\(\int \frac {x \sqrt {a+b x^2} (A+B x+C x^2)}{(c+d x)^2} \, dx\) [40]

Optimal result

Integrand size = 30, antiderivative size = 309 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\left (a C d^2+3 b \left (3 c^2 C-2 B c d+A d^2\right )\right ) \sqrt {a+b x^2}}{3 b d^4}-\frac {(2 c C-B d) x \sqrt {a+b x^2}}{2 d^3}+\frac {C x^2 \sqrt {a+b x^2}}{3 d^2}+\frac {c \left (c^2 C-B c d+A d^2\right ) \sqrt {a+b x^2}}{d^4 (c+d x)}-\frac {\left (a d^2 (2 c C-B d)+2 b c \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^5}-\frac {\left (a d^2 \left (3 c^2 C-2 B c d+A d^2\right )+b c^2 \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^5 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.91 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (2 a C d^2 (c+d x)+b \left (24 c^3 C-6 c^2 d (3 B-2 C x)+c d^2 \left (12 A-9 B x-4 C x^2\right )+d^3 x \left (6 A+3 B x+2 C x^2\right )\right )\right )}{b (c+d x)}-\frac {12 \left (a d^2 \left (3 c^2 C-2 B c d+A d^2\right )+b c^2 \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {3 \left (a d^2 (2 c C-B d)+2 b c \left (4 c^2 C-3 B c d+2 A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{6 d^5} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(2*a*C*d^2*(c + d*x) + b*(24*c^3*C - 6*c^2*d*(3*B - 2* 
C*x) + c*d^2*(12*A - 9*B*x - 4*C*x^2) + d^3*x*(6*A + 3*B*x + 2*C*x^2))))/( 
b*(c + d*x)) - (12*(a*d^2*(3*c^2*C - 2*B*c*d + A*d^2) + b*c^2*(4*c^2*C - 3 
*B*c*d + 2*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] + (3*(a*d^2*(2*c*C - B*d) + 2*b*c* 
(4*c^2*C - 3*B*c*d + 2*A*d^2))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b 
])/(6*d^5)
 

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2182, 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.

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(c*(c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(3/2))/(d^2*(b*c^2 + a*d^2)*(c + d* 
x)) - (-1/3*(C*(a/b + c^2/d^2)*(a + b*x^2)^(3/2)) - (((2*(a*d^2*(3*c^2*C - 
 2*B*c*d + A*d^2) + b*c^2*(4*c^2*C - 3*B*c*d + 2*A*d^2)) - d*(a*d^2*(2*c*C 
 - B*d) + b*c*(4*c^2*C - 3*B*c*d + 2*A*d^2))*x)*Sqrt[a + b*x^2])/(2*d^2) + 
 ((b*c^2 + a*d^2)*(-(((a*d^2*(2*c*C - B*d) + 2*b*c*(4*c^2*C - 3*B*c*d + 2* 
A*d^2))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (2*(a*d^2*(3* 
c^2*C - 2*B*c*d + A*d^2) + b*c^2*(4*c^2*C - 3*B*c*d + 2*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^2))/d^2)/(b*c^2 + a*d^2)
 

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 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. \(578\) vs. \(2(281)=562\).

Time = 0.25 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.87

method	result	size
risch	\(\frac {\left (2 C \,d^{2} b \,x^{2}+3 B b \,d^{2} x -6 C b c d x +6 A b \,d^{2}-12 B b c d +2 a C \,d^{2}+18 C b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{6 b \,d^{4}}-\frac {\frac {\left (4 A b c \,d^{2}-B a \,d^{3}-6 B b \,c^{2} d +2 C a c \,d^{2}+8 C b \,c^{3}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 \left (A a \,d^{4}+3 A b \,c^{2} d^{2}-2 B a c \,d^{3}-4 c^{3} B b d +3 C a \,c^{2} d^{2}+5 c^{4} C b \right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {2 c \left (A a \,d^{4}+A b \,c^{2} d^{2}-B a c \,d^{3}-c^{3} B b d +C a \,c^{2} d^{2}+c^{4} C b \right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}}{2 d^{4}}\)	\(579\)
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}-2 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^{3}}+\frac {\left (A \,d^{2}-2 B c d +3 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}}-\frac {c \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^{5}}\)	\(930\)

Input:

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

Output:

1/6*(2*C*b*d^2*x^2+3*B*b*d^2*x-6*C*b*c*d*x+6*A*b*d^2-12*B*b*c*d+2*C*a*d^2+ 
18*C*b*c^2)*(b*x^2+a)^(1/2)/b/d^4-1/2/d^4*((4*A*b*c*d^2-B*a*d^3-6*B*b*c^2* 
d+2*C*a*c*d^2+8*C*b*c^3)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+2/d^2*(A* 
a*d^4+3*A*b*c^2*d^2-2*B*a*c*d^3-4*B*b*c^3*d+3*C*a*c^2*d^2+5*C*b*c^4)/((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))+2*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)/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. \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral(x*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. 617 vs. \(2 (282) = 564\).

Time = 0.10 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.00 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

integrate(x*(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^3/(d^5*x + c*d^4) - sqrt(b*x^2 + a)*B*c^2/(d^4*x + c*d 
^3) + sqrt(b*x^2 + a)*A*c/(d^3*x + c*d^2) - sqrt(b*x^2 + a)*C*c*x/d^3 + 1/ 
2*sqrt(b*x^2 + a)*B*x/d^2 - 4*C*sqrt(b)*c^3*arcsinh(b*x/sqrt(a*b))/d^5 + 3 
*B*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^4 - C*a*c*arcsinh(b*x/sqrt(a*b))/( 
sqrt(b)*d^3) - 2*A*sqrt(b)*c*arcsinh(b*x/sqrt(a*b))/d^3 + 1/2*B*a*arcsinh( 
b*x/sqrt(a*b))/(sqrt(b)*d^2) + C*b*c^4*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^6) - B*b*c^3*ar 
csinh(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) + 3*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^4 + A*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*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^3 + 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^2 + 3*sqrt( 
b*x^2 + a)*C*c^2/d^4 - 2*sqrt(b*x^2 + a)*B*c/d^3 + sqrt(b*x^2 + a)*A/d^2 + 
 1/3*(b*x^2 + a)^(3/2)*C/(b*d^2)
 

Giac [F(-1)]

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

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

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

Output:

int((x*(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 = 2296, normalized size of antiderivative = 7.43 \[ \int \frac {x \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(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)*a**2*b*c*d**4 + 12*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sq 
rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*d**5*x + 24*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**3*d* 
*2 + 24*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**3*x - 24*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**3 - 24*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**4*x + 36*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**4*d**2 + 36*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**3*d**3*x - 36*sqr 
t(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**4*d - 36*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**2*x + 48*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**6 + 48*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*d*x - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**4 - 12* 
sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*d**5*x - 24*sqrt(a*d**2 + b*c**2 
)*log(c + d*x)*a*b**2*c**3*d**2 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a* 
b**2*c**2*d**3*x + 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**2*d*...