\(\int x^3 (a+b x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [74]

Optimal result

Integrand size = 30, antiderivative size = 260 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {3 a^3 (2 b B-a D) x \sqrt {a+b x^2}}{256 b^3}+\frac {a^2 (2 b B-a D) x^3 \sqrt {a+b x^2}}{128 b^2}+\frac {a (2 b B-a D) x^5 \sqrt {a+b x^2}}{32 b}+\frac {(2 b B-a D) x^5 \left (a+b x^2\right )^{3/2}}{16 b}-\frac {a (A b-a C) \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac {(A b-2 a C) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {C \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {3 a^4 (2 b B-a D) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}} \] Output:

-3/256*a^3*(2*B*b-D*a)*x*(b*x^2+a)^(1/2)/b^3+1/128*a^2*(2*B*b-D*a)*x^3*(b* 
x^2+a)^(1/2)/b^2+1/32*a*(2*B*b-D*a)*x^5*(b*x^2+a)^(1/2)/b+1/16*(2*B*b-D*a) 
*x^5*(b*x^2+a)^(3/2)/b-1/5*a*(A*b-C*a)*(b*x^2+a)^(5/2)/b^3+1/10*D*x^5*(b*x 
^2+a)^(5/2)/b+1/7*(A*b-2*C*a)*(b*x^2+a)^(7/2)/b^3+1/9*C*(b*x^2+a)^(9/2)/b^ 
3+3/256*a^4*(2*B*b-D*a)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.73 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (a^4 (2048 C+945 D x)+32 b^4 x^6 \left (360 A+7 x \left (45 B+40 C x+36 D x^2\right )\right )+12 a^2 b^2 x^2 \left (192 A+x \left (105 B+64 C x+42 D x^2\right )\right )-2 a^3 b \left (2304 A+x \left (945 B+512 C x+315 D x^2\right )\right )+16 a b^3 x^4 \left (1152 A+x \left (945 B+800 C x+693 D x^2\right )\right )\right )+945 a^4 (-2 b B+a D) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{80640 b^{7/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(a^4*(2048*C + 945*D*x) + 32*b^4*x^6*(360*A + 7*x 
*(45*B + 40*C*x + 36*D*x^2)) + 12*a^2*b^2*x^2*(192*A + x*(105*B + 64*C*x + 
 42*D*x^2)) - 2*a^3*b*(2304*A + x*(945*B + 512*C*x + 315*D*x^2)) + 16*a*b^ 
3*x^4*(1152*A + x*(945*B + 800*C*x + 693*D*x^2))) + 945*a^4*(-2*b*B + a*D) 
*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(80640*b^(7/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2340, 27, 2340, 27, 533, 533, 25, 27, 533, 27, 455, 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.

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

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\int 5 x^3 \left (b x^2+a\right )^{3/2} \left (2 b C x^2+(2 b B-a D) x+2 A b\right )dx}{10 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x^3 \left (b x^2+a\right )^{3/2} \left (2 b C x^2+(2 b B-a D) x+2 A b\right )dx}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int b x^3 (2 (9 A b-4 a C)+9 (2 b B-a D) x) \left (b x^2+a\right )^{3/2}dx}{9 b}+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \int x^3 (2 (9 A b-4 a C)+9 (2 b B-a D) x) \left (b x^2+a\right )^{3/2}dx+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\int x^2 (27 a (2 b B-a D)-16 b (9 A b-4 a C) x) \left (b x^2+a\right )^{3/2}dx}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {-\frac {\int -a b x (32 (9 A b-4 a C)+189 (2 b B-a D) x) \left (b x^2+a\right )^{3/2}dx}{7 b}-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {\int a b x (32 (9 A b-4 a C)+189 (2 b B-a D) x) \left (b x^2+a\right )^{3/2}dx}{7 b}-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \int x (32 (9 A b-4 a C)+189 (2 b B-a D) x) \left (b x^2+a\right )^{3/2}dx-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {\int 3 (63 a (2 b B-a D)-64 b (9 A b-4 a C) x) \left (b x^2+a\right )^{3/2}dx}{6 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {\int (63 a (2 b B-a D)-64 b (9 A b-4 a C) x) \left (b x^2+a\right )^{3/2}dx}{2 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {63 a (2 b B-a D) \int \left (b x^2+a\right )^{3/2}dx-\frac {64}{5} \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{2 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {63 a (2 b B-a D) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {64}{5} \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{2 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {63 a (2 b B-a D) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {64}{5} \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{2 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {63 a (2 b B-a D) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {64}{5} \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{2 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 x^3 \left (a+b x^2\right )^{5/2} (2 b B-a D)}{8 b}-\frac {\frac {1}{7} a \left (\frac {63 x \left (a+b x^2\right )^{5/2} (2 b B-a D)}{2 b}-\frac {63 a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) (2 b B-a D)-\frac {64}{5} \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{2 b}\right )-\frac {16}{7} x^2 \left (a+b x^2\right )^{5/2} (9 A b-4 a C)}{8 b}\right )+\frac {2}{9} C x^4 \left (a+b x^2\right )^{5/2}}{2 b}+\frac {D x^5 \left (a+b x^2\right )^{5/2}}{10 b}\)

Input:

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

Output:

(D*x^5*(a + b*x^2)^(5/2))/(10*b) + ((2*C*x^4*(a + b*x^2)^(5/2))/9 + ((9*(2 
*b*B - a*D)*x^3*(a + b*x^2)^(5/2))/(8*b) - ((-16*(9*A*b - 4*a*C)*x^2*(a + 
b*x^2)^(5/2))/7 + (a*((63*(2*b*B - a*D)*x*(a + b*x^2)^(5/2))/(2*b) - ((-64 
*(9*A*b - 4*a*C)*(a + b*x^2)^(5/2))/5 + 63*a*(2*b*B - 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))/(2*b)))/7)/(8*b))/9)/(2*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 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 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 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 = 0.84 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.22

method	result	size
default	\(A \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+B \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+C \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )+D \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )\)	\(318\)

Input:

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

Output:

A*(1/7*x^2*(b*x^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+B*(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)) 
))))+C*(1/9*x^4*(b*x^2+a)^(5/2)/b-4/9*a/b*(1/7*x^2*(b*x^2+a)^(5/2)/b-2/35* 
a/b^2*(b*x^2+a)^(5/2)))+D*(1/10*x^5*(b*x^2+a)^(5/2)/b-1/2*a/b*(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 = 477, normalized size of antiderivative = 1.83 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\left [-\frac {945 \, {\left (D a^{5} - 2 \, B a^{4} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8064 \, D b^{5} x^{9} + 8960 \, C b^{5} x^{8} + 1008 \, {\left (11 \, D a b^{4} + 10 \, B b^{5}\right )} x^{7} + 1280 \, {\left (10 \, C a b^{4} + 9 \, A b^{5}\right )} x^{6} + 2048 \, C a^{4} b - 4608 \, A a^{3} b^{2} + 504 \, {\left (D a^{2} b^{3} + 30 \, B a b^{4}\right )} x^{5} + 768 \, {\left (C a^{2} b^{3} + 24 \, A a b^{4}\right )} x^{4} - 630 \, {\left (D a^{3} b^{2} - 2 \, B a^{2} b^{3}\right )} x^{3} - 256 \, {\left (4 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 945 \, {\left (D a^{4} b - 2 \, B a^{3} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{161280 \, b^{4}}, \frac {945 \, {\left (D a^{5} - 2 \, B a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (8064 \, D b^{5} x^{9} + 8960 \, C b^{5} x^{8} + 1008 \, {\left (11 \, D a b^{4} + 10 \, B b^{5}\right )} x^{7} + 1280 \, {\left (10 \, C a b^{4} + 9 \, A b^{5}\right )} x^{6} + 2048 \, C a^{4} b - 4608 \, A a^{3} b^{2} + 504 \, {\left (D a^{2} b^{3} + 30 \, B a b^{4}\right )} x^{5} + 768 \, {\left (C a^{2} b^{3} + 24 \, A a b^{4}\right )} x^{4} - 630 \, {\left (D a^{3} b^{2} - 2 \, B a^{2} b^{3}\right )} x^{3} - 256 \, {\left (4 \, C a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{2} + 945 \, {\left (D a^{4} b - 2 \, B a^{3} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{80640 \, b^{4}}\right ] \] Input:

