\(\int \frac {A+B x+C x^2+D x^3}{(c+d x) (a+b x^2)^{3/2}} \, dx\) [112]

Optimal result

Integrand size = 34, antiderivative size = 193 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a (b B c-A b d+a C d-a c D)-\left (A b^2 c-a (b c C-b B d+a d D)\right ) x}{a b \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}+\frac {D \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2} d}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {A b^2 c x+a b (-B c+A d-c C x+B d x)-a^2 (C d-c D+d D x)}{a b \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {2 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{d \left (-b c^2-a d^2\right )^{3/2}}-\frac {D \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2} d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2178, 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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(463\) vs. \(2(179)=358\).

Time = 1.47 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.40

method	result	size
default	\(\frac {\frac {B \,d^{2} x}{\sqrt {b \,x^{2}+a}\, a}+\frac {D c^{2} x}{\sqrt {b \,x^{2}+a}\, a}-\frac {d \left (C d -D c \right )}{b \sqrt {b \,x^{2}+a}}+D d^{2} \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )-\frac {C c d x}{\sqrt {b \,x^{2}+a}\, a}}{d^{3}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (\frac {d^{2}}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 b c d \left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b c}{d}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}-\frac {d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}\)	\(464\)

Input:

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

Output:

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((A + B*x + C*x**2 + D*x**3)/((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. 670 vs. \(2 (177) = 354\).

Time = 0.11 (sec) , antiderivative size = 670, normalized size of antiderivative = 3.47 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=-\frac {D b c^{4} x}{\sqrt {b x^{2} + a} a b c^{2} d^{3} + \sqrt {b x^{2} + a} a^{2} d^{5}} + \frac {C b c^{3} x}{\sqrt {b x^{2} + a} a b c^{2} d^{2} + \sqrt {b x^{2} + a} a^{2} d^{4}} - \frac {B b c^{2} x}{\sqrt {b x^{2} + a} a b c^{2} d + \sqrt {b x^{2} + a} a^{2} d^{3}} - \frac {D c^{3}}{\sqrt {b x^{2} + a} b c^{2} d^{2} + \sqrt {b x^{2} + a} a d^{4}} + \frac {A b c x}{\sqrt {b x^{2} + a} a b c^{2} + \sqrt {b x^{2} + a} a^{2} d^{2}} + \frac {C c^{2}}{\sqrt {b x^{2} + a} b c^{2} d + \sqrt {b x^{2} + a} a d^{3}} - \frac {B c}{\sqrt {b x^{2} + a} b c^{2} + \sqrt {b x^{2} + a} a d^{2}} + \frac {A}{\frac {\sqrt {b x^{2} + a} b c^{2}}{d} + \sqrt {b x^{2} + a} a d} + \frac {D c^{2} x}{\sqrt {b x^{2} + a} a d^{3}} - \frac {C c x}{\sqrt {b x^{2} + a} a d^{2}} + \frac {B x}{\sqrt {b x^{2} + a} a d} - \frac {D x}{\sqrt {b x^{2} + a} b d} + \frac {D \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d} - \frac {D c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{4}} + \frac {C c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} - \frac {B c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{2}} + \frac {A \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d} + \frac {D c}{\sqrt {b x^{2} + a} b d^{2}} - \frac {C}{\sqrt {b x^{2} + a} b d} \] Input:

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

Output:

-D*b*c^4*x/(sqrt(b*x^2 + a)*a*b*c^2*d^3 + sqrt(b*x^2 + a)*a^2*d^5) + 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) - D*c^3/(sqrt(b*x^2 + 
 a)*b*c^2*d^2 + sqrt(b*x^2 + a)*a*d^4) + 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) + D*c^2*x/(sqrt(b*x^2 + a)*a*d^ 
3) - C*c*x/(sqrt(b*x^2 + a)*a*d^2) + B*x/(sqrt(b*x^2 + a)*a*d) - D*x/(sqrt 
(b*x^2 + a)*b*d) + D*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d) - D*c^3*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^4) + C*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) - 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) + D*c/(sqrt(b*x^2 + a)*b*d^2) - C/(sqrt(b* 
x^2 + a)*b*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 4998, normalized size of antiderivative = 25.90 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)* 
sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt( 
b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b**2*c*d + 2*sqrt(b) 
*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + 
 b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d** 
2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*b**3*c**2 - 2*sqrt(b)*sqrt(2*sqrt(b) 
*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2)*atan(( 
sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - 
 a*d**2 - 2*b*c**2))*a*b**3*c*d*x**2 + 2*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d** 
2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b* 
x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2 
*b*c**2))*b**4*c**2*x**2 - 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d* 
*2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt 
(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a**3*b**2*d**3 - 2*sqrt(2*sqrt(b 
)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + 
sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))* 
a**2*b**3*c**2*d + 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b 
*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 
+ b*c**2)*c - a*d**2 - 2*b*c**2))*a**2*b**3*c*d**2 - 2*sqrt(2*sqrt(b)*sqrt 
(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqr...