\(\int (A+B x) (c+d x)^3 (a+b x^2)^{3/2} \, dx\) [155]

Optimal result

Integrand size = 24, antiderivative size = 317 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {3 a \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) x \left (a+b x^2\right )^{3/2}}{64 b^2}+\frac {(3 B c+8 A d) (c+d x)^2 \left (a+b x^2\right )^{5/2}}{56 b}+\frac {B (c+d x)^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {\left (4 \left (8 a d^2 (3 B c+A d)-b c^2 (3 B c+64 A d)\right )+5 d \left (7 a B d^2-2 b c (B c+12 A d)\right ) x\right ) \left (a+b x^2\right )^{5/2}}{560 b^2}+\frac {3 a^2 \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:

3/128*a*(8*A*b*c*(-a*d^2+2*b*c^2)-a*B*d*(-a*d^2+8*b*c^2))*x*(b*x^2+a)^(1/2 
)/b^2+1/64*(8*A*b*c*(-a*d^2+2*b*c^2)-a*B*d*(-a*d^2+8*b*c^2))*x*(b*x^2+a)^( 
3/2)/b^2+1/56*(8*A*d+3*B*c)*(d*x+c)^2*(b*x^2+a)^(5/2)/b+1/8*B*(d*x+c)^3*(b 
*x^2+a)^(5/2)/b-1/560*(32*a*d^2*(A*d+3*B*c)-4*b*c^2*(64*A*d+3*B*c)+5*d*(7* 
a*B*d^2-2*b*c*(12*A*d+B*c))*x)*(b*x^2+a)^(5/2)/b^2+3/128*a^2*(8*A*b*c*(-a* 
d^2+2*b*c^2)-a*B*d*(-a*d^2+8*b*c^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^ 
(5/2)
 

Mathematica [A] (verified)

Time = 2.11 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.02 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (-a^3 d^2 (768 B c+256 A d+105 B d x)+16 b^3 x^3 \left (2 A \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+B x \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )\right )+2 a^2 b \left (4 A d \left (336 c^2+105 c d x+16 d^2 x^2\right )+B \left (448 c^3+420 c^2 d x+192 c d^2 x^2+35 d^3 x^3\right )\right )+8 a b^2 x \left (2 A \left (175 c^3+336 c^2 d x+245 c d^2 x^2+64 d^3 x^3\right )+B x \left (224 c^3+490 c^2 d x+384 c d^2 x^2+105 d^3 x^3\right )\right )\right )-105 a^2 \left (8 A b c \left (2 b c^2-a d^2\right )+a B d \left (-8 b c^2+a d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{4480 b^{5/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(-(a^3*d^2*(768*B*c + 256*A*d + 105*B*d*x)) + 16* 
b^3*x^3*(2*A*(35*c^3 + 84*c^2*d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + B*x*(56*c 
^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3)) + 2*a^2*b*(4*A*d*(336*c^2 
+ 105*c*d*x + 16*d^2*x^2) + B*(448*c^3 + 420*c^2*d*x + 192*c*d^2*x^2 + 35* 
d^3*x^3)) + 8*a*b^2*x*(2*A*(175*c^3 + 336*c^2*d*x + 245*c*d^2*x^2 + 64*d^3 
*x^3) + B*x*(224*c^3 + 490*c^2*d*x + 384*c*d^2*x^2 + 105*d^3*x^3))) - 105* 
a^2*(8*A*b*c*(2*b*c^2 - a*d^2) + a*B*d*(-8*b*c^2 + a*d^2))*Log[-(Sqrt[b]*x 
) + Sqrt[a + b*x^2]])/(4480*b^(5/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {687, 687, 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.

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

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {\int b (c+d x) \left (56 A b c^2-27 a B d c-16 a A d^2-3 \left (7 a B d^2-2 b c (B c+12 A d)\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b}+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \int (c+d x) \left (56 A b c^2-27 a B d c-16 a A d^2-3 \left (7 a B d^2-2 b c (B c+12 A d)\right ) x\right ) \left (b x^2+a\right )^{3/2}dx+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) \int \left (b x^2+a\right )^{3/2}dx}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (A d+3 B c)-b c^2 (64 A d+3 B c)\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (7 a B d^2-2 b c (12 A d+B c)\right )}{2 b}\right )+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (A d+3 B c)-b c^2 (64 A d+3 B c)\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (7 a B d^2-2 b c (12 A d+B c)\right )}{2 b}\right )+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (A d+3 B c)-b c^2 (64 A d+3 B c)\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (7 a B d^2-2 b c (12 A d+B c)\right )}{2 b}\right )+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (A d+3 B c)-b c^2 (64 A d+3 B c)\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (7 a B d^2-2 b c (12 A d+B c)\right )}{2 b}\right )+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{7} \left (\frac {7 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (8 A b c \left (2 b c^2-a d^2\right )-a B d \left (8 b c^2-a d^2\right )\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (8 a d^2 (A d+3 B c)-b c^2 (64 A d+3 B c)\right )}{5 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (7 a B d^2-2 b c (12 A d+B c)\right )}{2 b}\right )+\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 (8 A d+3 B c)}{8 b}+\frac {B \left (a+b x^2\right )^{5/2} (c+d x)^3}{8 b}\)

Input:

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

Output:

(B*(c + d*x)^3*(a + b*x^2)^(5/2))/(8*b) + (((3*B*c + 8*A*d)*(c + d*x)^2*(a 
 + b*x^2)^(5/2))/7 + ((-2*(8*a*d^2*(3*B*c + A*d) - b*c^2*(3*B*c + 64*A*d)) 
*(a + b*x^2)^(5/2))/(5*b) - (d*(7*a*B*d^2 - 2*b*c*(B*c + 12*A*d))*x*(a + b 
*x^2)^(5/2))/(2*b) + (7*(8*A*b*c*(2*b*c^2 - a*d^2) - a*B*d*(8*b*c^2 - a*d^ 
2))*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(S 
qrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/(2*b))/7)/(8*b)
 

Defintions of rubi rules used

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 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.99

method	result	size
default	\(A \,c^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+d^{2} \left (A d +3 B c \right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+3 c d \left (A d +B c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+\frac {c^{2} \left (3 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+B \,d^{3} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )\)	\(314\)
risch	\(-\frac {\left (-560 B \,b^{3} d^{3} x^{7}-640 A \,b^{3} d^{3} x^{6}-1920 B \,b^{3} c \,d^{2} x^{6}-2240 A \,b^{3} c \,d^{2} x^{5}-840 B a \,b^{2} d^{3} x^{5}-2240 B \,b^{3} c^{2} d \,x^{5}-1024 A \,x^{4} a \,b^{2} d^{3}-2688 A \,b^{3} c^{2} d \,x^{4}-3072 B a \,b^{2} c \,d^{2} x^{4}-896 B \,b^{3} c^{3} x^{4}-3920 A a \,b^{2} c \,d^{2} x^{3}-1120 A \,b^{3} c^{3} x^{3}-70 a^{2} B \,d^{3} b \,x^{3}-3920 B a \,b^{2} c^{2} d \,x^{3}-128 A \,a^{2} b \,d^{3} x^{2}-5376 A a \,b^{2} c^{2} d \,x^{2}-384 B \,a^{2} b c \,d^{2} x^{2}-1792 B a \,b^{2} c^{3} x^{2}-840 A \,a^{2} b c \,d^{2} x -2800 A a \,b^{2} c^{3} x +105 a^{3} B \,d^{3} x -840 B \,a^{2} b \,c^{2} d x +256 A \,a^{3} d^{3}-2688 A \,a^{2} b \,c^{2} d +768 B \,a^{3} c \,d^{2}-896 B \,a^{2} b \,c^{3}\right ) \sqrt {b \,x^{2}+a}}{4480 b^{2}}-\frac {3 a^{2} \left (8 A a b c \,d^{2}-16 A \,b^{2} c^{3}-a^{2} B \,d^{3}+8 B a b \,c^{2} d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}\)	\(398\)

Input:

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

Output:

A*c^3*(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))))+d^2*(A*d+3*B*c)*(1/7*x^2*(b*x^2+a)^(5/2)/b-2 
/35*a/b^2*(b*x^2+a)^(5/2))+3*c*d*(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)))))+1/5*c^2*(3*A*d+B*c)/b*(b*x^2+a)^(5/2)+B*d^3*(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 = 0.13 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.54 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/8960*(105*(16*A*a^2*b^2*c^3 - 8*B*a^3*b*c^2*d - 8*A*a^3*b*c*d^2 + B*a^4 
*d^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(560*B*b 
^4*d^3*x^7 + 896*B*a^2*b^2*c^3 + 2688*A*a^2*b^2*c^2*d - 768*B*a^3*b*c*d^2 
- 256*A*a^3*b*d^3 + 640*(3*B*b^4*c*d^2 + A*b^4*d^3)*x^6 + 280*(8*B*b^4*c^2 
*d + 8*A*b^4*c*d^2 + 3*B*a*b^3*d^3)*x^5 + 128*(7*B*b^4*c^3 + 21*A*b^4*c^2* 
d + 24*B*a*b^3*c*d^2 + 8*A*a*b^3*d^3)*x^4 + 70*(16*A*b^4*c^3 + 56*B*a*b^3* 
c^2*d + 56*A*a*b^3*c*d^2 + B*a^2*b^2*d^3)*x^3 + 128*(14*B*a*b^3*c^3 + 42*A 
*a*b^3*c^2*d + 3*B*a^2*b^2*c*d^2 + A*a^2*b^2*d^3)*x^2 + 35*(80*A*a*b^3*c^3 
 + 24*B*a^2*b^2*c^2*d + 24*A*a^2*b^2*c*d^2 - 3*B*a^3*b*d^3)*x)*sqrt(b*x^2 
+ a))/b^3, -1/4480*(105*(16*A*a^2*b^2*c^3 - 8*B*a^3*b*c^2*d - 8*A*a^3*b*c* 
d^2 + B*a^4*d^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (560*B*b^4* 
d^3*x^7 + 896*B*a^2*b^2*c^3 + 2688*A*a^2*b^2*c^2*d - 768*B*a^3*b*c*d^2 - 2 
56*A*a^3*b*d^3 + 640*(3*B*b^4*c*d^2 + A*b^4*d^3)*x^6 + 280*(8*B*b^4*c^2*d 
+ 8*A*b^4*c*d^2 + 3*B*a*b^3*d^3)*x^5 + 128*(7*B*b^4*c^3 + 21*A*b^4*c^2*d + 
 24*B*a*b^3*c*d^2 + 8*A*a*b^3*d^3)*x^4 + 70*(16*A*b^4*c^3 + 56*B*a*b^3*c^2 
*d + 56*A*a*b^3*c*d^2 + B*a^2*b^2*d^3)*x^3 + 128*(14*B*a*b^3*c^3 + 42*A*a* 
b^3*c^2*d + 3*B*a^2*b^2*c*d^2 + A*a^2*b^2*d^3)*x^2 + 35*(80*A*a*b^3*c^3 + 
24*B*a^2*b^2*c^2*d + 24*A*a^2*b^2*c*d^2 - 3*B*a^3*b*d^3)*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. 921 vs. \(2 (301) = 602\).

Time = 0.72 (sec) , antiderivative size = 921, normalized size of antiderivative = 2.91 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.18 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B d^{3} x^{3}}{8 \, b} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A c^{3} x + \frac {3}{8} \, \sqrt {b x^{2} + a} A a c^{3} x - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a d^{3} x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} d^{3} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{3} d^{3} x}{128 \, b^{2}} + \frac {3 \, A a^{2} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {3 \, B a^{4} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B c^{3}}{5 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A c^{2} d}{5 \, b} + \frac {{\left (3 \, B c d^{2} + A d^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{2}}{7 \, b} + \frac {{\left (B c^{2} d + A c d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x}{2 \, b} - \frac {{\left (B c^{2} d + A c d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{8 \, b} - \frac {3 \, {\left (B c^{2} d + A c d^{2}\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b} - \frac {3 \, {\left (B c^{2} d + A c d^{2}\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, B c d^{2} + A d^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a}{35 \, b^{2}} \] Input:

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

Output:

