\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [113]

Optimal result

Integrand size = 34, antiderivative size = 281 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a \left (b^2 c (B c-2 A d)+a^2 d^2 D+a b \left (2 c C d-B d^2-c^2 D\right )\right )-b \left (A b \left (b c^2-a d^2\right )-a (b c (c C-2 B d)-a d (C d-2 c D))\right ) x}{a b \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{\left (b c^2+a d^2\right )^2 (c+d x)}-\frac {\left (b c \left (c^2 C-2 B c d+3 A d^2\right )-a d \left (2 c C d-B d^2-3 c^2 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{5/2}} \] Output:

-(a*(b^2*c*(-2*A*d+B*c)+a^2*d^2*D+a*b*(-B*d^2+2*C*c*d-D*c^2))-b*(A*b*(-a*d 
^2+b*c^2)-a*(b*c*(-2*B*d+C*c)-a*d*(C*d-2*D*c)))*x)/a/b/(a*d^2+b*c^2)^2/(b* 
x^2+a)^(1/2)-(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^2 
/(d*x+c)-(b*c*(3*A*d^2-2*B*c*d+C*c^2)-a*d*(-B*d^2+2*C*c*d-3*D*c^2))*arctan 
h((-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)^(5/2)
 

Mathematica [A] (verified)

Time = 2.32 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-a^3 d^2 D (c+d x)+A b^3 c^2 x (c+d x)+a b^2 \left (c^2 x (-c C-2 C d x+c D x)+A d \left (2 c^2+c d x-2 d^2 x^2\right )+B c \left (-c^2+c d x+3 d^2 x^2\right )\right )+a^2 b \left (2 c^3 D-c^2 d (3 C+D x)+d^3 (-A+x (B+C x))+c d^2 (2 B-x (C+2 D x))\right )}{a b \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}-\frac {2 \left (b c \left (c^2 C-2 B c d+3 A d^2\right )+a d \left (-2 c C d+B d^2+3 c^2 D\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2178, 25, 27, 679, 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.

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

\(\Big \downarrow \) 2178

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((a*(b^2*c*(B*c - 2*A*d) + a^2*d^2*D + a*b*(2*c*C*d - B*d^2 - c^2*D)) - b 
*(A*b*(b*c^2 - a*d^2) - a*(b*c*(c*C - 2*B*d) - a*d*(C*d - 2*c*D)))*x)/(a*b 
*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^2])) + (-(((c^2*C*d - B*c*d^2 + A*d^3 - c^ 
3*D)*Sqrt[a + b*x^2])/(c + d*x)) - ((b*c*(c^2*C - 2*B*c*d + 3*A*d^2) - a*d 
*(2*c*C*d - B*d^2 - 3*c^2*D))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*S 
qrt[a + b*x^2])])/Sqrt[b*c^2 + a*d^2])/(b*c^2 + a*d^2)^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 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 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. \(925\) vs. \(2(271)=542\).

Time = 1.44 (sec) , antiderivative size = 926, normalized size of antiderivative = 3.30

method	result	size
default	\(\frac {\frac {C d x}{\sqrt {b \,x^{2}+a}\, a}-\frac {D d}{b \sqrt {b \,x^{2}+a}}-\frac {2 D c x}{\sqrt {b \,x^{2}+a}\, a}}{d^{3}}+\frac {\left (B \,d^{2}-2 C c d +3 D 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^{4}}+\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 ) \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^{5}}\)	\(926\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 959 vs. \(2 (267) = 534\).

