\(\int (A+B x) (c+d x)^3 (a+b x^2)^{5/2} \, dx\) [162]

Optimal result

Integrand size = 24, antiderivative size = 377 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {a^2 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {(3 B c+10 A d) (c+d x)^2 \left (a+b x^2\right )^{7/2}}{90 b}+\frac {B (c+d x)^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {\left (16 \left (10 a d^2 (3 B c+A d)-b c^2 (3 B c+100 A d)\right )+7 d \left (27 a B d^2-2 b c (3 B c+55 A d)\right ) x\right ) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac {a^3 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \] Output:

1/256*a^2*(10*A*b*c*(-3*a*d^2+8*b*c^2)-3*a*B*d*(-a*d^2+10*b*c^2))*x*(b*x^2 
+a)^(1/2)/b^2+1/384*a*(10*A*b*c*(-3*a*d^2+8*b*c^2)-3*a*B*d*(-a*d^2+10*b*c^ 
2))*x*(b*x^2+a)^(3/2)/b^2+1/480*(10*A*b*c*(-3*a*d^2+8*b*c^2)-3*a*B*d*(-a*d 
^2+10*b*c^2))*x*(b*x^2+a)^(5/2)/b^2+1/90*(10*A*d+3*B*c)*(d*x+c)^2*(b*x^2+a 
)^(7/2)/b+1/10*B*(d*x+c)^3*(b*x^2+a)^(7/2)/b-1/5040*(160*a*d^2*(A*d+3*B*c) 
-16*b*c^2*(100*A*d+3*B*c)+7*d*(27*a*B*d^2-2*b*c*(55*A*d+3*B*c))*x)*(b*x^2+ 
a)^(7/2)/b^2+1/256*a^3*(10*A*b*c*(-3*a*d^2+8*b*c^2)-3*a*B*d*(-a*d^2+10*b*c 
^2))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 3.25 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.08 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (-5 a^4 d^2 (512 A d+3 B (512 c+63 d x))+10 a^3 b \left (A d \left (3456 c^2+945 c d x+128 d^2 x^2\right )+3 B \left (384 c^3+315 c^2 d x+128 c d^2 x^2+21 d^3 x^3\right )\right )+32 b^4 x^5 \left (5 A \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )+3 B x \left (120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right )\right )+12 a^2 b^2 x \left (5 A \left (924 c^3+1728 c^2 d x+1239 c d^2 x^2+320 d^3 x^3\right )+3 B x \left (960 c^3+2065 c^2 d x+1600 c d^2 x^2+434 d^3 x^3\right )\right )+16 a b^3 x^3 \left (5 A \left (546 c^3+1296 c^2 d x+1071 c d^2 x^2+304 d^3 x^3\right )+3 B x \left (720 c^3+1785 c^2 d x+1520 c d^2 x^2+441 d^3 x^3\right )\right )\right )-315 a^3 \left (10 A b c \left (8 b c^2-3 a d^2\right )+3 a B d \left (-10 b c^2+a d^2\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{80640 b^{5/2}} \] Input:

Integrate[(A + B*x)*(c + d*x)^3*(a + b*x^2)^(5/2),x]
 

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(-5*a^4*d^2*(512*A*d + 3*B*(512*c + 63*d*x)) + 10 
*a^3*b*(A*d*(3456*c^2 + 945*c*d*x + 128*d^2*x^2) + 3*B*(384*c^3 + 315*c^2* 
d*x + 128*c*d^2*x^2 + 21*d^3*x^3)) + 32*b^4*x^5*(5*A*(84*c^3 + 216*c^2*d*x 
 + 189*c*d^2*x^2 + 56*d^3*x^3) + 3*B*x*(120*c^3 + 315*c^2*d*x + 280*c*d^2* 
x^2 + 84*d^3*x^3)) + 12*a^2*b^2*x*(5*A*(924*c^3 + 1728*c^2*d*x + 1239*c*d^ 
2*x^2 + 320*d^3*x^3) + 3*B*x*(960*c^3 + 2065*c^2*d*x + 1600*c*d^2*x^2 + 43 
4*d^3*x^3)) + 16*a*b^3*x^3*(5*A*(546*c^3 + 1296*c^2*d*x + 1071*c*d^2*x^2 + 
 304*d^3*x^3) + 3*B*x*(720*c^3 + 1785*c^2*d*x + 1520*c*d^2*x^2 + 441*d^3*x 
^3))) - 315*a^3*(10*A*b*c*(8*b*c^2 - 3*a*d^2) + 3*a*B*d*(-10*b*c^2 + a*d^2 
))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(80640*b^(5/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.78, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {687, 687, 27, 676, 211, 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.

