{2} + a} A d^{2}}{a x} + \frac {{\left (B c^{2} + 2 \, A c d\right )} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {\sqrt {b x^{2} + a} A c^{2}}{3 \, a x^{3}} - \frac {{\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a}}{2 \, a x^{2}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (160) = 320\).

Time = 0.20 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.57 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx=-\frac {C d^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{\sqrt {b}} - \frac {{\left (B b c^{2} - 4 \, C a c d + 2 \, A b c d - 2 \, B a d^{2}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B b c^{2} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A b c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a \sqrt {b} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a \sqrt {b} d^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{2} \sqrt {b} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} c^{2} - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} c d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} \sqrt {b} d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b c d + 6 \, C a^{3} \sqrt {b} c^{2} - 4 \, A a^{2} b^{\frac {3}{2}} c^{2} + 12 \, B a^{3} \sqrt {b} c d + 6 \, A a^{3} \sqrt {b} d^{2}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a} \] Input:

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

Output:

-C*d^2*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) - (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) 
)/(sqrt(-a)*a) + 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 + 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 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^(3/2)*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 - 4*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*a)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.74 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2\right )}{x^4 \sqrt {a+b x^2}} \, dx=\frac {-2 \sqrt {b \,x^{2}+a}\, a^{2} b \,c^{2}-6 \sqrt {b \,x^{2}+a}\, a^{2} b c d x -6 \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{2} x^{2}+4 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x^{2}-3 \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} x -12 \sqrt {b \,x^{2}+a}\, a \,b^{2} c d \,x^{2}-6 \sqrt {b \,x^{2}+a}\, a b \,c^{3} x^{2}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c d \,x^{3}+6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d^{2} x^{3}+12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} d \,x^{3}-3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{3}+6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c d \,x^{3}-6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d^{2} x^{3}-12 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} d \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{3} c^{2} x^{3}+6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} c \,d^{2} x^{3}+2 \sqrt {b}\, a^{2} b \,d^{2} x^{3}-4 \sqrt {b}\, a \,b^{2} c^{2} x^{3}+4 \sqrt {b}\, a \,b^{2} c d \,x^{3}+2 \sqrt {b}\, a b \,c^{3} x^{3}}{6 a^{2} b \,x^{3}} \] Input:

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

Output:

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