\(\int \frac {x^2 (A+B x+C x^2+D x^3)}{(a+b x^2)^{3/2}} \, dx\) [101]

Optimal result

Integrand size = 30, antiderivative size = 138 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {a (b B-a D)-b (A b-a C) x}{b^3 \sqrt {a+b x^2}}+\frac {(b B-2 a D) \sqrt {a+b x^2}}{b^3}+\frac {C x \sqrt {a+b x^2}}{2 b^2}+\frac {D \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac {(2 A b-3 a C) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.81 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-16 a^2 D+a b (12 B+x (9 C-8 D x))+b^2 x \left (-6 A+x \left (6 B+3 C x+2 D x^2\right )\right )}{6 b^3 \sqrt {a+b x^2}}+\frac {(2 A b-3 a C) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Input:

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

Output:

(-16*a^2*D + a*b*(12*B + x*(9*C - 8*D*x)) + b^2*x*(-6*A + x*(6*B + 3*C*x + 
 2*D*x^2)))/(6*b^3*Sqrt[a + b*x^2]) + ((2*A*b - 3*a*C)*ArcTanh[(Sqrt[b]*x) 
/(-Sqrt[a] + Sqrt[a + b*x^2])])/b^(5/2)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.28, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2335, 25, 2340, 533, 25, 27, 455, 224, 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.

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

\(\Big \downarrow \) 2335

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 533

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

rule 2335

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq 
, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] 
+ Simp[c/(2*a*b*(p + 1))   Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu 
m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, 
 b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.41

method	result	size
default	\(A \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+B \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )+C \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+D \left (\frac {x^{4}}{3 b \sqrt {b \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )}{3 b}\right )\)	\(194\)

Input:

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

Output:

A*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+B*(x^2/b/ 
(b*x^2+a)^(1/2)+2*a/b^2/(b*x^2+a)^(1/2))+C*(1/2*x^3/b/(b*x^2+a)^(1/2)-3/2* 
a/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))+D*(1/3 
*x^4/b/(b*x^2+a)^(1/2)-4/3*a/b*(x^2/b/(b*x^2+a)^(1/2)+2*a/b^2/(b*x^2+a)^(1 
/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.12 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (3 \, C a^{2} - 2 \, A a b + {\left (3 \, C a b - 2 \, A b^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2 \, D b^{2} x^{4} + 3 \, C b^{2} x^{3} - 16 \, D a^{2} + 12 \, B a b - 2 \, {\left (4 \, D a b - 3 \, B b^{2}\right )} x^{2} + 3 \, {\left (3 \, C a b - 2 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {3 \, {\left (3 \, C a^{2} - 2 \, A a b + {\left (3 \, C a b - 2 \, A b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, D b^{2} x^{4} + 3 \, C b^{2} x^{3} - 16 \, D a^{2} + 12 \, B a b - 2 \, {\left (4 \, D a b - 3 \, B b^{2}\right )} x^{2} + 3 \, {\left (3 \, C a b - 2 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \] Input:

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

Output:

[-1/12*(3*(3*C*a^2 - 2*A*a*b + (3*C*a*b - 2*A*b^2)*x^2)*sqrt(b)*log(-2*b*x 
^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(2*D*b^2*x^4 + 3*C*b^2*x^3 - 16* 
D*a^2 + 12*B*a*b - 2*(4*D*a*b - 3*B*b^2)*x^2 + 3*(3*C*a*b - 2*A*b^2)*x)*sq 
rt(b*x^2 + a))/(b^4*x^2 + a*b^3), 1/6*(3*(3*C*a^2 - 2*A*a*b + (3*C*a*b - 2 
*A*b^2)*x^2)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (2*D*b^2*x^4 + 
3*C*b^2*x^3 - 16*D*a^2 + 12*B*a*b - 2*(4*D*a*b - 3*B*b^2)*x^2 + 3*(3*C*a*b 
 - 2*A*b^2)*x)*sqrt(b*x^2 + a))/(b^4*x^2 + a*b^3)]
 

Sympy [A] (verification not implemented)

Time = 7.39 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.67 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=A \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + C \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + D \left (\begin {cases} - \frac {8 a^{2}}{3 b^{3} \sqrt {a + b x^{2}}} - \frac {4 a x^{2}}{3 b^{2} \sqrt {a + b x^{2}}} + \frac {x^{4}}{3 b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate(x**2*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**(3/2),x)
 