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {\int (c+d x)^2 (10 A b c-3 a B d+b (3 B c+10 A d) x) \left (b x^2+a\right )^{5/2}dx}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {\int b (c+d x) \left (90 A b c^2-33 a B d c-20 a A d^2-\left (27 a B d^2-2 b c (3 B c+55 A d)\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{9 b}+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \int (c+d x) \left (90 A b c^2-33 a B d c-20 a A d^2-\left (27 a B d^2-2 b c (3 B c+55 A d)\right ) x\right ) \left (b x^2+a\right )^{5/2}dx+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) \int \left (b x^2+a\right )^{5/2}dx}{8 b}-\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a d^2 (A d+3 B c)-b c^2 (100 A d+3 B c)\right )}{7 b}-\frac {d x \left (a+b x^2\right )^{7/2} \left (27 a B d^2-2 b c (55 A d+3 B c)\right )}{8 b}\right )+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}-\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a d^2 (A d+3 B c)-b c^2 (100 A d+3 B c)\right )}{7 b}-\frac {d x \left (a+b x^2\right )^{7/2} \left (27 a B d^2-2 b c (55 A d+3 B c)\right )}{8 b}\right )+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}-\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a d^2 (A d+3 B c)-b c^2 (100 A d+3 B c)\right )}{7 b}-\frac {d x \left (a+b x^2\right )^{7/2} \left (27 a B d^2-2 b c (55 A d+3 B c)\right )}{8 b}\right )+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}-\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a d^2 (A d+3 B c)-b c^2 (100 A d+3 B c)\right )}{7 b}-\frac {d x \left (a+b x^2\right )^{7/2} \left (27 a B d^2-2 b c (55 A d+3 B c)\right )}{8 b}\right )+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}-\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a d^2 (A d+3 B c)-b c^2 (100 A d+3 B c)\right )}{7 b}-\frac {d x \left (a+b x^2\right )^{7/2} \left (27 a B d^2-2 b c (55 A d+3 B c)\right )}{8 b}\right )+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{9} \left (\frac {9 \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right ) \left (10 A b c \left (8 b c^2-3 a d^2\right )-3 a B d \left (10 b c^2-a d^2\right )\right )}{8 b}-\frac {2 \left (a+b x^2\right )^{7/2} \left (10 a d^2 (A d+3 B c)-b c^2 (100 A d+3 B c)\right )}{7 b}-\frac {d x \left (a+b x^2\right )^{7/2} \left (27 a B d^2-2 b c (55 A d+3 B c)\right )}{8 b}\right )+\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 (10 A d+3 B c)}{10 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^3}{10 b}\)

Input:

Int[(A + B*x)*(c + d*x)^3*(a + b*x^2)^(5/2),x]
 

Output:

(B*(c + d*x)^3*(a + b*x^2)^(7/2))/(10*b) + (((3*B*c + 10*A*d)*(c + d*x)^2* 
(a + b*x^2)^(7/2))/9 + ((-2*(10*a*d^2*(3*B*c + A*d) - b*c^2*(3*B*c + 100*A 
*d))*(a + b*x^2)^(7/2))/(7*b) - (d*(27*a*B*d^2 - 2*b*c*(3*B*c + 55*A*d))*x 
*(a + b*x^2)^(7/2))/(8*b) + (9*(10*A*b*c*(8*b*c^2 - 3*a*d^2) - 3*a*B*d*(10 
*b*c^2 - a*d^2))*((x*(a + b*x^2)^(5/2))/6 + (5*a*((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))/(8*b))/9)/(10*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 = 1.22 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.96

