\(\int \frac {x^2 \sqrt {a+b x^2} (A+B x+C x^2)}{c+d x} \, dx\) [30]

Optimal result

Integrand size = 32, antiderivative size = 367 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\frac {\left (8 b c^2 \left (c^2 C-B c d+A d^2\right )+d \left (a d^2 (c C-B d)-4 b c \left (c^2 C-B c d+A d^2\right )\right ) x\right ) \sqrt {a+b x^2}}{8 b d^5}-\frac {\left (8 a C d^2-b \left (47 c^2 C-35 B c d+20 A d^2\right )\right ) \left (a+b x^2\right )^{3/2}}{60 b^2 d^3}-\frac {(13 c C-5 B d) (c+d x) \left (a+b x^2\right )^{3/2}}{20 b d^3}+\frac {C (c+d x)^2 \left (a+b x^2\right )^{3/2}}{5 b d^3}+\frac {\left (a^2 d^4 (c C-B d)-8 b^2 c^3 \left (c^2 C-B c d+A d^2\right )-4 a b c d^2 \left (c^2 C-B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2} d^6}-\frac {c^2 \sqrt {b c^2+a d^2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^6} \] Output:

1/8*(8*b*c^2*(A*d^2-B*c*d+C*c^2)+d*(a*d^2*(-B*d+C*c)-4*b*c*(A*d^2-B*c*d+C* 
c^2))*x)*(b*x^2+a)^(1/2)/b/d^5-1/60*(8*a*C*d^2-b*(20*A*d^2-35*B*c*d+47*C*c 
^2))*(b*x^2+a)^(3/2)/b^2/d^3-1/20*(-5*B*d+13*C*c)*(d*x+c)*(b*x^2+a)^(3/2)/ 
b/d^3+1/5*C*(d*x+c)^2*(b*x^2+a)^(3/2)/b/d^3+1/8*(a^2*d^4*(-B*d+C*c)-8*b^2* 
c^3*(A*d^2-B*c*d+C*c^2)-4*a*b*c*d^2*(A*d^2-B*c*d+C*c^2))*arctanh(b^(1/2)*x 
/(b*x^2+a)^(1/2))/b^(3/2)/d^6-c^2*(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^6
 

Mathematica [A] (verified)

Time = 1.81 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (-16 a^2 C d^4+a b d^2 \left (40 c^2 C-5 c d (8 B+3 C x)+d^2 \left (40 A+15 B x+8 C x^2\right )\right )+2 b^2 \left (60 c^4 C-30 c^3 d (2 B+C x)+10 c^2 d^2 (6 A+x (3 B+2 C x))-5 c d^3 x (6 A+x (4 B+3 C x))+d^4 x^2 (20 A+3 x (5 B+4 C x))\right )\right )}{b^2}+240 c^2 \sqrt {-b c^2-a d^2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {15 \left (a^2 d^4 (-c C+B d)+8 b^2 c^3 \left (c^2 C-B c d+A d^2\right )+4 a b c d^2 \left (c^2 C-B c d+A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}}}{120 d^6} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(-16*a^2*C*d^4 + a*b*d^2*(40*c^2*C - 5*c*d*(8*B + 3*C* 
x) + d^2*(40*A + 15*B*x + 8*C*x^2)) + 2*b^2*(60*c^4*C - 30*c^3*d*(2*B + C* 
x) + 10*c^2*d^2*(6*A + x*(3*B + 2*C*x)) - 5*c*d^3*x*(6*A + x*(4*B + 3*C*x) 
) + d^4*x^2*(20*A + 3*x*(5*B + 4*C*x)))))/b^2 + 240*c^2*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]] + (15*(a^2*d^4*(-(c*C) + B*d) + 8*b^2*c^3*(c^2*C 
 - B*c*d + A*d^2) + 4*a*b*c*d^2*(c^2*C - B*c*d + A*d^2))*Log[-(Sqrt[b]*x) 
+ Sqrt[a + b*x^2]])/b^(3/2))/(120*d^6)
 

Rubi [A] (verified)

