\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2)}{c+d x} \, dx\) [32]

Optimal result

Integrand size = 29, antiderivative size = 205 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=-\frac {\left (2 \left (B c-\frac {c^2 C}{d}-A d\right )+(c C-B d) x\right ) \sqrt {a+b x^2}}{2 d^2}+\frac {C \left (a+b x^2\right )^{3/2}}{3 b d}-\frac {\left (2 A b c d-\left (B-\frac {c C}{d}\right ) \left (2 b c^2+a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^3}-\frac {\sqrt {b c^2+a d^2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (2 a C d^2+b \left (6 c^2 C-3 c d (2 B+C x)+d^2 \left (6 A+3 B x+2 C x^2\right )\right )\right )}{b}+12 \sqrt {-b c^2-a d^2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {3 \left (a d^2 (c C-B d)+2 b c \left (c^2 C-B c d+A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{6 d^4} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(2*a*C*d^2 + b*(6*c^2*C - 3*c*d*(2*B + C*x) + d^2*(6*A 
 + 3*B*x + 2*C*x^2))))/b + 12*Sqrt[-(b*c^2) - a*d^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]] 
+ (3*(a*d^2*(c*C - B*d) + 2*b*c*(c^2*C - B*c*d + A*d^2))*Log[-(Sqrt[b]*x) 
+ Sqrt[a + b*x^2]])/Sqrt[b])/(6*d^4)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2185, 27, 682, 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.

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 [A] (verified)

Time = 0.19 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.52

method	result	size
risch	\(\frac {\left (2 C \,d^{2} b \,x^{2}+3 B b \,d^{2} x -3 C b c d x +6 A b \,d^{2}-6 B b c d +2 a C \,d^{2}+6 C b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{6 b \,d^{3}}-\frac {\frac {\left (2 A b c \,d^{2}-B a \,d^{3}-2 B b \,c^{2} d +C a c \,d^{2}+2 C b \,c^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 \left (A a \,d^{4}+A b \,c^{2} d^{2}-B a c \,d^{3}-c^{3} B b d +C a \,c^{2} d^{2}+c^{4} C b \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 d^{3}}\)	\(312\)
default	\(\frac {B d \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+\frac {C d \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}-C c \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{d^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}\)	\(376\)

Input:

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

Output:

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=-\frac {\sqrt {b x^{2} + a} C c x}{2 \, d^{2}} + \frac {\sqrt {b x^{2} + a} B x}{2 \, d} - \frac {C \sqrt {b} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {B \sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} - \frac {C a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d^{2}} - \frac {A \sqrt {b} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{2}} + \frac {B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d} + \frac {C \sqrt {a + \frac {b c^{2}}{d^{2}}} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{3}} - \frac {B \sqrt {a + \frac {b c^{2}}{d^{2}}} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} + \frac {A \sqrt {a + \frac {b c^{2}}{d^{2}}} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d} + \frac {\sqrt {b x^{2} + a} C c^{2}}{d^{3}} - \frac {\sqrt {b x^{2} + a} B c}{d^{2}} + \frac {\sqrt {b x^{2} + a} A}{d} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C}{3 \, b d} \] Input:

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

Output:

-1/2*sqrt(b*x^2 + a)*C*c*x/d^2 + 1/2*sqrt(b*x^2 + a)*B*x/d - C*sqrt(b)*c^3 
*arcsinh(b*x/sqrt(a*b))/d^4 + B*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^3 - 1 
/2*C*a*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^2) - A*sqrt(b)*c*arcsinh(b*x/sq 
rt(a*b))/d^2 + 1/2*B*a*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d) + C*sqrt(a + b*c 
^2/d^2)*c^2*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d* 
x + c)))/d^3 - B*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^2 + A*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 + sqrt(b*x^ 
2 + a)*C*c^2/d^3 - sqrt(b*x^2 + a)*B*c/d^2 + sqrt(b*x^2 + a)*A/d + 1/3*(b* 
x^2 + a)^(3/2)*C/(b*d)
 

Giac [F(-2)]

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

integrate((b*x^2+a)^(1/2)*(C*x^2+B*x+A)/(d*x+c),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. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (C\,x^2+B\,x+A\right )}{c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3439, normalized size of antiderivative = 16.78 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*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*c*d**2 + 6*sqrt(b)*sq 
rt(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**2*c**2*d - 6*sqrt(b)*sqrt(2*sqrt(b)*sq 
rt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*sqrt(a*d**2 + b*c**2)*atan((sqr 
t(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*c**4 - 6*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*d**4 - 6*sqrt(2*sqrt(b)*s 
qrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqr 
t(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*b 
**2*c**2*d**2 + 6*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*b**2*c*d**3 - 6*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*b*c**3*d** 
2 + 6*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sq 
rt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c ...