\(\int \frac {x^3 (A+B x+C x^2)}{(c+d x) \sqrt {a+b x^2}} \, dx\) [115]

Optimal result

Integrand size = 32, antiderivative size = 326 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {\left (2 a d^2 (c C-B d)-3 b c \left (c^2 C-B c d+A d^2\right )\right ) \sqrt {a+b x^2}}{3 b^2 d^4}-\frac {\left (3 a C d^2-4 b \left (c^2 C-B c d+A d^2\right )\right ) x \sqrt {a+b x^2}}{8 b^2 d^3}-\frac {(c C-B d) x^2 \sqrt {a+b x^2}}{3 b d^2}+\frac {C x^3 \sqrt {a+b x^2}}{4 b d}+\frac {\left (3 a^2 C d^4+8 b^2 c^2 \left (c^2 C-B c d+A d^2\right )-4 a b d^2 \left (c^2 C-B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2} d^5}+\frac {c^3 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^5 \sqrt {b c^2+a d^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.56 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (16 c C-16 B d-9 C d x)-2 b \left (12 c^3 C-6 c^2 d (2 B+C x)+2 c d^2 \left (6 A+3 B x+2 C x^2\right )-d^3 x \left (6 A+4 B x+3 C x^2\right )\right )\right )}{b^2}+\frac {48 c^3 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}-\frac {3 \left (3 a^2 C d^4+8 b^2 c^2 \left (c^2 C-B c d+A d^2\right )-4 a b d^2 \left (c^2 C-B c d+A d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{5/2}}}{24 d^5} \] Input:

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

Output:

((d*Sqrt[a + b*x^2]*(a*d^2*(16*c*C - 16*B*d - 9*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^2 + (48*c^3*(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]])/Sqrt[-(b*c^2) - a*d^2] - 
 (3*(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^(5/2))/(24*d^5)
 

Rubi [A] (verified)

Time = 2.73 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2185, 25, 2185, 25, 2185, 2185, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int -\frac {b d^4 (13 c C-4 B d) x^4+d^3 \left (15 b C c^2-4 A b d^2+3 a C d^2\right ) x^3+c C d^2 \left (7 b c^2+9 a d^2\right ) x^2+c^2 C d \left (b c^2+9 a d^2\right ) x+3 a c^3 C d^2}{(c+d x) \sqrt {b x^2+a}}dx}{4 b d^5}+\frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\int \frac {b d^4 (13 c C-4 B d) x^4+d^3 \left (15 b C c^2-4 A b d^2+3 a C d^2\right ) x^3+c C d^2 \left (7 b c^2+9 a d^2\right ) x^2+c^2 C d \left (b c^2+9 a d^2\right ) x+3 a c^3 C d^2}{(c+d x) \sqrt {b x^2+a}}dx}{4 b d^5}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {\int -\frac {-b \left (9 a C d^2-2 b \left (23 C c^2-14 B d c+6 A d^2\right )\right ) x^3 d^7+b \left (4 b c^2 (11 c C-5 B d)-a d^2 (c C+8 B d)\right ) x^2 d^6+a b c^2 (17 c C-8 B d) d^6+b c \left (2 b (5 c C-2 B d) c^2+a d^2 (25 c C-16 B d)\right ) x d^5}{(c+d x) \sqrt {b x^2+a}}dx}{3 b d^4}+\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)}{4 b d^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\int \frac {-b \left (9 a C d^2-2 b \left (23 C c^2-14 B d c+6 A d^2\right )\right ) x^3 d^7+b \left (4 b c^2 (11 c C-5 B d)-a d^2 (c C+8 B d)\right ) x^2 d^6+a b c^2 (17 c C-8 B d) d^6+b c \left (2 b (5 c C-2 B d) c^2+a d^2 (25 c C-16 B d)\right ) x d^5}{(c+d x) \sqrt {b x^2+a}}dx}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {\int \frac {b^2 \left (a d^2 (25 c C-16 B d)-2 b c \left (25 C c^2-22 B d c+18 A d^2\right )\right ) x^2 d^9+3 a b c \left (3 a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right ) d^9+b \left (9 a^2 C d^4+a b \left (13 C c^2-4 B d c-12 A d^2\right ) d^2-2 b^2 c^2 \left (13 C c^2-10 B d c+6 A d^2\right )\right ) x d^8}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {\frac {\int \frac {3 b^2 d^{10} \left (a c d \left (3 a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right )+\left (3 a^2 C d^4-4 a b \left (C c^2-B d c+A d^2\right ) d^2+8 b^2 c^2 \left (C c^2-B d c+A d^2\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {3 b d^8 \int \frac {a c d \left (3 a C d^2-4 b \left (C c^2-B d c+A d^2\right )\right )+\left (3 a^2 C d^4-4 a b \left (C c^2-B d c+A d^2\right ) d^2+8 b^2 c^2 \left (C c^2-B d c+A d^2\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {3 b d^8 \left (\frac {\left (3 a^2 C d^4-4 a b d^2 \left (A d^2-B c d+c^2 C\right )+8 b^2 c^2 \left (A d^2-B c d+c^2 C\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}-\frac {8 b^2 c^3 \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {3 b d^8 \left (\frac {\left (3 a^2 C d^4-4 a b d^2 \left (A d^2-B c d+c^2 C\right )+8 b^2 c^2 \left (A d^2-B c d+c^2 C\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}-\frac {8 b^2 c^3 \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {3 b d^8 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 C d^4-4 a b d^2 \left (A d^2-B c d+c^2 C\right )+8 b^2 c^2 \left (A d^2-B c d+c^2 C\right )\right )}{\sqrt {b} d}-\frac {8 b^2 c^3 \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {3 b d^8 \left (\frac {8 b^2 c^3 \left (A d^2-B c d+c^2 C\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 C d^4-4 a b d^2 \left (A d^2-B c d+c^2 C\right )+8 b^2 c^2 \left (A d^2-B c d+c^2 C\right )\right )}{\sqrt {b} d}\right )+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \sqrt {a+b x^2} (c+d x)^3}{4 b d^4}-\frac {\frac {1}{3} d \sqrt {a+b x^2} (c+d x)^2 (13 c C-4 B d)-\frac {\frac {3 b d^8 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 C d^4-4 a b d^2 \left (A d^2-B c d+c^2 C\right )+8 b^2 c^2 \left (A d^2-B c d+c^2 C\right )\right )}{\sqrt {b} d}+\frac {8 b^2 c^3 \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}\right )+b d^8 \sqrt {a+b x^2} \left (a d^2 (25 c C-16 B d)-2 b c \left (18 A d^2-22 B c d+25 c^2 C\right )\right )}{2 b d^3}+\frac {1}{2} d^5 \sqrt {a+b x^2} (c+d x) \left (-9 a C d^2+12 A b d^2-28 b B c d+46 b c^2 C\right )}{3 b d^4}}{4 b d^5}\)