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

Output:

[-1/161280*(945*(D*a^5 - 2*B*a^4*b)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + 
a)*sqrt(b)*x - a) - 2*(8064*D*b^5*x^9 + 8960*C*b^5*x^8 + 1008*(11*D*a*b^4 
+ 10*B*b^5)*x^7 + 1280*(10*C*a*b^4 + 9*A*b^5)*x^6 + 2048*C*a^4*b - 4608*A* 
a^3*b^2 + 504*(D*a^2*b^3 + 30*B*a*b^4)*x^5 + 768*(C*a^2*b^3 + 24*A*a*b^4)* 
x^4 - 630*(D*a^3*b^2 - 2*B*a^2*b^3)*x^3 - 256*(4*C*a^3*b^2 - 9*A*a^2*b^3)* 
x^2 + 945*(D*a^4*b - 2*B*a^3*b^2)*x)*sqrt(b*x^2 + a))/b^4, 1/80640*(945*(D 
*a^5 - 2*B*a^4*b)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (8064*D*b^ 
5*x^9 + 8960*C*b^5*x^8 + 1008*(11*D*a*b^4 + 10*B*b^5)*x^7 + 1280*(10*C*a*b 
^4 + 9*A*b^5)*x^6 + 2048*C*a^4*b - 4608*A*a^3*b^2 + 504*(D*a^2*b^3 + 30*B* 
a*b^4)*x^5 + 768*(C*a^2*b^3 + 24*A*a*b^4)*x^4 - 630*(D*a^3*b^2 - 2*B*a^2*b 
^3)*x^3 - 256*(4*C*a^3*b^2 - 9*A*a^2*b^3)*x^2 + 945*(D*a^4*b - 2*B*a^3*b^2 
)*x)*sqrt(b*x^2 + a))/b^4]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (236) = 472\).