Time = 2.16 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2185, 25, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int -\frac {\sqrt {b x^2+a} \left (b d^3 (13 c C-5 B d) x^3+d^2 \left (11 b C c^2-5 A b d^2+2 a C d^2\right ) x^2+c C d \left (3 b c^2+4 a d^2\right ) x+2 a c^2 C d^2\right )}{c+d x}dx}{5 b d^4}+\frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\int \frac {\sqrt {b x^2+a} \left (b d^3 (13 c C-5 B d) x^3+d^2 \left (11 b C c^2-5 A b d^2+2 a C d^2\right ) x^2+c C d \left (3 b c^2+4 a d^2\right ) x+2 a c^2 C d^2\right )}{c+d x}dx}{5 b d^4}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {\int -\frac {\sqrt {b x^2+a} \left (-b \left (8 a C d^2-b \left (47 C c^2-35 B d c+20 A d^2\right )\right ) x^2 d^5+5 a b c (c C-B d) d^5+b \left (3 b c^2 (9 c C-5 B d)-a d^2 (3 c C+5 B d)\right ) x d^4\right )}{c+d x}dx}{4 b d^3}+\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)}{5 b d^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {\int \frac {\sqrt {b x^2+a} \left (-b \left (8 a C d^2-b \left (47 C c^2-35 B d c+20 A d^2\right )\right ) x^2 d^5+5 a b c (c C-B d) d^5+b \left (3 b c^2 (9 c C-5 B d)-a d^2 (3 c C+5 B d)\right ) x d^4\right )}{c+d x}dx}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {\frac {\int \frac {15 b^2 d^6 \left (a c d (c C-B d)+\left (a d^2 (c C-B d)-4 b c \left (C c^2-B d c+A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{3 b d^2}-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \int \frac {\left (a c d (c C-B d)+\left (a d^2 (c C-B d)-4 b c \left (C c^2-B d c+A d^2\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\int \frac {b \left (a c d \left (a (c C-B d) d^2+4 b c \left (C c^2-B d c+A d^2\right )\right )+\left (a^2 (c C-B d) d^4-4 a b c \left (C c^2-B d c+A d^2\right ) d^2-8 b^2 c^3 \left (C c^2-B d c+A d^2\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\int \frac {a c d \left (a (c C-B d) d^2+4 b c \left (C c^2-B d c+A d^2\right )\right )+\left (a^2 (c C-B d) d^4-4 a b c \left (C c^2-B d c+A d^2\right ) d^2-8 b^2 c^3 \left (C c^2-B d c+A d^2\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\frac {\left (a^2 d^4 (c C-B d)-4 a b c d^2 \left (A d^2-B c d+c^2 C\right )-8 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}+\frac {8 b c^2 \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\frac {\left (a^2 d^4 (c C-B d)-4 a b c d^2 \left (A d^2-B c d+c^2 C\right )-8 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}+\frac {8 b c^2 \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\frac {8 b c^2 \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4 (c C-B d)-4 a b c d^2 \left (A d^2-B c d+c^2 C\right )-8 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{\sqrt {b} d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4 (c C-B d)-4 a b c d^2 \left (A d^2-B c d+c^2 C\right )-8 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{\sqrt {b} d}-\frac {8 b c^2 \left (a d^2+b c^2\right ) \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \left (a+b x^2\right )^{3/2} (c+d x)^2}{5 b d^3}-\frac {\frac {1}{4} d \left (a+b x^2\right )^{3/2} (c+d x) (13 c C-5 B d)-\frac {5 b d^4 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a^2 d^4 (c C-B d)-4 a b c d^2 \left (A d^2-B c d+c^2 C\right )-8 b^2 c^3 \left (A d^2-B c d+c^2 C\right )\right )}{\sqrt {b} d}-\frac {8 b c^2 \sqrt {a d^2+b c^2} \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d}}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (a d^2 (c C-B d)-4 b c \left (A d^2-B c d+c^2 C\right )\right )+8 b c^2 \left (A d^2-B c d+c^2 C\right )\right )}{2 d^2}\right )-\frac {1}{3} d^4 \left (a+b x^2\right )^{3/2} \left (8 a C d^2-b \left (20 A d^2-35 B c d+47 c^2 C\right )\right )}{4 b d^3}}{5 b d^4}\)

Input:

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

Output:

(C*(c + d*x)^2*(a + b*x^2)^(3/2))/(5*b*d^3) - ((d*(13*c*C - 5*B*d)*(c + d* 
x)*(a + b*x^2)^(3/2))/4 - (-1/3*(d^4*(8*a*C*d^2 - b*(47*c^2*C - 35*B*c*d + 
 20*A*d^2))*(a + b*x^2)^(3/2)) + 5*b*d^4*(((8*b*c^2*(c^2*C - B*c*d + A*d^2 
) + d*(a*d^2*(c*C - B*d) - 4*b*c*(c^2*C - B*c*d + A*d^2))*x)*Sqrt[a + b*x^ 
2])/(2*d^2) + (((a^2*d^4*(c*C - B*d) - 8*b^2*c^3*(c^2*C - B*c*d + A*d^2) - 
 4*a*b*c*d^2*(c^2*C - B*c*d + A*d^2))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]] 
)/(Sqrt[b]*d) - (8*b*c^2*Sqrt[b*c^2 + a*d^2]*(c^2*C - B*c*d + A*d^2)*ArcTa 
nh[(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))/(5*b*d^4)
 