Input:

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

Output:

(C*(c + d*x)^3*Sqrt[a + b*x^2])/(4*b*d^4) - ((d*(13*c*C - 4*B*d)*(c + d*x) 
^2*Sqrt[a + b*x^2])/3 - ((d^5*(46*b*c^2*C - 28*b*B*c*d + 12*A*b*d^2 - 9*a* 
C*d^2)*(c + d*x)*Sqrt[a + b*x^2])/2 + (b*d^8*(a*d^2*(25*c*C - 16*B*d) - 2* 
b*c*(25*c^2*C - 22*B*c*d + 18*A*d^2))*Sqrt[a + b*x^2] + 3*b*d^8*(((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))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) + (8*b^2*c^3*(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*Sqrt[b*c^2 + a*d^2])))/(2*b*d^3))/(3*b*d^4))/(4*b*d^5)
 

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 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 = 0.28 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\frac {\left (-6 C \,d^{3} b \,x^{3}-8 B b \,d^{3} x^{2}+8 C b c \,d^{2} x^{2}-12 A b \,d^{3} x +12 B b c \,d^{2} x +9 C a \,d^{3} x -12 C b \,c^{2} d x +24 A b c \,d^{2}+16 B a \,d^{3}-24 B b \,c^{2} d -16 C a c \,d^{2}+24 C b \,c^{3}\right ) \sqrt {b \,x^{2}+a}}{24 b^{2} d^{4}}-\frac {\frac {\left (4 A a b \,d^{4}-8 A \,b^{2} c^{2} d^{2}-4 B a b c \,d^{3}+8 B \,b^{2} c^{3} d -3 a^{2} C \,d^{4}+4 C a b \,c^{2} d^{2}-8 C \,b^{2} c^{4}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}-\frac {8 b^{2} c^{3} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{8 b^{2} d^{4}}\)	\(372\)
default	\(\frac {\frac {C \,c^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+d^{3} \left (B d -C c \right ) \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )+d^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )-\frac {c d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{b}+\frac {A \,c^{2} d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+C \,d^{4} \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )-\frac {B \,c^{3} d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}}{d^{5}}+\frac {c^{3} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{6} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\)	\(430\)

