\(\int x (c+d x) (a+b x^2)^{3/2} (A+B x+C x^2) \, dx\) [49]

Optimal result

Integrand size = 28, antiderivative size = 242 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=-\frac {a^2 (3 a C d-8 b (B c+A d)) x \sqrt {a+b x^2}}{128 b^2}-\frac {a (3 a C d-8 b (B c+A d)) x^3 \sqrt {a+b x^2}}{64 b}-\frac {(3 a C d-8 b (B c+A d)) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {(A b c-a (c C+B d)) \left (a+b x^2\right )^{5/2}}{5 b^2}+\frac {C d x^3 \left (a+b x^2\right )^{5/2}}{8 b}+\frac {(c C+B d) \left (a+b x^2\right )^{7/2}}{7 b^2}+\frac {a^3 (3 a C d-8 b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:

-1/128*a^2*(3*a*C*d-8*b*(A*d+B*c))*x*(b*x^2+a)^(1/2)/b^2-1/64*a*(3*a*C*d-8 
*b*(A*d+B*c))*x^3*(b*x^2+a)^(1/2)/b-1/48*(3*a*C*d-8*b*(A*d+B*c))*x^3*(b*x^ 
2+a)^(3/2)/b+1/5*(A*b*c-a*(B*d+C*c))*(b*x^2+a)^(5/2)/b^2+1/8*C*d*x^3*(b*x^ 
2+a)^(5/2)/b+1/7*(B*d+C*c)*(b*x^2+a)^(7/2)/b^2+1/128*a^3*(3*a*C*d-8*b*(A*d 
+B*c))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.92 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (-3 a^3 (256 c C+256 B d+105 C d x)+6 a^2 b \left (28 A (16 c+5 d x)+x \left (140 B c+64 c C x+64 B d x+35 C d x^2\right )\right )+8 a b^2 x^2 \left (14 A (48 c+35 d x)+x \left (490 B c+384 c C x+384 B d x+315 C d x^2\right )\right )+16 b^3 x^4 (28 A (6 c+5 d x)+5 x (4 B (7 c+6 d x)+3 C x (8 c+7 d x)))\right )-105 a^3 (3 a C d-8 b (B c+A d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{13440 b^{5/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(-3*a^3*(256*c*C + 256*B*d + 105*C*d*x) + 6*a^2*b 
*(28*A*(16*c + 5*d*x) + x*(140*B*c + 64*c*C*x + 64*B*d*x + 35*C*d*x^2)) + 
8*a*b^2*x^2*(14*A*(48*c + 35*d*x) + x*(490*B*c + 384*c*C*x + 384*B*d*x + 3 
15*C*d*x^2)) + 16*b^3*x^4*(28*A*(6*c + 5*d*x) + 5*x*(4*B*(7*c + 6*d*x) + 3 
*C*x*(8*c + 7*d*x)))) - 105*a^3*(3*a*C*d - 8*b*(B*c + A*d))*Log[-(Sqrt[b]* 
x) + Sqrt[a + b*x^2]])/(13440*b^(5/2))
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2185, 25, 2185, 25, 27, 676, 211, 211, 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.

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {\int -b d^3 (c+d x) \left (a d (5 c C-16 B d)-\left (21 a C d^2-b \left (30 C c^2-40 B d c+56 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b d^2}+\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)}{8 b d^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {\int b d^3 (c+d x) \left (a d (5 c C-16 B d)-\left (21 a C d^2-b \left (30 C c^2-40 B d c+56 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b d^2}}{8 b d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {1}{7} d \int (c+d x) \left (a d (5 c C-16 B d)-\left (21 a C d^2-b \left (30 C c^2-40 B d c+56 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b d^3}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {1}{7} d \left (\frac {7 a d^2 (3 a C d-8 b (A d+B c)) \int \left (b x^2+a\right )^{3/2}dx}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (B d+c C)-b c \left (28 A d^2-20 B c d+15 c^2 C\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a C d^2-b \left (56 A d^2-40 B c d+30 c^2 C\right )\right )}{6 b}\right )}{8 b d^3}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {1}{7} d \left (\frac {7 a d^2 (3 a C d-8 b (A d+B c)) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (B d+c C)-b c \left (28 A d^2-20 B c d+15 c^2 C\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a C d^2-b \left (56 A d^2-40 B c d+30 c^2 C\right )\right )}{6 b}\right )}{8 b d^3}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {1}{7} d \left (\frac {7 a d^2 (3 a C d-8 b (A d+B c)) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (B d+c C)-b c \left (28 A d^2-20 B c d+15 c^2 C\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a C d^2-b \left (56 A d^2-40 B c d+30 c^2 C\right )\right )}{6 b}\right )}{8 b d^3}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {1}{7} d \left (\frac {7 a d^2 (3 a C d-8 b (A d+B c)) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (B d+c C)-b c \left (28 A d^2-20 B c d+15 c^2 C\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a C d^2-b \left (56 A d^2-40 B c d+30 c^2 C\right )\right )}{6 b}\right )}{8 b d^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b d^2}-\frac {\frac {1}{7} d \left (a+b x^2\right )^{5/2} (c+d x)^2 (13 c C-8 B d)-\frac {1}{7} d \left (\frac {7 a d^2 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) (3 a C d-8 b (A d+B c))}{6 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (B d+c C)-b c \left (28 A d^2-20 B c d+15 c^2 C\right )\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (21 a C d^2-b \left (56 A d^2-40 B c d+30 c^2 C\right )\right )}{6 b}\right )}{8 b d^3}\)

Input:

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

Output:

(C*(c + d*x)^3*(a + b*x^2)^(5/2))/(8*b*d^2) - ((d*(13*c*C - 8*B*d)*(c + d* 
x)^2*(a + b*x^2)^(5/2))/7 - (d*((-2*(8*a*d^2*(c*C + B*d) - b*c*(15*c^2*C - 
 20*B*c*d + 28*A*d^2))*(a + b*x^2)^(5/2))/(5*b) - (d*(21*a*C*d^2 - b*(30*c 
^2*C - 40*B*c*d + 56*A*d^2))*x*(a + b*x^2)^(5/2))/(6*b) + (7*a*d^2*(3*a*C* 
d - 8*b*(B*c + A*d))*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/ 
2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/(6*b)))/7)/ 
(8*b*d^3)
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -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 [A] (verified)

Time = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.99

method	result	size
default	\(\left (A d +B c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+\left (B d +C c \right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+\frac {A c \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+d C \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )\)	\(240\)
risch	\(\frac {\left (1680 d C \,b^{3} x^{7}+1920 B \,b^{3} d \,x^{6}+1920 C \,b^{3} c \,x^{6}+2240 A \,b^{3} d \,x^{5}+2240 B \,b^{3} c \,x^{5}+2520 C a \,b^{2} d \,x^{5}+2688 A \,b^{3} c \,x^{4}+3072 B a \,b^{2} d \,x^{4}+3072 C a \,b^{2} c \,x^{4}+3920 A a \,b^{2} d \,x^{3}+3920 B a \,b^{2} c \,x^{3}+210 C \,a^{2} b d \,x^{3}+5376 A a \,b^{2} c \,x^{2}+384 B \,a^{2} b d \,x^{2}+384 C \,a^{2} b c \,x^{2}+840 A \,a^{2} b d x +840 B \,a^{2} b c x -315 C \,a^{3} d x +2688 A \,a^{2} b c -768 B \,a^{3} d -768 C \,a^{3} c \right ) \sqrt {b \,x^{2}+a}}{13440 b^{2}}-\frac {a^{3} \left (8 A b d +8 B b c -3 a C d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}\)	\(264\)

Input:

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

Output:

(A*d+B*c)*(1/6*x*(b*x^2+a)^(5/2)/b-1/6*a/b*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1 
/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))))+(B*d+C 
*c)*(1/7*x^2*(b*x^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+1/5*A*c*(b*x^2+ 
a)^(5/2)/b+d*C*(1/8*x^3*(b*x^2+a)^(5/2)/b-3/8*a/b*(1/6*x*(b*x^2+a)^(5/2)/b 
-1/6*a/b*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2) 
*ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.37 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\left [\frac {105 \, {\left (8 \, B a^{3} b c - {\left (3 \, C a^{4} - 8 \, A a^{3} b\right )} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (1680 \, C b^{4} d x^{7} + 1920 \, {\left (C b^{4} c + B b^{4} d\right )} x^{6} - 768 \, B a^{3} b d + 280 \, {\left (8 \, B b^{4} c + {\left (9 \, C a b^{3} + 8 \, A b^{4}\right )} d\right )} x^{5} + 384 \, {\left (8 \, B a b^{3} d + {\left (8 \, C a b^{3} + 7 \, A b^{4}\right )} c\right )} x^{4} + 70 \, {\left (56 \, B a b^{3} c + {\left (3 \, C a^{2} b^{2} + 56 \, A a b^{3}\right )} d\right )} x^{3} + 384 \, {\left (B a^{2} b^{2} d + {\left (C a^{2} b^{2} + 14 \, A a b^{3}\right )} c\right )} x^{2} - 384 \, {\left (2 \, C a^{3} b - 7 \, A a^{2} b^{2}\right )} c + 105 \, {\left (8 \, B a^{2} b^{2} c - {\left (3 \, C a^{3} b - 8 \, A a^{2} b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{26880 \, b^{3}}, \frac {105 \, {\left (8 \, B a^{3} b c - {\left (3 \, C a^{4} - 8 \, A a^{3} b\right )} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (1680 \, C b^{4} d x^{7} + 1920 \, {\left (C b^{4} c + B b^{4} d\right )} x^{6} - 768 \, B a^{3} b d + 280 \, {\left (8 \, B b^{4} c + {\left (9 \, C a b^{3} + 8 \, A b^{4}\right )} d\right )} x^{5} + 384 \, {\left (8 \, B a b^{3} d + {\left (8 \, C a b^{3} + 7 \, A b^{4}\right )} c\right )} x^{4} + 70 \, {\left (56 \, B a b^{3} c + {\left (3 \, C a^{2} b^{2} + 56 \, A a b^{3}\right )} d\right )} x^{3} + 384 \, {\left (B a^{2} b^{2} d + {\left (C a^{2} b^{2} + 14 \, A a b^{3}\right )} c\right )} x^{2} - 384 \, {\left (2 \, C a^{3} b - 7 \, A a^{2} b^{2}\right )} c + 105 \, {\left (8 \, B a^{2} b^{2} c - {\left (3 \, C a^{3} b - 8 \, A a^{2} b^{2}\right )} d\right )} x\right )} \sqrt {b x^{2} + a}}{13440 \, b^{3}}\right ] \] Input:

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

Output:

[1/26880*(105*(8*B*a^3*b*c - (3*C*a^4 - 8*A*a^3*b)*d)*sqrt(b)*log(-2*b*x^2 
 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(1680*C*b^4*d*x^7 + 1920*(C*b^4*c 
+ B*b^4*d)*x^6 - 768*B*a^3*b*d + 280*(8*B*b^4*c + (9*C*a*b^3 + 8*A*b^4)*d) 
*x^5 + 384*(8*B*a*b^3*d + (8*C*a*b^3 + 7*A*b^4)*c)*x^4 + 70*(56*B*a*b^3*c 
+ (3*C*a^2*b^2 + 56*A*a*b^3)*d)*x^3 + 384*(B*a^2*b^2*d + (C*a^2*b^2 + 14*A 
*a*b^3)*c)*x^2 - 384*(2*C*a^3*b - 7*A*a^2*b^2)*c + 105*(8*B*a^2*b^2*c - (3 
*C*a^3*b - 8*A*a^2*b^2)*d)*x)*sqrt(b*x^2 + a))/b^3, 1/13440*(105*(8*B*a^3* 
b*c - (3*C*a^4 - 8*A*a^3*b)*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) 
 + (1680*C*b^4*d*x^7 + 1920*(C*b^4*c + B*b^4*d)*x^6 - 768*B*a^3*b*d + 280* 
(8*B*b^4*c + (9*C*a*b^3 + 8*A*b^4)*d)*x^5 + 384*(8*B*a*b^3*d + (8*C*a*b^3 
+ 7*A*b^4)*c)*x^4 + 70*(56*B*a*b^3*c + (3*C*a^2*b^2 + 56*A*a*b^3)*d)*x^3 + 
 384*(B*a^2*b^2*d + (C*a^2*b^2 + 14*A*a*b^3)*c)*x^2 - 384*(2*C*a^3*b - 7*A 
*a^2*b^2)*c + 105*(8*B*a^2*b^2*c - (3*C*a^3*b - 8*A*a^2*b^2)*d)*x)*sqrt(b* 
x^2 + a))/b^3]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (224) = 448\).

Time = 0.63 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.47 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\begin {cases} - \frac {a \left (A a^{2} d + B a^{2} c - \frac {3 a \left (2 A a b d + 2 B a b c + C a^{2} d - \frac {5 a \left (A b^{2} d + B b^{2} c + \frac {9 C a b d}{8}\right )}{6 b}\right )}{4 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2 b} + \sqrt {a + b x^{2}} \left (\frac {C b d x^{7}}{8} + \frac {x^{6} \left (B b^{2} d + C b^{2} c\right )}{7 b} + \frac {x^{5} \left (A b^{2} d + B b^{2} c + \frac {9 C a b d}{8}\right )}{6 b} + \frac {x^{4} \left (A b^{2} c + 2 B a b d + 2 C a b c - \frac {6 a \left (B b^{2} d + C b^{2} c\right )}{7 b}\right )}{5 b} + \frac {x^{3} \cdot \left (2 A a b d + 2 B a b c + C a^{2} d - \frac {5 a \left (A b^{2} d + B b^{2} c + \frac {9 C a b d}{8}\right )}{6 b}\right )}{4 b} + \frac {x^{2} \cdot \left (2 A a b c + B a^{2} d + C a^{2} c - \frac {4 a \left (A b^{2} c + 2 B a b d + 2 C a b c - \frac {6 a \left (B b^{2} d + C b^{2} c\right )}{7 b}\right )}{5 b}\right )}{3 b} + \frac {x \left (A a^{2} d + B a^{2} c - \frac {3 a \left (2 A a b d + 2 B a b c + C a^{2} d - \frac {5 a \left (A b^{2} d + B b^{2} c + \frac {9 C a b d}{8}\right )}{6 b}\right )}{4 b}\right )}{2 b} + \frac {A a^{2} c - \frac {2 a \left (2 A a b c + B a^{2} d + C a^{2} c - \frac {4 a \left (A b^{2} c + 2 B a b d + 2 C a b c - \frac {6 a \left (B b^{2} d + C b^{2} c\right )}{7 b}\right )}{5 b}\right )}{3 b}}{b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A c x^{2}}{2} + \frac {C d x^{5}}{5} + \frac {x^{4} \left (B d + C c\right )}{4} + \frac {x^{3} \left (A d + B c\right )}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-a*(A*a**2*d + B*a**2*c - 3*a*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d 
 - 5*a*(A*b**2*d + B*b**2*c + 9*C*a*b*d/8)/(6*b))/(4*b))*Piecewise((log(2* 
sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x** 
2), True))/(2*b) + sqrt(a + b*x**2)*(C*b*d*x**7/8 + x**6*(B*b**2*d + C*b** 
2*c)/(7*b) + x**5*(A*b**2*d + B*b**2*c + 9*C*a*b*d/8)/(6*b) + x**4*(A*b**2 
*c + 2*B*a*b*d + 2*C*a*b*c - 6*a*(B*b**2*d + C*b**2*c)/(7*b))/(5*b) + x**3 
*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d - 5*a*(A*b**2*d + B*b**2*c + 9*C*a*b*d/ 
8)/(6*b))/(4*b) + x**2*(2*A*a*b*c + B*a**2*d + C*a**2*c - 4*a*(A*b**2*c + 
2*B*a*b*d + 2*C*a*b*c - 6*a*(B*b**2*d + C*b**2*c)/(7*b))/(5*b))/(3*b) + x* 
(A*a**2*d + B*a**2*c - 3*a*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d - 5*a*(A*b**2 
*d + B*b**2*c + 9*C*a*b*d/8)/(6*b))/(4*b))/(2*b) + (A*a**2*c - 2*a*(2*A*a* 
b*c + B*a**2*d + C*a**2*c - 4*a*(A*b**2*c + 2*B*a*b*d + 2*C*a*b*c - 6*a*(B 
*b**2*d + C*b**2*c)/(7*b))/(5*b))/(3*b))/b), Ne(b, 0)), (a**(3/2)*(A*c*x** 
2/2 + C*d*x**5/5 + x**4*(B*d + C*c)/4 + x**3*(A*d + B*c)/3), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d x^{3}}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C a d x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C a^{2} d x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} C a^{3} d x}{128 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} x^{2}}{7 \, b} + \frac {3 \, C a^{4} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c}{5 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} {\left (B c + A d\right )} a^{2} x}{16 \, b} - \frac {{\left (B c + A d\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} a}{35 \, b^{2}} \] Input:

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

Output:

1/8*(b*x^2 + a)^(5/2)*C*d*x^3/b - 1/16*(b*x^2 + a)^(5/2)*C*a*d*x/b^2 + 1/6 
4*(b*x^2 + a)^(3/2)*C*a^2*d*x/b^2 + 3/128*sqrt(b*x^2 + a)*C*a^3*d*x/b^2 + 
1/7*(b*x^2 + a)^(5/2)*(C*c + B*d)*x^2/b + 3/128*C*a^4*d*arcsinh(b*x/sqrt(a 
*b))/b^(5/2) + 1/5*(b*x^2 + a)^(5/2)*A*c/b + 1/6*(b*x^2 + a)^(5/2)*(B*c + 
A*d)*x/b - 1/24*(b*x^2 + a)^(3/2)*(B*c + A*d)*a*x/b - 1/16*sqrt(b*x^2 + a) 
*(B*c + A*d)*a^2*x/b - 1/16*(B*c + A*d)*a^3*arcsinh(b*x/sqrt(a*b))/b^(3/2) 
 - 2/35*(b*x^2 + a)^(5/2)*(C*c + B*d)*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.24 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {1}{13440} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, C b d x + \frac {8 \, {\left (C b^{7} c + B b^{7} d\right )}}{b^{6}}\right )} x + \frac {7 \, {\left (8 \, B b^{7} c + 9 \, C a b^{6} d + 8 \, A b^{7} d\right )}}{b^{6}}\right )} x + \frac {48 \, {\left (8 \, C a b^{6} c + 7 \, A b^{7} c + 8 \, B a b^{6} d\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (56 \, B a b^{6} c + 3 \, C a^{2} b^{5} d + 56 \, A a b^{6} d\right )}}{b^{6}}\right )} x + \frac {192 \, {\left (C a^{2} b^{5} c + 14 \, A a b^{6} c + B a^{2} b^{5} d\right )}}{b^{6}}\right )} x + \frac {105 \, {\left (8 \, B a^{2} b^{5} c - 3 \, C a^{3} b^{4} d + 8 \, A a^{2} b^{5} d\right )}}{b^{6}}\right )} x - \frac {384 \, {\left (2 \, C a^{3} b^{4} c - 7 \, A a^{2} b^{5} c + 2 \, B a^{3} b^{4} d\right )}}{b^{6}}\right )} + \frac {{\left (8 \, B a^{3} b c - 3 \, C a^{4} d + 8 \, A a^{3} b d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:

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

Output:

1/13440*sqrt(b*x^2 + a)*((2*((4*(5*(6*(7*C*b*d*x + 8*(C*b^7*c + B*b^7*d)/b 
^6)*x + 7*(8*B*b^7*c + 9*C*a*b^6*d + 8*A*b^7*d)/b^6)*x + 48*(8*C*a*b^6*c + 
 7*A*b^7*c + 8*B*a*b^6*d)/b^6)*x + 35*(56*B*a*b^6*c + 3*C*a^2*b^5*d + 56*A 
*a*b^6*d)/b^6)*x + 192*(C*a^2*b^5*c + 14*A*a*b^6*c + B*a^2*b^5*d)/b^6)*x + 
 105*(8*B*a^2*b^5*c - 3*C*a^3*b^4*d + 8*A*a^2*b^5*d)/b^6)*x - 384*(2*C*a^3 
*b^4*c - 7*A*a^2*b^5*c + 2*B*a^3*b^4*d)/b^6) + 1/128*(8*B*a^3*b*c - 3*C*a^ 
4*d + 8*A*a^3*b*d)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.99 \[ \int x (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {2688 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c +840 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d x -768 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d -768 \sqrt {b \,x^{2}+a}\, a^{3} b \,c^{2}-315 \sqrt {b \,x^{2}+a}\, a^{3} b c d x +5376 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,x^{2}+840 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c x +3920 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{3}+384 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{2}+384 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c^{2} x^{2}+210 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c d \,x^{3}+2688 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{4}+3920 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{3}+2240 \sqrt {b \,x^{2}+a}\, a \,b^{4} d \,x^{5}+3072 \sqrt {b \,x^{2}+a}\, a \,b^{4} d \,x^{4}+3072 \sqrt {b \,x^{2}+a}\, a \,b^{3} c^{2} x^{4}+2520 \sqrt {b \,x^{2}+a}\, a \,b^{3} c d \,x^{5}+2240 \sqrt {b \,x^{2}+a}\, b^{5} c \,x^{5}+1920 \sqrt {b \,x^{2}+a}\, b^{5} d \,x^{6}+1920 \sqrt {b \,x^{2}+a}\, b^{4} c^{2} x^{6}+1680 \sqrt {b \,x^{2}+a}\, b^{4} c d \,x^{7}-840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b d +315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} c d -840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} c}{13440 b^{3}} \] Input:

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

Output:

(2688*sqrt(a + b*x**2)*a**3*b**2*c + 840*sqrt(a + b*x**2)*a**3*b**2*d*x - 
768*sqrt(a + b*x**2)*a**3*b**2*d - 768*sqrt(a + b*x**2)*a**3*b*c**2 - 315* 
sqrt(a + b*x**2)*a**3*b*c*d*x + 5376*sqrt(a + b*x**2)*a**2*b**3*c*x**2 + 8 
40*sqrt(a + b*x**2)*a**2*b**3*c*x + 3920*sqrt(a + b*x**2)*a**2*b**3*d*x**3 
 + 384*sqrt(a + b*x**2)*a**2*b**3*d*x**2 + 384*sqrt(a + b*x**2)*a**2*b**2* 
c**2*x**2 + 210*sqrt(a + b*x**2)*a**2*b**2*c*d*x**3 + 2688*sqrt(a + b*x**2 
)*a*b**4*c*x**4 + 3920*sqrt(a + b*x**2)*a*b**4*c*x**3 + 2240*sqrt(a + b*x* 
*2)*a*b**4*d*x**5 + 3072*sqrt(a + b*x**2)*a*b**4*d*x**4 + 3072*sqrt(a + b* 
x**2)*a*b**3*c**2*x**4 + 2520*sqrt(a + b*x**2)*a*b**3*c*d*x**5 + 2240*sqrt 
(a + b*x**2)*b**5*c*x**5 + 1920*sqrt(a + b*x**2)*b**5*d*x**6 + 1920*sqrt(a 
 + b*x**2)*b**4*c**2*x**6 + 1680*sqrt(a + b*x**2)*b**4*c*d*x**7 - 840*sqrt 
(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*d + 315*sqrt(b)*log 
((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*c*d - 840*sqrt(b)*log((sqrt( 
a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**3*b**2*c)/(13440*b**3)