method	result	size
default	\(A \,c^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+d^{2} \left (A d +3 B c \right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )+3 c d \left (A d +B c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )+\frac {c^{2} \left (3 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+B \,d^{3} \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )\)	\(362\)
risch	\(-\frac {\left (-8064 B \,b^{4} d^{3} x^{9}-8960 A \,b^{4} d^{3} x^{8}-26880 B \,b^{4} c \,d^{2} x^{8}-30240 A \,b^{4} c \,d^{2} x^{7}-21168 B a \,b^{3} d^{3} x^{7}-30240 B \,b^{4} c^{2} d \,x^{7}-24320 A a \,b^{3} d^{3} x^{6}-34560 A \,b^{4} c^{2} d \,x^{6}-72960 B a \,b^{3} c \,d^{2} x^{6}-11520 B \,b^{4} c^{3} x^{6}-85680 A a \,b^{3} c \,d^{2} x^{5}-13440 A \,b^{4} c^{3} x^{5}-15624 B \,a^{2} b^{2} d^{3} x^{5}-85680 B a \,b^{3} c^{2} d \,x^{5}-19200 A \,a^{2} b^{2} d^{3} x^{4}-103680 A a \,b^{3} c^{2} d \,x^{4}-57600 B \,a^{2} b^{2} c \,d^{2} x^{4}-34560 B a \,b^{3} c^{3} x^{4}-74340 A \,a^{2} b^{2} c \,d^{2} x^{3}-43680 A a \,b^{3} c^{3} x^{3}-630 B \,a^{3} b \,d^{3} x^{3}-74340 B \,a^{2} b^{2} c^{2} d \,x^{3}-1280 A \,a^{3} b \,d^{3} x^{2}-103680 A \,a^{2} b^{2} c^{2} d \,x^{2}-3840 B \,a^{3} b c \,d^{2} x^{2}-34560 B \,a^{2} b^{2} c^{3} x^{2}-9450 A \,a^{3} b c \,d^{2} x -55440 A \,a^{2} b^{2} c^{3} x +945 B \,a^{4} d^{3} x -9450 B \,a^{3} b \,c^{2} d x +2560 A \,a^{4} d^{3}-34560 A \,a^{3} b \,c^{2} d +7680 B \,a^{4} c \,d^{2}-11520 B \,a^{3} b \,c^{3}\right ) \sqrt {b \,x^{2}+a}}{80640 b^{2}}-\frac {a^{3} \left (30 A a b c \,d^{2}-80 A \,b^{2} c^{3}-3 a^{2} B \,d^{3}+30 B a b \,c^{2} d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}\)	\(522\)

Input:

int((B*x+A)*(d*x+c)^3*(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

A*c^3*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(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 
)*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2))+3*c*d*(A*d+B*c)*( 
1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(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))))))+1/7*c^2*(3*A*d+B*c)*(b*x^2+a)^(7/2)/b+B*d^3*(1/10*x^3*(b*x^2+ 
a)^(7/2)/b-3/10*a/b*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2 
)+5/6*a*(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 = 0.19 (sec) , antiderivative size = 1032, normalized size of antiderivative = 2.74 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(d*x+c)^3*(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/161280*(315*(80*A*a^3*b^2*c^3 - 30*B*a^4*b*c^2*d - 30*A*a^4*b*c*d^2 + 3 
*B*a^5*d^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(8 
064*B*b^5*d^3*x^9 + 11520*B*a^3*b^2*c^3 + 34560*A*a^3*b^2*c^2*d - 7680*B*a 
^4*b*c*d^2 - 2560*A*a^4*b*d^3 + 8960*(3*B*b^5*c*d^2 + A*b^5*d^3)*x^8 + 302 
4*(10*B*b^5*c^2*d + 10*A*b^5*c*d^2 + 7*B*a*b^4*d^3)*x^7 + 1280*(9*B*b^5*c^ 
3 + 27*A*b^5*c^2*d + 57*B*a*b^4*c*d^2 + 19*A*a*b^4*d^3)*x^6 + 168*(80*A*b^ 
5*c^3 + 510*B*a*b^4*c^2*d + 510*A*a*b^4*c*d^2 + 93*B*a^2*b^3*d^3)*x^5 + 38 
40*(9*B*a*b^4*c^3 + 27*A*a*b^4*c^2*d + 15*B*a^2*b^3*c*d^2 + 5*A*a^2*b^3*d^ 
3)*x^4 + 210*(208*A*a*b^4*c^3 + 354*B*a^2*b^3*c^2*d + 354*A*a^2*b^3*c*d^2 
+ 3*B*a^3*b^2*d^3)*x^3 + 1280*(27*B*a^2*b^3*c^3 + 81*A*a^2*b^3*c^2*d + 3*B 
*a^3*b^2*c*d^2 + A*a^3*b^2*d^3)*x^2 + 315*(176*A*a^2*b^3*c^3 + 30*B*a^3*b^ 
2*c^2*d + 30*A*a^3*b^2*c*d^2 - 3*B*a^4*b*d^3)*x)*sqrt(b*x^2 + a))/b^3, -1/ 
80640*(315*(80*A*a^3*b^2*c^3 - 30*B*a^4*b*c^2*d - 30*A*a^4*b*c*d^2 + 3*B*a 
^5*d^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8064*B*b^5*d^3*x^9 
+ 11520*B*a^3*b^2*c^3 + 34560*A*a^3*b^2*c^2*d - 7680*B*a^4*b*c*d^2 - 2560* 
A*a^4*b*d^3 + 8960*(3*B*b^5*c*d^2 + A*b^5*d^3)*x^8 + 3024*(10*B*b^5*c^2*d 
+ 10*A*b^5*c*d^2 + 7*B*a*b^4*d^3)*x^7 + 1280*(9*B*b^5*c^3 + 27*A*b^5*c^2*d 
 + 57*B*a*b^4*c*d^2 + 19*A*a*b^4*d^3)*x^6 + 168*(80*A*b^5*c^3 + 510*B*a*b^ 
4*c^2*d + 510*A*a*b^4*c*d^2 + 93*B*a^2*b^3*d^3)*x^5 + 3840*(9*B*a*b^4*c^3 
+ 27*A*a*b^4*c^2*d + 15*B*a^2*b^3*c*d^2 + 5*A*a^2*b^3*d^3)*x^4 + 210*(2...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1486 vs. \(2 (366) = 732\).

