\(\int \frac {x \sqrt {a+b x^2} (A+B x+C x^2)}{c+d x} \, dx\) [31]

Optimal result

Integrand size = 30, antiderivative size = 280 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=-\frac {\left (8 b c \left (c^2 C-B c d+A d^2\right )-d^3 \left (4 A b-a C+\frac {4 b c (c C-B d)}{d^2}\right ) x\right ) \sqrt {a+b x^2}}{8 b d^4}-\frac {(7 c C-4 B d) \left (a+b x^2\right )^{3/2}}{12 b d^2}+\frac {C (c+d x) \left (a+b x^2\right )^{3/2}}{4 b d^2}+\frac {\left (2 a b c^2 C+\left (2 b c^2+a d^2\right ) \left (4 A b-a C+\frac {4 b c (c C-B d)}{d^2}\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2} d^3}+\frac {c \sqrt {b c^2+a d^2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^5} \] Output:

-1/8*(8*b*c*(A*d^2-B*c*d+C*c^2)-d^3*(4*A*b-a*C+4*b*c*(-B*d+C*c)/d^2)*x)*(b 
*x^2+a)^(1/2)/b/d^4-1/12*(-4*B*d+7*C*c)*(b*x^2+a)^(3/2)/b/d^2+1/4*C*(d*x+c 
)*(b*x^2+a)^(3/2)/b/d^2+1/8*(2*a*b*c^2*C+(a*d^2+2*b*c^2)*(4*A*b-a*C+4*b*c* 
(-B*d+C*c)/d^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3/2)/d^3+c*(a*d^2+b 
*c^2)^(1/2)*(A*d^2-B*c*d+C*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/( 
b*x^2+a)^(1/2))/d^5
 

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.98 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (-8 c C+8 B d+3 C d x)-2 b \left (12 c^3 C-6 c^2 d (2 B+C x)+2 c d^2 \left (6 A+3 B x+2 C x^2\right )-d^3 x \left (6 A+4 B x+3 C x^2\right )\right )\right )}{b}-48 c \sqrt {-b c^2-a d^2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )-\frac {3 \left (-a^2 C d^4+8 b^2 c^2 \left (c^2 C-B c d+A d^2\right )+4 a b d^2 \left (c^2 C-B c d+A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{24 d^5} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(a*d^2*(-8*c*C + 8*B*d + 3*C*d*x) - 2*b*(12*c^3*C - 6* 
c^2*d*(2*B + C*x) + 2*c*d^2*(6*A + 3*B*x + 2*C*x^2) - d^3*x*(6*A + 4*B*x + 
 3*C*x^2))))/b - 48*c*Sqrt[-(b*c^2) - a*d^2]*(c^2*C - B*c*d + A*d^2)*ArcTa 
n[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] - (3*(-( 
a^2*C*d^4) + 8*b^2*c^2*(c^2*C - B*c*d + A*d^2) + 4*a*b*d^2*(c^2*C - B*c*d 
+ A*d^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(3/2))/(24*d^5)
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2185, 25, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int -\frac {\sqrt {b x^2+a} \left (b (7 c C-4 B d) x^2 d^2+a c C d^2+\left (3 b C c^2-4 A b d^2+a C d^2\right ) x d\right )}{c+d x}dx}{4 b d^3}+\frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {\int \frac {\sqrt {b x^2+a} \left (b (7 c C-4 B d) x^2 d^2+a c C d^2+\left (3 b C c^2-4 A b d^2+a C d^2\right ) x d\right )}{c+d x}dx}{4 b d^3}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {\frac {\int \frac {3 b d^3 \left (a c C d+\left (a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{3 b d^2}+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \int \frac {\left (a c C d+\left (a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {\int \frac {b \left (a c d \left (a C d^2+4 b \left (C c^2-B d c+A d^2\right )\right )-\left (2 a b c^2 C d^2-\left (2 b c^2+a d^2\right ) \left (a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {\int \frac {a c d \left (a C d^2+4 b \left (C c^2-B d c+A d^2\right )\right )-\left (2 a b c^2 C d^2-\left (2 b c^2+a d^2\right ) \left (a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {\frac {8 b c \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (2 a b c^2 C d^2-\left (a d^2+2 b c^2\right ) \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {\frac {8 b c \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (2 a b c^2 C d^2-\left (a d^2+2 b c^2\right ) \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {\frac {8 b c \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a b c^2 C d^2-\left (a d^2+2 b c^2\right ) \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )\right )}{\sqrt {b} d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {-\frac {8 b c \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a b c^2 C d^2-\left (a d^2+2 b c^2\right ) \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )\right )}{\sqrt {b} d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)}{4 b d^2}-\frac {d \left (\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 a b c^2 C d^2-\left (a d^2+2 b c^2\right ) \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )\right )}{\sqrt {b} d}-\frac {8 b c \sqrt {a d^2+b c^2} \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a C d^2-4 b \left (A d^2-B c d+c^2 C\right )\right )+8 b c \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )+\frac {1}{3} d \left (a+b x^2\right )^{3/2} (7 c C-4 B d)}{4 b d^3}\)

Input:

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

Output:

