\(\int \frac {x^2 (A+B x+C x^2)}{(c+d x) (a+b x^2)^{3/2}} \, dx\) [151]

Optimal result

Integrand size = 32, antiderivative size = 196 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {a (b B c-A b d+a C d)-b (A b c-a c C+a B d) x}{b^2 \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {C \sqrt {a+b x^2}}{b^2 d}-\frac {(c C-B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d^2}-\frac {c^2 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^2 \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.13 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {d \left (2 a^2 C d^2+b^2 c x (-A d+c C x)+a b \left (c^2 C+c d (B+C x)-d^2 (A+x (B-C x))\right )\right )}{b^2 \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {2 c^2 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {2 (-c C+B d) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}}}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2178, 25, 2185, 27, 719, 224, 219, 488, 219}

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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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 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 488

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

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(416\) vs. \(2(180)=360\).

Time = 0.29 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.13

method	result	size
risch	\(\frac {C \sqrt {b \,x^{2}+a}}{b^{2} d}+\frac {\frac {\left (B d -C c \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {d \left (A b -B \sqrt {-a b}-a C \right ) \sqrt {b \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 \left (d \sqrt {-a b}-b c \right ) b \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {d \left (A b +B \sqrt {-a b}-a C \right ) \sqrt {b \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 \left (d \sqrt {-a b}+b c \right ) b \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} c^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \left (d \sqrt {-a b}+b c \right ) \left (d \sqrt {-a b}-b c \right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{d b}\)	\(417\)
default	\(\frac {c^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}-\frac {\frac {C \,c^{3} x}{a \sqrt {b \,x^{2}+a}}-d^{2} \left (B d -C c \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right )}{b \sqrt {b \,x^{2}+a}}+\frac {A c \,d^{2} x}{a \sqrt {b \,x^{2}+a}}-\frac {B \,c^{2} d x}{a \sqrt {b \,x^{2}+a}}-C \,d^{3} \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )}{d^{4}}\)	\(515\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (181) = 362\).

Time = 0.17 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.33 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {C b c^{5} x}{\sqrt {b x^{2} + a} a b c^{2} d^{4} + \sqrt {b x^{2} + a} a^{2} d^{6}} - \frac {B b c^{4} x}{\sqrt {b x^{2} + a} a b c^{2} d^{3} + \sqrt {b x^{2} + a} a^{2} d^{5}} + \frac {A b c^{3} x}{\sqrt {b x^{2} + a} a b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{2} d^{4}} + \frac {C c^{4}}{\sqrt {b x^{2} + a} b c^{2} d^{3} + \sqrt {b x^{2} + a} a d^{5}} - \frac {B c^{3}}{\sqrt {b x^{2} + a} b c^{2} d^{2} + \sqrt {b x^{2} + a} a d^{4}} + \frac {A c^{2}}{\sqrt {b x^{2} + a} b c^{2} d + \sqrt {b x^{2} + a} a d^{3}} + \frac {C x^{2}}{\sqrt {b x^{2} + a} b d} - \frac {C c^{3} x}{\sqrt {b x^{2} + a} a d^{4}} + \frac {B c^{2} x}{\sqrt {b x^{2} + a} a d^{3}} - \frac {A c x}{\sqrt {b x^{2} + a} a d^{2}} + \frac {C c x}{\sqrt {b x^{2} + a} b d^{2}} - \frac {B x}{\sqrt {b x^{2} + a} b d} - \frac {C c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d^{2}} + \frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d} + \frac {C c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{5}} - \frac {B c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{4}} + \frac {A c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} - \frac {C c^{2}}{\sqrt {b x^{2} + a} b d^{3}} + \frac {B c}{\sqrt {b x^{2} + a} b d^{2}} + \frac {2 \, C a}{\sqrt {b x^{2} + a} b^{2} d} - \frac {A}{\sqrt {b x^{2} + a} b d} \] Input:

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

Output:

C*b*c^5*x/(sqrt(b*x^2 + a)*a*b*c^2*d^4 + sqrt(b*x^2 + a)*a^2*d^6) - B*b*c^ 
4*x/(sqrt(b*x^2 + a)*a*b*c^2*d^3 + sqrt(b*x^2 + a)*a^2*d^5) + A*b*c^3*x/(s 
qrt(b*x^2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)*a^2*d^4) + C*c^4/(sqrt(b*x^2 
+ a)*b*c^2*d^3 + sqrt(b*x^2 + a)*a*d^5) - B*c^3/(sqrt(b*x^2 + a)*b*c^2*d^2 
 + sqrt(b*x^2 + a)*a*d^4) + A*c^2/(sqrt(b*x^2 + a)*b*c^2*d + sqrt(b*x^2 + 
a)*a*d^3) + C*x^2/(sqrt(b*x^2 + a)*b*d) - C*c^3*x/(sqrt(b*x^2 + a)*a*d^4) 
+ B*c^2*x/(sqrt(b*x^2 + a)*a*d^3) - A*c*x/(sqrt(b*x^2 + a)*a*d^2) + C*c*x/ 
(sqrt(b*x^2 + a)*b*d^2) - B*x/(sqrt(b*x^2 + a)*b*d) - C*c*arcsinh(b*x/sqrt 
(a*b))/(b^(3/2)*d^2) + B*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d) + C*c^4*arcsin 
h(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c 
^2/d^2)^(3/2)*d^5) - B*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(s 
qrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d^4) + A*c^2*arcsinh(b*c*x/ 
(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^ 
(3/2)*d^3) - C*c^2/(sqrt(b*x^2 + a)*b*d^3) + B*c/(sqrt(b*x^2 + a)*b*d^2) + 
 2*C*a/(sqrt(b*x^2 + a)*b^2*d) - A/(sqrt(b*x^2 + a)*b*d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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