Input:

int(x^3*(C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-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+9*C*a*d^3*x-12*C*b*c^2*d*x+24*A*b*c*d^2+16*B*a*d^3-24*B*b*c^2*d-16*C 
*a*c*d^2+24*C*b*c^3)*(b*x^2+a)^(1/2)/b^2/d^4-1/8/b^2/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-3*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*b^2*c^3*(A*d^2-B*c*d+C*c 
^2)/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. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate(x^3*(C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {x^{3} \left (A + B x + C x^{2}\right )}{\sqrt {a + b x^{2}} \left (c + d x\right )}\, dx \] Input:

integrate(x**3*(C*x**2+B*x+A)/(d*x+c)/(b*x**2+a)**(1/2),x)
 

Output:

Integral(x**3*(A + B*x + C*x**2)/(sqrt(a + b*x**2)*(c + d*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.71 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} C x^{3}}{4 \, b d} - \frac {\sqrt {b x^{2} + a} C c x^{2}}{3 \, b d^{2}} + \frac {\sqrt {b x^{2} + a} B x^{2}}{3 \, b d} + \frac {\sqrt {b x^{2} + a} C c^{2} x}{2 \, b d^{3}} - \frac {\sqrt {b x^{2} + a} B c x}{2 \, b d^{2}} - \frac {3 \, \sqrt {b x^{2} + a} C a x}{8 \, b^{2} d} + \frac {\sqrt {b x^{2} + a} A x}{2 \, b d} + \frac {C c^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{5}} - \frac {B c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{4}} - \frac {C a c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}} d^{3}} + \frac {A c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b} d^{3}} + \frac {B a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}} d^{2}} + \frac {3 \, C a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}} d} - \frac {A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}} d} - \frac {C c^{5} \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^{6}} + \frac {B c^{4} \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 {A 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^{4}} - \frac {\sqrt {b x^{2} + a} C c^{3}}{b d^{4}} + \frac {\sqrt {b x^{2} + a} B c^{2}}{b d^{3}} + \frac {2 \, \sqrt {b x^{2} + a} C a c}{3 \, b^{2} d^{2}} - \frac {\sqrt {b x^{2} + a} A c}{b d^{2}} - \frac {2 \, \sqrt {b x^{2} + a} B a}{3 \, b^{2} d} \] Input:

integrate(x^3*(C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

1/4*sqrt(b*x^2 + a)*C*x^3/(b*d) - 1/3*sqrt(b*x^2 + a)*C*c*x^2/(b*d^2) + 1/ 
3*sqrt(b*x^2 + a)*B*x^2/(b*d) + 1/2*sqrt(b*x^2 + a)*C*c^2*x/(b*d^3) - 1/2* 
sqrt(b*x^2 + a)*B*c*x/(b*d^2) - 3/8*sqrt(b*x^2 + a)*C*a*x/(b^2*d) + 1/2*sq 
rt(b*x^2 + a)*A*x/(b*d) + C*c^4*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^5) - B*c 
^3*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^4) - 1/2*C*a*c^2*arcsinh(b*x/sqrt(a*b 
))/(b^(3/2)*d^3) + A*c^2*arcsinh(b*x/sqrt(a*b))/(sqrt(b)*d^3) + 1/2*B*a*c* 
arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^2) + 3/8*C*a^2*arcsinh(b*x/sqrt(a*b))/(b 
^(5/2)*d) - 1/2*A*a*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d) - C*c^5*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*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^5) - A*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^4) 
- sqrt(b*x^2 + a)*C*c^3/(b*d^4) + sqrt(b*x^2 + a)*B*c^2/(b*d^3) + 2/3*sqrt 
(b*x^2 + a)*C*a*c/(b^2*d^2) - sqrt(b*x^2 + a)*A*c/(b*d^2) - 2/3*sqrt(b*x^2 
 + a)*B*a/(b^2*d)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

int((x^3*(A + B*x + C*x^2))/((a + b*x^2)^(1/2)*(c + d*x)),x)
 

Output:

int((x^3*(A + B*x + C*x^2))/((a + b*x^2)^(1/2)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{(c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {x^{3} \left (C \,x^{2}+B x +A \right )}{\left (d x +c \right ) \sqrt {b \,x^{2}+a}}d x \] Input:

int(x^3*(C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x)
 

Output:

int(x^3*(C*x^2+B*x+A)/(d*x+c)/(b*x^2+a)^(1/2),x)