1/8*(b*x^2 + a)^(5/2)*B*d^3*x^3/b + 1/4*(b*x^2 + a)^(3/2)*A*c^3*x + 3/8*sq 
rt(b*x^2 + a)*A*a*c^3*x - 1/16*(b*x^2 + a)^(5/2)*B*a*d^3*x/b^2 + 1/64*(b*x 
^2 + a)^(3/2)*B*a^2*d^3*x/b^2 + 3/128*sqrt(b*x^2 + a)*B*a^3*d^3*x/b^2 + 3/ 
8*A*a^2*c^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 3/128*B*a^4*d^3*arcsinh(b*x/s 
qrt(a*b))/b^(5/2) + 1/5*(b*x^2 + a)^(5/2)*B*c^3/b + 3/5*(b*x^2 + a)^(5/2)* 
A*c^2*d/b + 1/7*(3*B*c*d^2 + A*d^3)*(b*x^2 + a)^(5/2)*x^2/b + 1/2*(B*c^2*d 
 + A*c*d^2)*(b*x^2 + a)^(5/2)*x/b - 1/8*(B*c^2*d + A*c*d^2)*(b*x^2 + a)^(3 
/2)*a*x/b - 3/16*(B*c^2*d + A*c*d^2)*sqrt(b*x^2 + a)*a^2*x/b - 3/16*(B*c^2 
*d + A*c*d^2)*a^3*arcsinh(b*x/sqrt(a*b))/b^(3/2) - 2/35*(3*B*c*d^2 + A*d^3 
)*(b*x^2 + a)^(5/2)*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.34 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{3/2} \, dx=\frac {1}{4480} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, B b d^{3} x + \frac {8 \, {\left (3 \, B b^{7} c d^{2} + A b^{7} d^{3}\right )}}{b^{6}}\right )} x + \frac {7 \, {\left (8 \, B b^{7} c^{2} d + 8 \, A b^{7} c d^{2} + 3 \, B a b^{6} d^{3}\right )}}{b^{6}}\right )} x + \frac {16 \, {\left (7 \, B b^{7} c^{3} + 21 \, A b^{7} c^{2} d + 24 \, B a b^{6} c d^{2} + 8 \, A a b^{6} d^{3}\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (16 \, A b^{7} c^{3} + 56 \, B a b^{6} c^{2} d + 56 \, A a b^{6} c d^{2} + B a^{2} b^{5} d^{3}\right )}}{b^{6}}\right )} x + \frac {64 \, {\left (14 \, B a b^{6} c^{3} + 42 \, A a b^{6} c^{2} d + 3 \, B a^{2} b^{5} c d^{2} + A a^{2} b^{5} d^{3}\right )}}{b^{6}}\right )} x + \frac {35 \, {\left (80 \, A a b^{6} c^{3} + 24 \, B a^{2} b^{5} c^{2} d + 24 \, A a^{2} b^{5} c d^{2} - 3 \, B a^{3} b^{4} d^{3}\right )}}{b^{6}}\right )} x + \frac {128 \, {\left (7 \, B a^{2} b^{5} c^{3} + 21 \, A a^{2} b^{5} c^{2} d - 6 \, B a^{3} b^{4} c d^{2} - 2 \, A a^{3} b^{4} d^{3}\right )}}{b^{6}}\right )} - \frac {3 \, {\left (16 \, A a^{2} b^{2} c^{3} - 8 \, B a^{3} b c^{2} d - 8 \, A a^{3} b c d^{2} + B a^{4} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:

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

Output:

1/4480*sqrt(b*x^2 + a)*((2*((4*(5*(2*(7*B*b*d^3*x + 8*(3*B*b^7*c*d^2 + A*b 
^7*d^3)/b^6)*x + 7*(8*B*b^7*c^2*d + 8*A*b^7*c*d^2 + 3*B*a*b^6*d^3)/b^6)*x 
+ 16*(7*B*b^7*c^3 + 21*A*b^7*c^2*d + 24*B*a*b^6*c*d^2 + 8*A*a*b^6*d^3)/b^6 
)*x + 35*(16*A*b^7*c^3 + 56*B*a*b^6*c^2*d + 56*A*a*b^6*c*d^2 + B*a^2*b^5*d 
^3)/b^6)*x + 64*(14*B*a*b^6*c^3 + 42*A*a*b^6*c^2*d + 3*B*a^2*b^5*c*d^2 + A 
*a^2*b^5*d^3)/b^6)*x + 35*(80*A*a*b^6*c^3 + 24*B*a^2*b^5*c^2*d + 24*A*a^2* 
b^5*c*d^2 - 3*B*a^3*b^4*d^3)/b^6)*x + 128*(7*B*a^2*b^5*c^3 + 21*A*a^2*b^5* 
c^2*d - 6*B*a^3*b^4*c*d^2 - 2*A*a^3*b^4*d^3)/b^6) - 3/128*(16*A*a^2*b^2*c^ 
3 - 8*B*a^3*b*c^2*d - 8*A*a^3*b*c*d^2 + B*a^4*d^3)*log(abs(-sqrt(b)*x + sq 
rt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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