Time = 0.88 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.86 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \frac {3 a^{2} \left (B a^{2} - \frac {5 a \left (2 B a b + D a^{2} - \frac {7 a \left (B b^{2} + \frac {11 D a b}{10}\right )}{8 b}\right )}{6 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {a + b x^{2}} \left (\frac {C b x^{8}}{9} + \frac {D b x^{9}}{10} - \frac {3 a x \left (B a^{2} - \frac {5 a \left (2 B a b + D a^{2} - \frac {7 a \left (B b^{2} + \frac {11 D a b}{10}\right )}{8 b}\right )}{6 b}\right )}{8 b^{2}} - \frac {2 a \left (A a^{2} - \frac {4 a \left (2 A a b + C a^{2} - \frac {6 a \left (A b^{2} + \frac {10 C a b}{9}\right )}{7 b}\right )}{5 b}\right )}{3 b^{2}} + \frac {x^{7} \left (B b^{2} + \frac {11 D a b}{10}\right )}{8 b} + \frac {x^{6} \left (A b^{2} + \frac {10 C a b}{9}\right )}{7 b} + \frac {x^{5} \cdot \left (2 B a b + D a^{2} - \frac {7 a \left (B b^{2} + \frac {11 D a b}{10}\right )}{8 b}\right )}{6 b} + \frac {x^{4} \cdot \left (2 A a b + C a^{2} - \frac {6 a \left (A b^{2} + \frac {10 C a b}{9}\right )}{7 b}\right )}{5 b} + \frac {x^{3} \left (B a^{2} - \frac {5 a \left (2 B a b + D a^{2} - \frac {7 a \left (B b^{2} + \frac {11 D a b}{10}\right )}{8 b}\right )}{6 b}\right )}{4 b} + \frac {x^{2} \left (A a^{2} - \frac {4 a \left (2 A a b + C a^{2} - \frac {6 a \left (A b^{2} + \frac {10 C a b}{9}\right )}{7 b}\right )}{5 b}\right )}{3 b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{4}}{4} + \frac {B x^{5}}{5} + \frac {C x^{6}}{6} + \frac {D x^{7}}{7}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((3*a**2*(B*a**2 - 5*a*(2*B*a*b + D*a**2 - 7*a*(B*b**2 + 11*D*a*b 
/10)/(8*b))/(6*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))/(8*b**2) + sqrt(a + b*x**2) 
*(C*b*x**8/9 + D*b*x**9/10 - 3*a*x*(B*a**2 - 5*a*(2*B*a*b + D*a**2 - 7*a*( 
B*b**2 + 11*D*a*b/10)/(8*b))/(6*b))/(8*b**2) - 2*a*(A*a**2 - 4*a*(2*A*a*b 
+ C*a**2 - 6*a*(A*b**2 + 10*C*a*b/9)/(7*b))/(5*b))/(3*b**2) + x**7*(B*b**2 
 + 11*D*a*b/10)/(8*b) + x**6*(A*b**2 + 10*C*a*b/9)/(7*b) + x**5*(2*B*a*b + 
 D*a**2 - 7*a*(B*b**2 + 11*D*a*b/10)/(8*b))/(6*b) + x**4*(2*A*a*b + C*a**2 
 - 6*a*(A*b**2 + 10*C*a*b/9)/(7*b))/(5*b) + x**3*(B*a**2 - 5*a*(2*B*a*b + 
D*a**2 - 7*a*(B*b**2 + 11*D*a*b/10)/(8*b))/(6*b))/(4*b) + x**2*(A*a**2 - 4 
*a*(2*A*a*b + C*a**2 - 6*a*(A*b**2 + 10*C*a*b/9)/(7*b))/(5*b))/(3*b)), Ne( 
b, 0)), (a**(3/2)*(A*x**4/4 + B*x**5/5 + C*x**6/6 + D*x**7/7), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.13 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D x^{5}}{10 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C x^{4}}{9 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D a x^{3}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{3}}{8 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C a x^{2}}{63 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x^{2}}{7 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D a^{2} x}{32 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} D a^{3} x}{128 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a} D a^{4} x}{256 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{3} x}{128 \, b^{2}} - \frac {3 \, D a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {7}{2}}} + \frac {3 \, B a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C a^{2}}{315 \, b^{3}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a}{35 \, b^{2}} \] Input:

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

Output:

1/10*(b*x^2 + a)^(5/2)*D*x^5/b + 1/9*(b*x^2 + a)^(5/2)*C*x^4/b - 1/16*(b*x 
^2 + a)^(5/2)*D*a*x^3/b^2 + 1/8*(b*x^2 + a)^(5/2)*B*x^3/b - 4/63*(b*x^2 + 
a)^(5/2)*C*a*x^2/b^2 + 1/7*(b*x^2 + a)^(5/2)*A*x^2/b + 1/32*(b*x^2 + a)^(5 
/2)*D*a^2*x/b^3 - 1/128*(b*x^2 + a)^(3/2)*D*a^3*x/b^3 - 3/256*sqrt(b*x^2 + 
 a)*D*a^4*x/b^3 - 1/16*(b*x^2 + a)^(5/2)*B*a*x/b^2 + 1/64*(b*x^2 + a)^(3/2 
)*B*a^2*x/b^2 + 3/128*sqrt(b*x^2 + a)*B*a^3*x/b^2 - 3/256*D*a^5*arcsinh(b* 
x/sqrt(a*b))/b^(7/2) + 3/128*B*a^4*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 8/315* 
(b*x^2 + a)^(5/2)*C*a^2/b^3 - 2/35*(b*x^2 + a)^(5/2)*A*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{80640} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, D b x + 10 \, C b\right )} x + \frac {9 \, {\left (11 \, D a b^{8} + 10 \, B b^{9}\right )}}{b^{8}}\right )} x + \frac {80 \, {\left (10 \, C a b^{8} + 9 \, A b^{9}\right )}}{b^{8}}\right )} x + \frac {63 \, {\left (D a^{2} b^{7} + 30 \, B a b^{8}\right )}}{b^{8}}\right )} x + \frac {96 \, {\left (C a^{2} b^{7} + 24 \, A a b^{8}\right )}}{b^{8}}\right )} x - \frac {315 \, {\left (D a^{3} b^{6} - 2 \, B a^{2} b^{7}\right )}}{b^{8}}\right )} x - \frac {128 \, {\left (4 \, C a^{3} b^{6} - 9 \, A a^{2} b^{7}\right )}}{b^{8}}\right )} x + \frac {945 \, {\left (D a^{4} b^{5} - 2 \, B a^{3} b^{6}\right )}}{b^{8}}\right )} x + \frac {512 \, {\left (4 \, C a^{4} b^{5} - 9 \, A a^{3} b^{6}\right )}}{b^{8}}\right )} + \frac {3 \, {\left (D a^{5} - 2 \, B a^{4} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {7}{2}}} \] Input:

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

Output:

1/80640*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(9*D*b*x + 10*C*b)*x + 9*(11*D*a 
*b^8 + 10*B*b^9)/b^8)*x + 80*(10*C*a*b^8 + 9*A*b^9)/b^8)*x + 63*(D*a^2*b^7 
 + 30*B*a*b^8)/b^8)*x + 96*(C*a^2*b^7 + 24*A*a*b^8)/b^8)*x - 315*(D*a^3*b^ 
6 - 2*B*a^2*b^7)/b^8)*x - 128*(4*C*a^3*b^6 - 9*A*a^2*b^7)/b^8)*x + 945*(D* 
a^4*b^5 - 2*B*a^3*b^6)/b^8)*x + 512*(4*C*a^4*b^5 - 9*A*a^3*b^6)/b^8) + 3/2 
56*(D*a^5 - 2*B*a^4*b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.47 \[ \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {-4608 \sqrt {b \,x^{2}+a}\, a^{4} b^{2}+2048 \sqrt {b \,x^{2}+a}\, a^{4} b c +945 \sqrt {b \,x^{2}+a}\, a^{4} b d x +2304 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x^{2}-1890 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x -1024 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c \,x^{2}-630 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d \,x^{3}+18432 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{4}+1260 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{3}+768 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c \,x^{4}+504 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d \,x^{5}+11520 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{6}+15120 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{5}+12800 \sqrt {b \,x^{2}+a}\, a \,b^{4} c \,x^{6}+11088 \sqrt {b \,x^{2}+a}\, a \,b^{4} d \,x^{7}+10080 \sqrt {b \,x^{2}+a}\, b^{6} x^{7}+8960 \sqrt {b \,x^{2}+a}\, b^{5} c \,x^{8}+8064 \sqrt {b \,x^{2}+a}\, b^{5} d \,x^{9}-945 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} d +1890 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b^{2}}{80640 b^{4}} \] Input:

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

Output:

( - 4608*sqrt(a + b*x**2)*a**4*b**2 + 2048*sqrt(a + b*x**2)*a**4*b*c + 945 
*sqrt(a + b*x**2)*a**4*b*d*x + 2304*sqrt(a + b*x**2)*a**3*b**3*x**2 - 1890 
*sqrt(a + b*x**2)*a**3*b**3*x - 1024*sqrt(a + b*x**2)*a**3*b**2*c*x**2 - 6 
30*sqrt(a + b*x**2)*a**3*b**2*d*x**3 + 18432*sqrt(a + b*x**2)*a**2*b**4*x* 
*4 + 1260*sqrt(a + b*x**2)*a**2*b**4*x**3 + 768*sqrt(a + b*x**2)*a**2*b**3 
*c*x**4 + 504*sqrt(a + b*x**2)*a**2*b**3*d*x**5 + 11520*sqrt(a + b*x**2)*a 
*b**5*x**6 + 15120*sqrt(a + b*x**2)*a*b**5*x**5 + 12800*sqrt(a + b*x**2)*a 
*b**4*c*x**6 + 11088*sqrt(a + b*x**2)*a*b**4*d*x**7 + 10080*sqrt(a + b*x** 
2)*b**6*x**7 + 8960*sqrt(a + b*x**2)*b**5*c*x**8 + 8064*sqrt(a + b*x**2)*b 
**5*d*x**9 - 945*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**5* 
d + 1890*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b**2)/(8 
0640*b**4)