Time = 1.04 (sec) , antiderivative size = 1486, normalized size of antiderivative = 3.94 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)*(d*x+c)**3*(b*x**2+a)**(5/2),x)
 

Output:

Piecewise((sqrt(a + b*x**2)*(B*b**2*d**3*x**9/10 + x**8*(A*b**3*d**3 + 3*B 
*b**3*c*d**2)/(9*b) + x**7*(3*A*b**3*c*d**2 + 21*B*a*b**2*d**3/10 + 3*B*b* 
*3*c**2*d)/(8*b) + x**6*(3*A*a*b**2*d**3 + 3*A*b**3*c**2*d + 9*B*a*b**2*c* 
d**2 + B*b**3*c**3 - 8*a*(A*b**3*d**3 + 3*B*b**3*c*d**2)/(9*b))/(7*b) + x* 
*5*(9*A*a*b**2*c*d**2 + A*b**3*c**3 + 3*B*a**2*b*d**3 + 9*B*a*b**2*c**2*d 
- 7*a*(3*A*b**3*c*d**2 + 21*B*a*b**2*d**3/10 + 3*B*b**3*c**2*d)/(8*b))/(6* 
b) + x**4*(3*A*a**2*b*d**3 + 9*A*a*b**2*c**2*d + 9*B*a**2*b*c*d**2 + 3*B*a 
*b**2*c**3 - 6*a*(3*A*a*b**2*d**3 + 3*A*b**3*c**2*d + 9*B*a*b**2*c*d**2 + 
B*b**3*c**3 - 8*a*(A*b**3*d**3 + 3*B*b**3*c*d**2)/(9*b))/(7*b))/(5*b) + x* 
*3*(9*A*a**2*b*c*d**2 + 3*A*a*b**2*c**3 + B*a**3*d**3 + 9*B*a**2*b*c**2*d 
- 5*a*(9*A*a*b**2*c*d**2 + A*b**3*c**3 + 3*B*a**2*b*d**3 + 9*B*a*b**2*c**2 
*d - 7*a*(3*A*b**3*c*d**2 + 21*B*a*b**2*d**3/10 + 3*B*b**3*c**2*d)/(8*b))/ 
(6*b))/(4*b) + x**2*(A*a**3*d**3 + 9*A*a**2*b*c**2*d + 3*B*a**3*c*d**2 + 3 
*B*a**2*b*c**3 - 4*a*(3*A*a**2*b*d**3 + 9*A*a*b**2*c**2*d + 9*B*a**2*b*c*d 
**2 + 3*B*a*b**2*c**3 - 6*a*(3*A*a*b**2*d**3 + 3*A*b**3*c**2*d + 9*B*a*b** 
2*c*d**2 + B*b**3*c**3 - 8*a*(A*b**3*d**3 + 3*B*b**3*c*d**2)/(9*b))/(7*b)) 
/(5*b))/(3*b) + x*(3*A*a**3*c*d**2 + 3*A*a**2*b*c**3 + 3*B*a**3*c**2*d - 3 
*a*(9*A*a**2*b*c*d**2 + 3*A*a*b**2*c**3 + B*a**3*d**3 + 9*B*a**2*b*c**2*d 
- 5*a*(9*A*a*b**2*c*d**2 + A*b**3*c**3 + 3*B*a**2*b*d**3 + 9*B*a*b**2*c**2 
*d - 7*a*(3*A*b**3*c*d**2 + 21*B*a*b**2*d**3/10 + 3*B*b**3*c**2*d)/(8*b...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.18 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B d^{3} x^{3}}{10 \, b} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A c^{3} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a c^{3} x + \frac {5}{16} \, \sqrt {b x^{2} + a} A a^{2} c^{3} x - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a d^{3} x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} d^{3} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} d^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B a^{4} d^{3} x}{256 \, b^{2}} + \frac {5 \, A a^{3} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} + \frac {3 \, B a^{5} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B c^{3}}{7 \, b} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A c^{2} d}{7 \, b} + \frac {{\left (3 \, B c d^{2} + A d^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {7}{2}} x^{2}}{9 \, b} + \frac {3 \, {\left (B c^{2} d + A c d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {7}{2}} x}{8 \, b} - \frac {{\left (B c^{2} d + A c d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a x}{16 \, b} - \frac {5 \, {\left (B c^{2} d + A c d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{64 \, b} - \frac {15 \, {\left (B c^{2} d + A c d^{2}\right )} \sqrt {b x^{2} + a} a^{3} x}{128 \, b} - \frac {15 \, {\left (B c^{2} d + A c d^{2}\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, B c d^{2} + A d^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {7}{2}} a}{63 \, b^{2}} \] Input:

integrate((B*x+A)*(d*x+c)^3*(b*x^2+a)^(5/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/10*(b*x^2 + a)^(7/2)*B*d^3*x^3/b + 1/6*(b*x^2 + a)^(5/2)*A*c^3*x + 5/24* 
(b*x^2 + a)^(3/2)*A*a*c^3*x + 5/16*sqrt(b*x^2 + a)*A*a^2*c^3*x - 3/80*(b*x 
^2 + a)^(7/2)*B*a*d^3*x/b^2 + 1/160*(b*x^2 + a)^(5/2)*B*a^2*d^3*x/b^2 + 1/ 
128*(b*x^2 + a)^(3/2)*B*a^3*d^3*x/b^2 + 3/256*sqrt(b*x^2 + a)*B*a^4*d^3*x/ 
b^2 + 5/16*A*a^3*c^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 3/256*B*a^5*d^3*arcs 
inh(b*x/sqrt(a*b))/b^(5/2) + 1/7*(b*x^2 + a)^(7/2)*B*c^3/b + 3/7*(b*x^2 + 
a)^(7/2)*A*c^2*d/b + 1/9*(3*B*c*d^2 + A*d^3)*(b*x^2 + a)^(7/2)*x^2/b + 3/8 
*(B*c^2*d + A*c*d^2)*(b*x^2 + a)^(7/2)*x/b - 1/16*(B*c^2*d + A*c*d^2)*(b*x 
^2 + a)^(5/2)*a*x/b - 5/64*(B*c^2*d + A*c*d^2)*(b*x^2 + a)^(3/2)*a^2*x/b - 
 15/128*(B*c^2*d + A*c*d^2)*sqrt(b*x^2 + a)*a^3*x/b - 15/128*(B*c^2*d + A* 
c*d^2)*a^4*arcsinh(b*x/sqrt(a*b))/b^(3/2) - 2/63*(3*B*c*d^2 + A*d^3)*(b*x^ 
2 + a)^(7/2)*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.45 \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\frac {1}{80640} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, B b^{2} d^{3} x + \frac {10 \, {\left (3 \, B b^{10} c d^{2} + A b^{10} d^{3}\right )}}{b^{8}}\right )} x + \frac {27 \, {\left (10 \, B b^{10} c^{2} d + 10 \, A b^{10} c d^{2} + 7 \, B a b^{9} d^{3}\right )}}{b^{8}}\right )} x + \frac {80 \, {\left (9 \, B b^{10} c^{3} + 27 \, A b^{10} c^{2} d + 57 \, B a b^{9} c d^{2} + 19 \, A a b^{9} d^{3}\right )}}{b^{8}}\right )} x + \frac {21 \, {\left (80 \, A b^{10} c^{3} + 510 \, B a b^{9} c^{2} d + 510 \, A a b^{9} c d^{2} + 93 \, B a^{2} b^{8} d^{3}\right )}}{b^{8}}\right )} x + \frac {480 \, {\left (9 \, B a b^{9} c^{3} + 27 \, A a b^{9} c^{2} d + 15 \, B a^{2} b^{8} c d^{2} + 5 \, A a^{2} b^{8} d^{3}\right )}}{b^{8}}\right )} x + \frac {105 \, {\left (208 \, A a b^{9} c^{3} + 354 \, B a^{2} b^{8} c^{2} d + 354 \, A a^{2} b^{8} c d^{2} + 3 \, B a^{3} b^{7} d^{3}\right )}}{b^{8}}\right )} x + \frac {640 \, {\left (27 \, B a^{2} b^{8} c^{3} + 81 \, A a^{2} b^{8} c^{2} d + 3 \, B a^{3} b^{7} c d^{2} + A a^{3} b^{7} d^{3}\right )}}{b^{8}}\right )} x + \frac {315 \, {\left (176 \, A a^{2} b^{8} c^{3} + 30 \, B a^{3} b^{7} c^{2} d + 30 \, A a^{3} b^{7} c d^{2} - 3 \, B a^{4} b^{6} d^{3}\right )}}{b^{8}}\right )} x + \frac {1280 \, {\left (9 \, B a^{3} b^{7} c^{3} + 27 \, A a^{3} b^{7} c^{2} d - 6 \, B a^{4} b^{6} c d^{2} - 2 \, A a^{4} b^{6} d^{3}\right )}}{b^{8}}\right )} - \frac {{\left (80 \, A a^{3} b^{2} c^{3} - 30 \, B a^{4} b c^{2} d - 30 \, A a^{4} b c d^{2} + 3 \, B a^{5} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} \] Input:

integrate((B*x+A)*(d*x+c)^3*(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

1/80640*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(9*B*b^2*d^3*x + 10*(3*B*b^10*c* 
d^2 + A*b^10*d^3)/b^8)*x + 27*(10*B*b^10*c^2*d + 10*A*b^10*c*d^2 + 7*B*a*b 
^9*d^3)/b^8)*x + 80*(9*B*b^10*c^3 + 27*A*b^10*c^2*d + 57*B*a*b^9*c*d^2 + 1 
9*A*a*b^9*d^3)/b^8)*x + 21*(80*A*b^10*c^3 + 510*B*a*b^9*c^2*d + 510*A*a*b^ 
9*c*d^2 + 93*B*a^2*b^8*d^3)/b^8)*x + 480*(9*B*a*b^9*c^3 + 27*A*a*b^9*c^2*d 
 + 15*B*a^2*b^8*c*d^2 + 5*A*a^2*b^8*d^3)/b^8)*x + 105*(208*A*a*b^9*c^3 + 3 
54*B*a^2*b^8*c^2*d + 354*A*a^2*b^8*c*d^2 + 3*B*a^3*b^7*d^3)/b^8)*x + 640*( 
27*B*a^2*b^8*c^3 + 81*A*a^2*b^8*c^2*d + 3*B*a^3*b^7*c*d^2 + A*a^3*b^7*d^3) 
/b^8)*x + 315*(176*A*a^2*b^8*c^3 + 30*B*a^3*b^7*c^2*d + 30*A*a^3*b^7*c*d^2 
 - 3*B*a^4*b^6*d^3)/b^8)*x + 1280*(9*B*a^3*b^7*c^3 + 27*A*a^3*b^7*c^2*d - 
6*B*a^4*b^6*c*d^2 - 2*A*a^4*b^6*d^3)/b^8) - 1/256*(80*A*a^3*b^2*c^3 - 30*B 
*a^4*b*c^2*d - 30*A*a^4*b*c*d^2 + 3*B*a^5*d^3)*log(abs(-sqrt(b)*x + sqrt(b 
*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\int {\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )\,{\left (c+d\,x\right )}^3 \,d x \] Input:

int((a + b*x^2)^(5/2)*(A + B*x)*(c + d*x)^3,x)
 

Output:

int((a + b*x^2)^(5/2)*(A + B*x)*(c + d*x)^3, x)
 

Reduce [F]

\[ \int (A+B x) (c+d x)^3 \left (a+b x^2\right )^{5/2} \, dx=\int \left (B x +A \right ) \left (d x +c \right )^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}d x \] Input:

int((B*x+A)*(d*x+c)^3*(b*x^2+a)^(5/2),x)
 

Output:

int((B*x+A)*(d*x+c)^3*(b*x^2+a)^(5/2),x)