\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2)}{(c+d x)^2} \, dx\) [41]

Optimal result

Integrand size = 29, antiderivative size = 234 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=-\frac {(2 c C-B d) \sqrt {a+b x^2}}{d^3}+\frac {C x \sqrt {a+b x^2}}{2 d^2}-\frac {\left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d^3 (c+d x)}+\frac {\left (a C d^2+2 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^4}+\frac {\left (a d^2 (2 c C-B d)+b c \left (3 c^2 C-2 B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^4 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (-6 c^2 C+c d (4 B-3 C x)+d^2 (-2 A+x (2 B+C x))\right )}{c+d x}+\frac {4 \left (a d^2 (2 c C-B d)+b c \left (3 c^2 C-2 B c d+A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}-\frac {\left (a C d^2+2 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{2 d^4} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(-6*c^2*C + c*d*(4*B - 3*C*x) + d^2*(-2*A + x*(2*B + C 
*x))))/(c + d*x) + (4*(a*d^2*(2*c*C - B*d) + b*c*(3*c^2*C - 2*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]] 
)/Sqrt[-(b*c^2) - a*d^2] - ((a*C*d^2 + 2*b*(3*c^2*C - 2*B*c*d + A*d^2))*Lo 
g[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/(2*d^4)
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.38, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2182, 25, 682, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-(((c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(3/2))/(d*(b*c^2 + a*d^2)*(c + d*x) 
)) + (-1/2*((2*(a*d^2*(2*c*C - B*d) + b*c*(3*c^2*C - 2*B*c*d + A*d^2)) - d 
*(a*C*d^2 + b*(3*c^2*C - 2*B*c*d + 2*A*d^2))*x)*Sqrt[a + b*x^2])/d^3 - ((b 
*c^2 + a*d^2)*(-(((a*C*d^2 + 2*b*(3*c^2*C - 2*B*c*d + A*d^2))*ArcTanh[(Sqr 
t[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (2*(a*d^2*(2*c*C - B*d) + b*c*(3* 
c^2*C - 2*B*c*d + A*d^2))*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])))/(2*d^3))/(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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(511\) vs. \(2(212)=424\).

Time = 0.22 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.19

method	result	size
risch	\(\frac {\left (C x d +2 B d -4 C c \right ) \sqrt {b \,x^{2}+a}}{2 d^{3}}+\frac {\frac {\left (2 A b \,d^{2}-4 B b c d +a C \,d^{2}+6 C b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 \left (2 A b c \,d^{2}-B a \,d^{3}-3 B b \,c^{2} d +2 C a c \,d^{2}+4 C b \,c^{3}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 \left (A a \,d^{4}+A b \,c^{2} d^{2}-B a c \,d^{3}-c^{3} B b d +C a \,c^{2} d^{2}+c^{4} C b \right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{2 d^{3}}\)	\(512\)
default	\(\frac {C \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{d^{2}}+\frac {\left (B d -2 C c \right ) \left (\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 {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}-\frac {\left (a \,d^{2}+b \,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^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {d^{2} \left (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 )^{\frac {3}{2}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \left (\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 {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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}-\frac {\left (a \,d^{2}+b \,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^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{a \,d^{2}+b \,c^{2}}+\frac {2 b \,d^{2} \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )-\frac {2 b 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}}}}{4 b}+\frac {\left (\frac {4 b \left (a \,d^{2}+b \,c^{2}\right )}{d^{2}}-\frac {4 b^{2} c^{2}}{d^{2}}\right ) \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\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 )}{8 b^{\frac {3}{2}}}\right )}{a \,d^{2}+b \,c^{2}}\right )}{d^{4}}\)	\(861\)

Input:

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

Output:

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*x**2)*(A + B*x + C*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. 478 vs. \(2 (213) = 426\).

Time = 0.10 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=-\frac {\sqrt {b x^{2} + a} C c^{2}}{d^{4} x + c d^{3}} + \frac {\sqrt {b x^{2} + a} B c}{d^{3} x + c d^{2}} - \frac {\sqrt {b x^{2} + a} A}{d^{2} x + c d} + \frac {\sqrt {b x^{2} + a} C x}{2 \, d^{2}} + \frac {3 \, C \sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} - \frac {2 \, B \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} + \frac {C a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d^{2}} + \frac {A \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{2}} - \frac {C b c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{5}} + \frac {B b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} - \frac {2 \, C \sqrt {a + \frac {b c^{2}}{d^{2}}} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{3}} - \frac {A b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} + \frac {B \sqrt {a + \frac {b c^{2}}{d^{2}}} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} - \frac {2 \, \sqrt {b x^{2} + a} C c}{d^{3}} + \frac {\sqrt {b x^{2} + a} B}{d^{2}} \] Input:

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

Output:

-sqrt(b*x^2 + a)*C*c^2/(d^4*x + c*d^3) + sqrt(b*x^2 + a)*B*c/(d^3*x + c*d^ 
2) - sqrt(b*x^2 + a)*A/(d^2*x + c*d) + 1/2*sqrt(b*x^2 + a)*C*x/d^2 + 3*C*s 
qrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^4 - 2*B*sqrt(b)*c*arcsinh(b*x/sqrt(a*b 
))/d^3 + 1/2*C*a*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) + A*sqrt(b)*arcsinh( 
b*x/sqrt(a*b))/d^2 - C*b*c^3*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^5) + B*b*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) - 2*C*sqrt(a + b*c^2/d^2)*c*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + 
c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^3 - A*b*c*arcsinh(b*c*x/(sqrt(a*b)*a 
bs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqrt(a + b*c^2/d^2)*d^3) + B 
*sqrt(a + b*c^2/d^2)*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a* 
b)*abs(d*x + c)))/d^2 - 2*sqrt(b*x^2 + a)*C*c/d^3 + sqrt(b*x^2 + a)*B/d^2
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1847, normalized size of antiderivative = 7.89 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*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 + 4*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*x - 4*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 - 4*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 + 8*sqrt(a*d**2 + b*c**2)*log( - sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b*c**3*d**2 + 8*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 - 8*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**3*c**3*d - 8*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**3*c**2*d**2*x 
 + 12*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 + 12*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 - 4*sqrt(a*d**2 + b*c 
**2)*log(c + d*x)*a*b**2*c**2*d**2 - 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)* 
a*b**2*c*d**3*x + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c*d**3 + 4*s 
qrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*d**4*x - 8*sqrt(a*d**2 + b*c**2)* 
log(c + d*x)*a*b*c**3*d**2 - 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2 
*d**3*x + 8*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**3*d + 8*sqrt(a*d**2 
 + b*c**2)*log(c + d*x)*b**3*c**2*d**2*x - 12*sqrt(a*d**2 + b*c**2)*log...