\(\int \frac {x^2 (c+d x) (A+B x+C x^2)}{(a+b x^2)^{3/2}} \, dx\) [133]

Optimal result

Integrand size = 30, antiderivative size = 184 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {a (a C d-b (B c+A d))}{b^3 \sqrt {a+b x^2}}-\frac {(A b c-a c C-a B d) x}{b^2 \sqrt {a+b x^2}}+\frac {(b B c+A b d-2 a C d) \sqrt {a+b x^2}}{b^3}+\frac {(c C+B d) x \sqrt {a+b x^2}}{2 b^2}+\frac {C d \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac {(2 A b c-3 a (c C+B d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-16 a^2 C d+a b \left (12 B c+12 A d+9 c C x+9 B d x-8 C d x^2\right )+b^2 x \left (-6 A (c-d x)+x \left (6 B c+3 c C x+3 B d x+2 C d x^2\right )\right )}{6 b^3 \sqrt {a+b x^2}}+\frac {(2 A b c-3 a (c C+B d)) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2176, 25, 2346, 2346, 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.

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

\(\Big \downarrow \) 2176

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\int \frac {3 a b (c C+B d) x^2+a (3 b B c+6 A b d-8 a C d) x+3 a (A b c-a C c-a B d)}{\sqrt {b x^2+a}}dx}{3 b}+\frac {a C d x^2 \sqrt {a+b x^2}}{3 b}}{a b}+\frac {(c+d x) \left (a B-b x \left (A-\frac {a C}{b}\right )\right )}{b^2 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {\int \frac {a b (3 (2 A b c-3 a (c C+B d))+2 (3 b B c+6 A b d-8 a C d) x)}{\sqrt {b x^2+a}}dx}{2 b}+\frac {3}{2} a x \sqrt {a+b x^2} (B d+c C)}{3 b}+\frac {a C d x^2 \sqrt {a+b x^2}}{3 b}}{a b}+\frac {(c+d x) \left (a B-b x \left (A-\frac {a C}{b}\right )\right )}{b^2 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{2} a \int \frac {3 (2 A b c-3 a (c C+B d))+2 (3 b B c+6 A b d-8 a C d) x}{\sqrt {b x^2+a}}dx+\frac {3}{2} a x \sqrt {a+b x^2} (B d+c C)}{3 b}+\frac {a C d x^2 \sqrt {a+b x^2}}{3 b}}{a b}+\frac {(c+d x) \left (a B-b x \left (A-\frac {a C}{b}\right )\right )}{b^2 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {1}{2} a \left (3 (2 A b c-3 a (B d+c C)) \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {2 \sqrt {a+b x^2} (-8 a C d+6 A b d+3 b B c)}{b}\right )+\frac {3}{2} a x \sqrt {a+b x^2} (B d+c C)}{3 b}+\frac {a C d x^2 \sqrt {a+b x^2}}{3 b}}{a b}+\frac {(c+d x) \left (a B-b x \left (A-\frac {a C}{b}\right )\right )}{b^2 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {1}{2} a \left (3 (2 A b c-3 a (B d+c C)) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {2 \sqrt {a+b x^2} (-8 a C d+6 A b d+3 b B c)}{b}\right )+\frac {3}{2} a x \sqrt {a+b x^2} (B d+c C)}{3 b}+\frac {a C d x^2 \sqrt {a+b x^2}}{3 b}}{a b}+\frac {(c+d x) \left (a B-b x \left (A-\frac {a C}{b}\right )\right )}{b^2 \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {1}{2} a \left (\frac {3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (2 A b c-3 a (B d+c C))}{\sqrt {b}}+\frac {2 \sqrt {a+b x^2} (-8 a C d+6 A b d+3 b B c)}{b}\right )+\frac {3}{2} a x \sqrt {a+b x^2} (B d+c C)}{3 b}+\frac {a C d x^2 \sqrt {a+b x^2}}{3 b}}{a b}+\frac {(c+d x) \left (a B-b x \left (A-\frac {a C}{b}\right )\right )}{b^2 \sqrt {a+b x^2}}\)

Input:

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

Output:

((a*B - b*(A - (a*C)/b)*x)*(c + d*x))/(b^2*Sqrt[a + b*x^2]) + ((a*C*d*x^2* 
Sqrt[a + b*x^2])/(3*b) + ((3*a*(c*C + B*d)*x*Sqrt[a + b*x^2])/2 + (a*((2*( 
3*b*B*c + 6*A*b*d - 8*a*C*d)*Sqrt[a + b*x^2])/b + (3*(2*A*b*c - 3*a*(c*C + 
 B*d))*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 2176

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema 
inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 
, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p 
 + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p 
 + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + 
b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && R 
ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {\left (2 C b d \,x^{2}+3 B b d x +3 C c x b +6 A b d +6 B b c -10 a C d \right ) \sqrt {b \,x^{2}+a}}{6 b^{3}}+\frac {b \left (2 A b c -3 B a d -3 C a c \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+\frac {2 a \left (A b d +B b c -a C d \right )}{b \sqrt {b \,x^{2}+a}}-\frac {B a d x}{\sqrt {b \,x^{2}+a}}-\frac {C a c x}{\sqrt {b \,x^{2}+a}}}{2 b^{2}}\)	\(171\)
default	\(\left (A d +B c \right ) \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )+\left (B d +C c \right ) \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+A c \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )+d C \left (\frac {x^{4}}{3 b \sqrt {b \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{b \sqrt {b \,x^{2}+a}}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )}{3 b}\right )\)	\(208\)

Input:

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

Output:

1/6*(2*C*b*d*x^2+3*B*b*d*x+3*C*b*c*x+6*A*b*d+6*B*b*c-10*C*a*d)*(b*x^2+a)^( 
1/2)/b^3+1/2/b^2*(b*(2*A*b*c-3*B*a*d-3*C*a*c)*(-x/b/(b*x^2+a)^(1/2)+1/b^(3 
/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))+2*a*(A*b*d+B*b*c-C*a*d)/b/(b*x^2+a)^(1/ 
2)-B*a*d*x/(b*x^2+a)^(1/2)-C*a*c*x/(b*x^2+a)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.23 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (3 \, B a^{2} d + {\left (3 \, B a b d + {\left (3 \, C a b - 2 \, A b^{2}\right )} c\right )} x^{2} + {\left (3 \, C a^{2} - 2 \, A a b\right )} c\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, C b^{2} d x^{4} + 12 \, B a b c + 3 \, {\left (C b^{2} c + B b^{2} d\right )} x^{3} + 2 \, {\left (3 \, B b^{2} c - {\left (4 \, C a b - 3 \, A b^{2}\right )} d\right )} x^{2} - 4 \, {\left (4 \, C a^{2} - 3 \, A a b\right )} d + 3 \, {\left (3 \, B a b d + {\left (3 \, C a b - 2 \, A b^{2}\right )} c\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (b^{4} x^{2} + a b^{3}\right )}}, \frac {3 \, {\left (3 \, B a^{2} d + {\left (3 \, B a b d + {\left (3 \, C a b - 2 \, A b^{2}\right )} c\right )} x^{2} + {\left (3 \, C a^{2} - 2 \, A a b\right )} c\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, C b^{2} d x^{4} + 12 \, B a b c + 3 \, {\left (C b^{2} c + B b^{2} d\right )} x^{3} + 2 \, {\left (3 \, B b^{2} c - {\left (4 \, C a b - 3 \, A b^{2}\right )} d\right )} x^{2} - 4 \, {\left (4 \, C a^{2} - 3 \, A a b\right )} d + 3 \, {\left (3 \, B a b d + {\left (3 \, C a b - 2 \, A b^{2}\right )} c\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.95 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=A c \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + A d \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + B c \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + B d \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C c \left (\frac {3 \sqrt {a} x}{2 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{3}}{2 \sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C d \left (\begin {cases} - \frac {8 a^{2}}{3 b^{3} \sqrt {a + b x^{2}}} - \frac {4 a x^{2}}{3 b^{2} \sqrt {a + b x^{2}}} + \frac {x^{4}}{3 b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

A*c*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) 
 + A*d*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)) + B*c*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 
)) + B*d*(3*sqrt(a)*x/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/sq 
rt(a))/(2*b**(5/2)) + x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a))) + C*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))) + C*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*sqr 
t(a + b*x**2)), Ne(b, 0)), (x**6/(6*a**(3/2)), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.11 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {C d x^{4}}{3 \, \sqrt {b x^{2} + a} b} - \frac {4 \, C a d x^{2}}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {{\left (C c + B d\right )} x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {A c x}{\sqrt {b x^{2} + a} b} + \frac {{\left (B c + A d\right )} x^{2}}{\sqrt {b x^{2} + a} b} + \frac {A c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {8 \, C a^{2} d}{3 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, {\left (C c + B d\right )} a x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {3 \, {\left (C c + B d\right )} a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {2 \, {\left (B c + A d\right )} a}{\sqrt {b x^{2} + a} b^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left ({\left (\frac {2 \, C d x}{b} + \frac {3 \, {\left (C b^{4} c + B b^{4} d\right )}}{b^{5}}\right )} x + \frac {2 \, {\left (3 \, B b^{4} c - 4 \, C a b^{3} d + 3 \, A b^{4} d\right )}}{b^{5}}\right )} x + \frac {3 \, {\left (3 \, C a b^{3} c - 2 \, A b^{4} c + 3 \, B a b^{3} d\right )}}{b^{5}}\right )} x + \frac {4 \, {\left (3 \, B a b^{3} c - 4 \, C a^{2} b^{2} d + 3 \, A a b^{3} d\right )}}{b^{5}}}{6 \, \sqrt {b x^{2} + a}} + \frac {{\left (3 \, C a c - 2 \, A b c + 3 \, B a d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.48 \[ \int \frac {x^2 (c+d x) \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {48 \sqrt {b \,x^{2}+a}\, a^{2} b d -64 \sqrt {b \,x^{2}+a}\, a^{2} c d -24 \sqrt {b \,x^{2}+a}\, a \,b^{2} c x +48 \sqrt {b \,x^{2}+a}\, a \,b^{2} c +24 \sqrt {b \,x^{2}+a}\, a \,b^{2} d \,x^{2}+36 \sqrt {b \,x^{2}+a}\, a \,b^{2} d x +36 \sqrt {b \,x^{2}+a}\, a b \,c^{2} x -32 \sqrt {b \,x^{2}+a}\, a b c d \,x^{2}+24 \sqrt {b \,x^{2}+a}\, b^{3} c \,x^{2}+12 \sqrt {b \,x^{2}+a}\, b^{3} d \,x^{3}+12 \sqrt {b \,x^{2}+a}\, b^{2} c^{2} x^{3}+8 \sqrt {b \,x^{2}+a}\, b^{2} c d \,x^{4}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b c -36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b d -36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} c^{2}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} c \,x^{2}-36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{2} d \,x^{2}-36 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a b \,c^{2} x^{2}-24 \sqrt {b}\, a^{2} b c +27 \sqrt {b}\, a^{2} b d +27 \sqrt {b}\, a^{2} c^{2}-24 \sqrt {b}\, a \,b^{2} c \,x^{2}+27 \sqrt {b}\, a \,b^{2} d \,x^{2}+27 \sqrt {b}\, a b \,c^{2} x^{2}}{24 b^{3} \left (b \,x^{2}+a \right )} \] Input:

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

Output:

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