\(\int \frac {x^2 (A+B x+C x^2)}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [159]

Optimal result

Integrand size = 32, antiderivative size = 278 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {a (b c (B c-2 A d)+a d (2 c C-B d))-\left (A b \left (b c^2-a d^2\right )+a \left (a C d^2-b c (c C-2 B d)\right )\right ) x}{b \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {c^2 \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d^2}+\frac {c \left (a d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+b \left (c^4 C-A c^2 d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^2 \left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-\frac {d \left (a^2 d^2 (c+d x) (-2 c C+d (B+C x))+b^2 c^2 x (c (c C-B d) x+A d (c+2 d x))+a b \left (c^4 C-A d^4 x^2+c d^3 x (A+2 B x)-c^3 d (2 B+C x)+c^2 d^2 (3 A+x (B-C x))\right )\right )}{b \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}+\frac {2 c \left (a d^2 \left (4 c^2 C-3 B c d+2 A d^2\right )+b \left (c^4 C-A c^2 d^2\right )\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {2 C \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}}}{d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2178, 25, 2182, 25, 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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(a*(b*c*(B*c - 2*A*d) + a*d*(2*c*C - B*d)) - (A*b*(b*c^2 - a*d^2) + a*(a*C 
*d^2 - b*c*(c*C - 2*B*d)))*x)/(b*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2]) + (-(( 
a*b*c^2*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2])/(d*(b*c^2 + a*d^2)^2*(c + 
 d*x))) + (a*((C*(b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sq 
rt[b]*d^2) + (b*c*(a*d^2*(4*c^2*C - 3*B*c*d + 2*A*d^2) + b*(c^4*C - A*c^2* 
d^2))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d^2*( 
b*c^2 + a*d^2)^(3/2))))/(b*c^2 + a*d^2))/(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 2182

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(992\) vs. \(2(262)=524\).

Time = 0.28 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.57

method	result	size
default	\(\frac {\frac {A \,d^{2} x}{a \sqrt {b \,x^{2}+a}}-\frac {d \left (B d -2 C c \right )}{b \sqrt {b \,x^{2}+a}}+C \,d^{2} \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 {3 C \,c^{2} x}{a \sqrt {b \,x^{2}+a}}-\frac {2 B c d x}{a \sqrt {b \,x^{2}+a}}}{d^{4}}+\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 ) \left (x +\frac {c}{d}\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 {3 b c d \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 )}{a \,d^{2}+b \,c^{2}}-\frac {4 b \,d^{2} \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}}}}\right )}{d^{6}}-\frac {c \left (2 A \,d^{2}-3 B c d +4 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}}\)	\(993\)

Input:

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

Output:

1/d^4*(A*d^2*x/a/(b*x^2+a)^(1/2)-d*(B*d-2*C*c)/b/(b*x^2+a)^(1/2)+C*d^2*(-x 
/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+3*C*c^2*x/a/(b 
*x^2+a)^(1/2)-2*B*c*d*x/a/(b*x^2+a)^(1/2))+c^2*(A*d^2-B*c*d+C*c^2)/d^6*(-1 
/(a*d^2+b*c^2)*d^2/(x+c/d)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2) 
^(1/2)+3*b*c*d/(a*d^2+b*c^2)*(1/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+ 
c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/ 
(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+ 
b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2/((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)))-4*b/(a*d^2+b*c^2)*d^2*(2* 
b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b* 
c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-c/d^5*(2*A*d^2-3*B*c*d+4*C*c^2)*(1/( 
a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b 
*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/ 
d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2) 
*d^2/((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. \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1347 vs. \(2 (263) = 526\).

Time = 0.21 (sec) , antiderivative size = 1347, normalized size of antiderivative = 4.85 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

3*C*b^2*c^6*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^4 + 2*sqrt(b*x^2 + a)*a^2*b*c^2 
*d^6 + sqrt(b*x^2 + a)*a^3*d^8) - 3*B*b^2*c^5*x/(sqrt(b*x^2 + a)*a*b^2*c^4 
*d^3 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d^5 + sqrt(b*x^2 + a)*a^3*d^7) + 3*A*b^ 
2*c^4*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^2 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d^4 + 
 sqrt(b*x^2 + a)*a^3*d^6) + 3*C*b*c^5/(sqrt(b*x^2 + a)*b^2*c^4*d^3 + 2*sqr 
t(b*x^2 + a)*a*b*c^2*d^5 + sqrt(b*x^2 + a)*a^2*d^7) - 6*C*b*c^4*x/(sqrt(b* 
x^2 + a)*a*b*c^2*d^4 + sqrt(b*x^2 + a)*a^2*d^6) - 3*B*b*c^4/(sqrt(b*x^2 + 
a)*b^2*c^4*d^2 + 2*sqrt(b*x^2 + a)*a*b*c^2*d^4 + sqrt(b*x^2 + a)*a^2*d^6) 
+ 5*B*b*c^3*x/(sqrt(b*x^2 + a)*a*b*c^2*d^3 + sqrt(b*x^2 + a)*a^2*d^5) + 3* 
A*b*c^3/(sqrt(b*x^2 + a)*b^2*c^4*d + 2*sqrt(b*x^2 + a)*a*b*c^2*d^3 + sqrt( 
b*x^2 + a)*a^2*d^5) - C*c^4/(sqrt(b*x^2 + a)*b*c^2*d^4*x + sqrt(b*x^2 + a) 
*a*d^6*x + sqrt(b*x^2 + a)*b*c^3*d^3 + sqrt(b*x^2 + a)*a*c*d^5) - 4*A*b*c^ 
2*x/(sqrt(b*x^2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)*a^2*d^4) + B*c^3/(sqrt( 
b*x^2 + a)*b*c^2*d^3*x + sqrt(b*x^2 + a)*a*d^5*x + sqrt(b*x^2 + a)*b*c^3*d 
^2 + sqrt(b*x^2 + a)*a*c*d^4) - 4*C*c^3/(sqrt(b*x^2 + a)*b*c^2*d^3 + sqrt( 
b*x^2 + a)*a*d^5) - A*c^2/(sqrt(b*x^2 + a)*b*c^2*d^2*x + sqrt(b*x^2 + a)*a 
*d^4*x + sqrt(b*x^2 + a)*b*c^3*d + sqrt(b*x^2 + a)*a*c*d^3) + 3*B*c^2/(sqr 
t(b*x^2 + a)*b*c^2*d^2 + sqrt(b*x^2 + a)*a*d^4) - 2*A*c/(sqrt(b*x^2 + a)*b 
*c^2*d + sqrt(b*x^2 + a)*a*d^3) + 3*C*c^2*x/(sqrt(b*x^2 + a)*a*d^4) - 2*B* 
c*x/(sqrt(b*x^2 + a)*a*d^3) + A*x/(sqrt(b*x^2 + a)*a*d^2) - C*x/(sqrt(b...
 

Giac [F(-2)]

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

integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^2/(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:Recursive assumption sageVARa>=((-s 
ageVARb*sageVARc^2*sageVARd^2*t_nostep^2-2*sageVARb*sageVARc*sageVARd*t_no 
step-sage
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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