(C*(c + d*x)*(a + b*x^2)^(3/2))/(4*b*d^2) - ((d*(7*c*C - 4*B*d)*(a + b*x^2 
)^(3/2))/3 + d*(((8*b*c*(c^2*C - B*c*d + A*d^2) + d*(a*C*d^2 - 4*b*(c^2*C 
- B*c*d + A*d^2))*x)*Sqrt[a + b*x^2])/(2*d^2) + (-(((2*a*b*c^2*C*d^2 - (2* 
b*c^2 + a*d^2)*(a*C*d^2 - 4*b*(c^2*C - B*c*d + A*d^2)))*ArcTanh[(Sqrt[b]*x 
)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (8*b*c*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)))/(4*b*d^3)
 

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 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 = 0.22 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.42

method	result	size
risch	\(-\frac {\left (-6 C \,d^{3} b \,x^{3}-8 B b \,d^{3} x^{2}+8 C b c \,d^{2} x^{2}-12 A b \,d^{3} x +12 B b c \,d^{2} x -3 C a \,d^{3} x -12 C b \,c^{2} d x +24 A b c \,d^{2}-8 B a \,d^{3}-24 B b \,c^{2} d +8 C a c \,d^{2}+24 C b \,c^{3}\right ) \sqrt {b \,x^{2}+a}}{24 b \,d^{4}}+\frac {\frac {\left (4 A a b \,d^{4}+8 A \,b^{2} c^{2} d^{2}-4 B a b c \,d^{3}-8 B \,b^{2} c^{3} d -a^{2} C \,d^{4}+4 C a b \,c^{2} d^{2}+8 C \,b^{2} c^{4}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {8 c \left (A a \,d^{4}+A b \,c^{2} d^{2}-B a c \,d^{3}-c^{3} B b d +C a \,c^{2} d^{2}+c^{4} C b \right ) b \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{8 b \,d^{4}}\)	\(399\)
default	\(\frac {A \,d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+C \,c^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )+\frac {d \left (B d -C c \right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}+C \,d^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )-B c d \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{d^{3}}-\frac {c \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}\)	\(490\)

Input:

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

Output:

-1/24/b*(-6*C*b*d^3*x^3-8*B*b*d^3*x^2+8*C*b*c*d^2*x^2-12*A*b*d^3*x+12*B*b* 
c*d^2*x-3*C*a*d^3*x-12*C*b*c^2*d*x+24*A*b*c*d^2-8*B*a*d^3-24*B*b*c^2*d+8*C 
*a*c*d^2+24*C*b*c^3)*(b*x^2+a)^(1/2)/d^4+1/8/b/d^4*((4*A*a*b*d^4+8*A*b^2*c 
^2*d^2-4*B*a*b*c*d^3-8*B*b^2*c^3*d-C*a^2*d^4+4*C*a*b*c^2*d^2+8*C*b^2*c^4)/ 
d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+8*c*(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)*b/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. \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.75 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\frac {\sqrt {b x^{2} + a} C c^{2} x}{2 \, d^{3}} - \frac {\sqrt {b x^{2} + a} B c x}{2 \, d^{2}} + \frac {\sqrt {b x^{2} + a} A x}{2 \, d} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C x}{4 \, b d} - \frac {\sqrt {b x^{2} + a} C a x}{8 \, b d} + \frac {C \sqrt {b} c^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{5}} - \frac {B \sqrt {b} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{4}} + \frac {C a c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d^{3}} + \frac {A \sqrt {b} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} - \frac {B a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d^{2}} - \frac {C a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}} d} + \frac {A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d} - \frac {C \sqrt {a + \frac {b c^{2}}{d^{2}}} c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{4}} + \frac {B \sqrt {a + \frac {b c^{2}}{d^{2}}} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{3}} - \frac {A \sqrt {a + \frac {b c^{2}}{d^{2}}} c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{2}} - \frac {\sqrt {b x^{2} + a} C c^{3}}{d^{4}} + \frac {\sqrt {b x^{2} + a} B c^{2}}{d^{3}} - \frac {\sqrt {b x^{2} + a} A c}{d^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C c}{3 \, b d^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B}{3 \, b d} \] Input:

integrate(x*(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^2*x/d^3 - 1/2*sqrt(b*x^2 + a)*B*c*x/d^2 + 1/2*sqrt 
(b*x^2 + a)*A*x/d + 1/4*(b*x^2 + a)^(3/2)*C*x/(b*d) - 1/8*sqrt(b*x^2 + a)* 
C*a*x/(b*d) + C*sqrt(b)*c^4*arcsinh(b*x/sqrt(a*b))/d^5 - B*sqrt(b)*c^3*arc 
sinh(b*x/sqrt(a*b))/d^4 + 1/2*C*a*c^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) 
 + A*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^3 - 1/2*B*a*c*arcsinh(b*x/sqrt(a 
*b))/(sqrt(b)*d^2) - 1/8*C*a^2*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d) + 1/2*A* 
a*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d) - C*sqrt(a + b*c^2/d^2)*c^3*arcsinh(b 
*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^4 + B*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 - A*sqrt(a + b*c^2/d^2)*c*arcsinh(b*c*x/(sqrt(a*b)*ab 
s(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^2 - sqrt(b*x^2 + a)*C*c^3/d^ 
4 + sqrt(b*x^2 + a)*B*c^2/d^3 - sqrt(b*x^2 + a)*A*c/d^2 - 1/3*(b*x^2 + a)^ 
(3/2)*C*c/(b*d^2) + 1/3*(b*x^2 + a)^(3/2)*B/(b*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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