\(\int \frac {A+B x+C x^2}{(c+d x) (a+b x^2)^{3/2}} \, dx\) [153]

Optimal result

Integrand size = 29, antiderivative size = 138 \[ \int \frac {A+B x+C x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a (b B c-A b d+a C d)-b (A b c-a c C+a B d) x}{a b \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {\left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x+C x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a^2 C d+A b^2 c x+a b (-B c+A d-c C x+B d x)}{a b \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {2 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2178, 25, 27, 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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

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 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(130)=260\).

Time = 0.19 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.80

method	result	size
default	\(\frac {\frac {B d x}{a \sqrt {b \,x^{2}+a}}-\frac {d C}{b \sqrt {b \,x^{2}+a}}-\frac {C c x}{a \sqrt {b \,x^{2}+a}}}{d^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}\)	\(386\)

Input:

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

Output:

1/d^2*(B*d*x/a/(b*x^2+a)^(1/2)-d*C/b/(b*x^2+a)^(1/2)-C*c*x/a/(b*x^2+a)^(1/ 
2))+(A*d^2-B*c*d+C*c^2)/d^3*(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (131) = 262\).

Time = 0.86 (sec) , antiderivative size = 721, normalized size of antiderivative = 5.22 \[ \int \frac {A+B x+C x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (C a^{2} b c^{2} - B a^{2} b c d + A a^{2} b d^{2} + {\left (C a b^{2} c^{2} - B a b^{2} c d + A a b^{2} d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (B a b^{2} c^{3} + B a^{2} b c d^{2} + {\left (C a^{2} b - A a b^{2}\right )} c^{2} d + {\left (C a^{3} - A a^{2} b\right )} d^{3} - {\left (B a b^{2} c^{2} d + B a^{2} b d^{3} - {\left (C a b^{2} - A b^{3}\right )} c^{3} - {\left (C a^{2} b - A a b^{2}\right )} c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{2} b^{3} c^{4} + 2 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b d^{4} + {\left (a b^{4} c^{4} + 2 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} d^{4}\right )} x^{2}\right )}}, -\frac {{\left (C a^{2} b c^{2} - B a^{2} b c d + A a^{2} b d^{2} + {\left (C a b^{2} c^{2} - B a b^{2} c d + A a b^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) + {\left (B a b^{2} c^{3} + B a^{2} b c d^{2} + {\left (C a^{2} b - A a b^{2}\right )} c^{2} d + {\left (C a^{3} - A a^{2} b\right )} d^{3} - {\left (B a b^{2} c^{2} d + B a^{2} b d^{3} - {\left (C a b^{2} - A b^{3}\right )} c^{3} - {\left (C a^{2} b - A a b^{2}\right )} c d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{a^{2} b^{3} c^{4} + 2 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b d^{4} + {\left (a b^{4} c^{4} + 2 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} d^{4}\right )} x^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral((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. 453 vs. \(2 (131) = 262\).

Time = 0.10 (sec) , antiderivative size = 453, normalized size of antiderivative = 3.28 \[ \int \frac {A+B x+C x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {C b c^{3} x}{\sqrt {b x^{2} + a} a b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{2} d^{4}} - \frac {B b c^{2} x}{\sqrt {b x^{2} + a} a b c^{2} d + \sqrt {b x^{2} + a} a^{2} d^{3}} + \frac {A b c x}{\sqrt {b x^{2} + a} a b c^{2} + \sqrt {b x^{2} + a} a^{2} d^{2}} + \frac {C c^{2}}{\sqrt {b x^{2} + a} b c^{2} d + \sqrt {b x^{2} + a} a d^{3}} - \frac {B c}{\sqrt {b x^{2} + a} b c^{2} + \sqrt {b x^{2} + a} a d^{2}} + \frac {A}{\frac {\sqrt {b x^{2} + a} b c^{2}}{d} + \sqrt {b x^{2} + a} a d} - \frac {C c x}{\sqrt {b x^{2} + a} a d^{2}} + \frac {B x}{\sqrt {b x^{2} + a} a d} + \frac {C c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} - \frac {B c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{2}} + \frac {A \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d} - \frac {C}{\sqrt {b x^{2} + a} b d} \] Input:

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

Output:

C*b*c^3*x/(sqrt(b*x^2 + a)*a*b*c^2*d^2 + sqrt(b*x^2 + a)*a^2*d^4) - B*b*c^ 
2*x/(sqrt(b*x^2 + a)*a*b*c^2*d + sqrt(b*x^2 + a)*a^2*d^3) + A*b*c*x/(sqrt( 
b*x^2 + a)*a*b*c^2 + sqrt(b*x^2 + a)*a^2*d^2) + C*c^2/(sqrt(b*x^2 + a)*b*c 
^2*d + sqrt(b*x^2 + a)*a*d^3) - B*c/(sqrt(b*x^2 + a)*b*c^2 + sqrt(b*x^2 + 
a)*a*d^2) + A/(sqrt(b*x^2 + a)*b*c^2/d + sqrt(b*x^2 + a)*a*d) - C*c*x/(sqr 
t(b*x^2 + a)*a*d^2) + B*x/(sqrt(b*x^2 + a)*a*d) + C*c^2*arcsinh(b*c*x/(sqr 
t(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) - B*c*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^2) + A*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) - C/(sqrt 
(b*x^2 + a)*b*d)
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (131) = 262\).

Time = 0.14 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.10 \[ \int \frac {A+B x+C x^2}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {\frac {{\left (C a b^{2} c^{3} - A b^{3} c^{3} - B a b^{2} c^{2} d + C a^{2} b c d^{2} - A a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x}{a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}} + \frac {B a b^{2} c^{3} + C a^{2} b c^{2} d - A a b^{2} c^{2} d + B a^{2} b c d^{2} + C a^{3} d^{3} - A a^{2} b d^{3}}{a b^{3} c^{4} + 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b d^{4}}}{\sqrt {b x^{2} + a}} - \frac {2 \, {\left (C c^{2} - B c d + A d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt {-b c^{2} - a d^{2}}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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