Output:

A*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) + 
 B*Piecewise((2*a/(b**2*sqrt(a + b*x**2)) + x**2/(b*sqrt(a + b*x**2)), Ne( 
b, 0)), (x**4/(4*a**(3/2)), True)) + C*(3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x** 
2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(5/2)) + x**3/(2*sqrt(a)*b*sqrt 
(1 + b*x**2/a))) + D*Piecewise((-8*a**2/(3*b**3*sqrt(a + b*x**2)) - 4*a*x* 
*2/(3*b**2*sqrt(a + b*x**2)) + x**4/(3*b*sqrt(a + b*x**2)), Ne(b, 0)), (x* 
*6/(6*a**(3/2)), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.23 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {D x^{4}}{3 \, \sqrt {b x^{2} + a} b} + \frac {C x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {4 \, D a x^{2}}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {B x^{2}}{\sqrt {b x^{2} + a} b} + \frac {3 \, C a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {A x}{\sqrt {b x^{2} + a} b} - \frac {3 \, C a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {8 \, D a^{2}}{3 \, \sqrt {b x^{2} + a} b^{3}} + \frac {2 \, B a}{\sqrt {b x^{2} + a} b^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left ({\left (\frac {2 \, D x}{b} + \frac {3 \, C}{b}\right )} x - \frac {2 \, {\left (4 \, D a b^{3} - 3 \, B b^{4}\right )}}{b^{5}}\right )} x + \frac {3 \, {\left (3 \, C a b^{3} - 2 \, A b^{4}\right )}}{b^{5}}\right )} x - \frac {4 \, {\left (4 \, D a^{2} b^{2} - 3 \, B a b^{3}\right )}}{b^{5}}}{6 \, \sqrt {b x^{2} + a}} + \frac {{\left (3 \, C a - 2 \, A b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \] Input:

integrate(x^2*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

1/6*((((2*D*x/b + 3*C/b)*x - 2*(4*D*a*b^3 - 3*B*b^4)/b^5)*x + 3*(3*C*a*b^3 
 - 2*A*b^4)/b^5)*x - 4*(4*D*a^2*b^2 - 3*B*a*b^3)/b^5)/sqrt(b*x^2 + a) + 1/ 
2*(3*C*a - 2*A*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.09 \[ \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-64 \sqrt {b \,x^{2}+a}\, a^{2} d -24 \sqrt {b \,x^{2}+a}\, a \,b^{2} x +48 \sqrt {b \,x^{2}+a}\, a \,b^{2}+36 \sqrt {b \,x^{2}+a}\, a b c x -32 \sqrt {b \,x^{2}+a}\, a b d \,x^{2}+24 \sqrt {b \,x^{2}+a}\, b^{3} x^{2}+12 \sqrt {b \,x^{2}+a}\, b^{2} c \,x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{2} d \,x^{4}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b -36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} c +24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} x^{2}-36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b c \,x^{2}-24 \sqrt {b}\, a^{2} b +27 \sqrt {b}\, a^{2} c -24 \sqrt {b}\, a \,b^{2} x^{2}+27 \sqrt {b}\, a b c \,x^{2}}{24 b^{3} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

( - 64*sqrt(a + b*x**2)*a**2*d - 24*sqrt(a + b*x**2)*a*b**2*x + 48*sqrt(a 
+ b*x**2)*a*b**2 + 36*sqrt(a + b*x**2)*a*b*c*x - 32*sqrt(a + b*x**2)*a*b*d 
*x**2 + 24*sqrt(a + b*x**2)*b**3*x**2 + 12*sqrt(a + b*x**2)*b**2*c*x**3 + 
8*sqrt(a + b*x**2)*b**2*d*x**4 + 24*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b 
)*x)/sqrt(a))*a**2*b - 36*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt( 
a))*a**2*c + 24*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b**2 
*x**2 - 36*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b*c*x**2 
- 24*sqrt(b)*a**2*b + 27*sqrt(b)*a**2*c - 24*sqrt(b)*a*b**2*x**2 + 27*sqrt 
(b)*a*b*c*x**2)/(24*b**3*(a + b*x**2))