Time = 4.72 (sec) , antiderivative size = 1945, normalized size of antiderivative = 6.92 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*((C*a^2*b^2*c^4 + B*a^3*b*c*d^3 + (3*D*a^3*b - 2*B*a^2*b^2)*c^3*d - ( 
2*C*a^3*b - 3*A*a^2*b^2)*c^2*d^2 + (C*a*b^3*c^3*d + B*a^2*b^2*d^4 + (3*D*a 
^2*b^2 - 2*B*a*b^3)*c^2*d^2 - (2*C*a^2*b^2 - 3*A*a*b^3)*c*d^3)*x^3 + (C*a* 
b^3*c^4 + B*a^2*b^2*c*d^3 + (3*D*a^2*b^2 - 2*B*a*b^3)*c^3*d - (2*C*a^2*b^2 
 - 3*A*a*b^3)*c^2*d^2)*x^2 + (C*a^2*b^2*c^3*d + B*a^3*b*d^4 + (3*D*a^3*b - 
 2*B*a^2*b^2)*c^2*d^2 - (2*C*a^3*b - 3*A*a^2*b^2)*c*d^3)*x)*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)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + 
 c^2)) - 2*(A*a^3*b*d^5 - (2*D*a^2*b^2 - B*a*b^3)*c^5 + (3*C*a^2*b^2 - 2*A 
*a*b^3)*c^4*d - (D*a^3*b + B*a^2*b^2)*c^3*d^2 + (3*C*a^3*b - A*a^2*b^2)*c^ 
2*d^3 + (D*a^4 - 2*B*a^3*b)*c*d^4 - (D*a*b^3*c^5 - (2*C*a*b^3 - A*b^4)*c^4 
*d - (D*a^2*b^2 - 3*B*a*b^3)*c^3*d^2 - (C*a^2*b^2 + A*a*b^3)*c^2*d^3 - (2* 
D*a^3*b - 3*B*a^2*b^2)*c*d^4 + (C*a^3*b - 2*A*a^2*b^2)*d^5)*x^2 + ((C*a*b^ 
3 - A*b^4)*c^5 + (D*a^2*b^2 - B*a*b^3)*c^4*d + 2*(C*a^2*b^2 - A*a*b^3)*c^3 
*d^2 + 2*(D*a^3*b - B*a^2*b^2)*c^2*d^3 + (C*a^3*b - A*a^2*b^2)*c*d^4 + (D* 
a^4 - B*a^3*b)*d^5)*x)*sqrt(b*x^2 + a))/(a^2*b^4*c^7 + 3*a^3*b^3*c^5*d^2 + 
 3*a^4*b^2*c^3*d^4 + a^5*b*c*d^6 + (a*b^5*c^6*d + 3*a^2*b^4*c^4*d^3 + 3*a^ 
3*b^3*c^2*d^5 + a^4*b^2*d^7)*x^3 + (a*b^5*c^7 + 3*a^2*b^4*c^5*d^2 + 3*a^3* 
b^3*c^3*d^4 + a^4*b^2*c*d^6)*x^2 + (a^2*b^4*c^6*d + 3*a^3*b^3*c^4*d^3 + 3* 
a^4*b^2*c^2*d^5 + a^5*b*d^7)*x), -((C*a^2*b^2*c^4 + B*a^3*b*c*d^3 + (3*...
 

Sympy [F]

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

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1530 vs. \(2 (267) = 534\).

Time = 0.14 (sec) , antiderivative size = 1530, normalized size of antiderivative = 5.44 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 4.56 (sec) , antiderivative size = 2441, normalized size of antiderivative = 8.69 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
+ b*c*x)*a**2*b**2*c**2*d**2 + 3*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2 
)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c*d**3*x + sqrt(a*d**2 + 
b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b** 
2*c*d**3 + sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2 
) - a*d + b*c*x)*a**2*b**2*d**4*x + sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x 
**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**3*d**2 + sqrt(a*d**2 + 
 b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b* 
c**2*d**3*x - 2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b 
*c**2) - a*d + b*c*x)*a*b**3*c**3*d + 3*sqrt(a*d**2 + b*c**2)*log(sqrt(a + 
 b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**2*d**2*x**2 - 2*sq 
rt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c 
*x)*a*b**3*c**2*d**2*x + 3*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt 
(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c*d**3*x**3 + sqrt(a*d**2 + b*c**2 
)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c*d**3* 
x**2 + sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a*b**3*d**4*x**3 + sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2) 
*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**5 + sqrt(a*d**2 + b*c**2)* 
log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**4*d*x 
+ sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*...