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 = 0.27 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.42

method	result	size
risch	\(\frac {\left (24 C \,d^{4} b^{2} x^{4}+30 B \,b^{2} d^{4} x^{3}-30 C \,b^{2} c \,d^{3} x^{3}+40 A \,b^{2} d^{4} x^{2}-40 B \,b^{2} c \,d^{3} x^{2}+8 C a b \,d^{4} x^{2}+40 C \,b^{2} c^{2} d^{2} x^{2}-60 A \,b^{2} c \,d^{3} x +15 B a b \,d^{4} x +60 B \,b^{2} c^{2} d^{2} x -15 C a b c \,d^{3} x -60 C \,b^{2} c^{3} d x +40 A a b \,d^{4}+120 A \,b^{2} c^{2} d^{2}-40 B a b c \,d^{3}-120 B \,b^{2} c^{3} d -16 a^{2} C \,d^{4}+40 C a b \,c^{2} d^{2}+120 C \,b^{2} c^{4}\right ) \sqrt {b \,x^{2}+a}}{120 b^{2} d^{5}}-\frac {\frac {\left (4 A a b c \,d^{4}+8 A \,b^{2} c^{3} d^{2}+a^{2} B \,d^{5}-4 B a b \,c^{2} d^{3}-8 B \,b^{2} c^{4} d -C \,a^{2} c \,d^{4}+4 C a b \,c^{3} d^{2}+8 C \,b^{2} c^{5}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {8 c^{2} \left (A a \,d^{4}+A b \,c^{2} d^{2}-B a c \,d^{3}-c^{3} B b d +C a \,c^{2} d^{2}+c^{4} C b \right ) b \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{8 d^{5} b}\)	\(520\)
default	\(\frac {c^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{5}}-\frac {C \,c^{3} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )-d^{2} \left (B d -C c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4 b}-\frac {a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4 b}\right )-\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 b}+A c \,d^{2} \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )-B \,c^{2} d \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )-C \,d^{3} \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{5 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{15 b^{2}}\right )}{d^{4}}\)	\(550\)

Input:

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

Output:

1/120*(24*C*b^2*d^4*x^4+30*B*b^2*d^4*x^3-30*C*b^2*c*d^3*x^3+40*A*b^2*d^4*x 
^2-40*B*b^2*c*d^3*x^2+8*C*a*b*d^4*x^2+40*C*b^2*c^2*d^2*x^2-60*A*b^2*c*d^3* 
x+15*B*a*b*d^4*x+60*B*b^2*c^2*d^2*x-15*C*a*b*c*d^3*x-60*C*b^2*c^3*d*x+40*A 
*a*b*d^4+120*A*b^2*c^2*d^2-40*B*a*b*c*d^3-120*B*b^2*c^3*d-16*C*a^2*d^4+40* 
C*a*b*c^2*d^2+120*C*b^2*c^4)*(b*x^2+a)^(1/2)/b^2/d^5-1/8/d^5/b*((4*A*a*b*c 
*d^4+8*A*b^2*c^3*d^2+B*a^2*d^5-4*B*a*b*c^2*d^3-8*B*b^2*c^4*d-C*a^2*c*d^4+4 
*C*a*b*c^3*d^2+8*C*b^2*c^5)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+8*c^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)*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. \[ \int \frac {x^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.70 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (A+B x+C x^2\right )}{c+d x} \, dx =\text {Too large to display} \] Input:

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

Output:

1/5*(b*x^2 + a)^(3/2)*C*x^2/(b*d) - 1/2*sqrt(b*x^2 + a)*C*c^3*x/d^4 + 1/2* 
sqrt(b*x^2 + a)*B*c^2*x/d^3 - 1/2*sqrt(b*x^2 + a)*A*c*x/d^2 - 1/4*(b*x^2 + 
 a)^(3/2)*C*c*x/(b*d^2) + 1/8*sqrt(b*x^2 + a)*C*a*c*x/(b*d^2) + 1/4*(b*x^2 
 + a)^(3/2)*B*x/(b*d) - 1/8*sqrt(b*x^2 + a)*B*a*x/(b*d) - C*sqrt(b)*c^5*ar 
csinh(b*x/sqrt(a*b))/d^6 + B*sqrt(b)*c^4*arcsinh(b*x/sqrt(a*b))/d^5 - 1/2* 
C*a*c^3*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^4) - A*sqrt(b)*c^3*arcsinh(b*x/s 
qrt(a*b))/d^4 + 1/2*B*a*c^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) + 1/8*C*a 
^2*c*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^2) - 1/2*A*a*c*arcsinh(b*x/sqrt(a*b 
))/(sqrt(b)*d^2) - 1/8*B*a^2*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d) + C*sqrt(a 
 + b*c^2/d^2)*c^4*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)* 
abs(d*x + c)))/d^5 - B*sqrt(a + b*c^2/d^2)*c^3*arcsinh(b*c*x/(sqrt(a*b)*ab 
s(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^4 + A*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 + sqrt(b*x^2 + a)*C*c^4/d^5 - sqrt(b*x^2 + a)*B*c^3/d^4 + sqrt(b*x^2 + 
a)*A*c^2/d^3 + 1/3*(b*x^2 + a)^(3/2)*C*c^2/(b*d^3) - 1/3*(b*x^2 + a)^(3/2) 
*B*c/(b*d^2) - 2/15*(b*x^2 + a)^(3/2)*C*a/(b^2*d) + 1/3*(b*x^2 + a)^(3/2)* 
A/(b*d)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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