\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx\) [106]

Optimal result

Integrand size = 34, antiderivative size = 219 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {D \sqrt {a+b x^2}}{b d^2}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}+\frac {(C d-2 c D) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b} d^3}+\frac {\left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b \left (c^3 C d-A c d^3-2 c^4 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^3 \left (b c^2+a d^2\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.77 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 D (c+d x)+b \left (B c d^2-A d^3+2 c^3 D+c^2 (-C d+d D x)\right )\right )}{b \left (b c^2+a d^2\right ) (c+d x)}+\frac {2 \left (a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-c^3 C d+A c d^3+2 c^4 D\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+\frac {(-C d+2 c D) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2182, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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. \(448\) vs. \(2(203)=406\).

Time = 1.47 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.05

method	result	size
default	\(\frac {\frac {C d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+\frac {D d \sqrt {b \,x^{2}+a}}{b}-\frac {2 D c \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}}{d^{3}}-\frac {\left (B \,d^{2}-2 C c d +3 D c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}\)	\(449\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^2/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**2/(b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 616 vs. \(2 (205) = 410\).

Time = 0.08 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.81 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} D c^{3}}{b c^{2} d^{3} x + a d^{5} x + b c^{3} d^{2} + a c d^{4}} - \frac {\sqrt {b x^{2} + a} C c^{2}}{b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}} + \frac {\sqrt {b x^{2} + a} B c}{b c^{2} d x + a d^{3} x + b c^{3} + a c d^{2}} - \frac {\sqrt {b x^{2} + a} A}{b c^{2} x + a d^{2} x + \frac {b c^{3}}{d} + a c d} - \frac {2 \, D c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{3}} + \frac {C \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{2}} - \frac {D b 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^{6}} + \frac {C 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^{5}} + \frac {3 \, D 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 )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} - \frac {B b 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^{4}} - \frac {2 \, C 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 )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} + \frac {A 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^{3}} + \frac {B \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 )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{2}} + \frac {\sqrt {b x^{2} + a} D}{b d^{2}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="maxi 
ma")
 

Output:

sqrt(b*x^2 + a)*D*c^3/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4) - sqrt 
(b*x^2 + a)*C*c^2/(b*c^2*d^2*x + a*d^4*x + b*c^3*d + a*c*d^3) + sqrt(b*x^2 
 + a)*B*c/(b*c^2*d*x + a*d^3*x + b*c^3 + a*c*d^2) - sqrt(b*x^2 + a)*A/(b*c 
^2*x + a*d^2*x + b*c^3/d + a*c*d) - 2*D*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)* 
d^3) + C*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) - D*b*c^4*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^6) + C*b*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*a 
bs(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d^5) + 3*D*c^2*arcsinh(b*c*x/(sqrt(a* 
b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^4) 
 - B*b*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^4) - 2*C*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d 
*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^3) + A*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^3) + B*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d 
/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^2) + sqrt(b*x^2 + a)*D/( 
b*d^2)
 

Giac [F(-1)]

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

integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^2/(b*x^2+a)^(1/2),x, algorithm="giac 
")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1213, normalized size of antiderivative = 5.54 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^2 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(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**2*c**2*d**2 + 2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*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**2*c*d** 
3 + 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**2*d**4*x + 2*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**3*d**2 + 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*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)*b**2*c**5 + 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)*b**2*c**4*d*x - 2*sqrt(a*d**2 + b*c**2)*lo 
g(c + d*x)*a*b**2*c**2*d**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2* 
c*d**3*x - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c*d**3 - 2*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*a*b**2*d**4*x - 2*sqrt(a*d**2 + b*c**2)*log(c + 
 d*x)*a*b*c**3*d**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2*d**3*x 
 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**2*c**5 - 2*sqrt(a*d**2 + b*c**2 
)*log(c + d*x)*b**2*c**4*d*x - 2*sqrt(a + b*x**2)*a**2*b*d**5 + 2*sqrt(a + 
 b*x**2)*a**2*c*d**5 + 2*sqrt(a + b*x**2)*a**2*d**6*x - 2*sqrt(a + b*x**2) 
*a*b**2*c**2*d**3 + 2*sqrt(a + b*x**2)*a*b**2*c*d**4 + 4*sqrt(a + b*x**2)* 
a*b*c**3*d**3 + 4*sqrt(a + b*x**2)*a*b*c**2*d**4*x + 2*sqrt(a + b*x**2)...