\(\int (A+B x) (c+d x)^2 (a+b x^2)^{5/2} \, dx\) [163]

Optimal result

Integrand size = 24, antiderivative size = 257 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx=\frac {5 a^2 \left (8 A b c^2-2 a B c d-a A d^2\right ) x \sqrt {a+b x^2}}{128 b}+\frac {5 a \left (8 A b c^2-2 a B c d-a A d^2\right ) x \left (a+b x^2\right )^{3/2}}{192 b}+\frac {\left (8 A b c^2-2 a B c d-a A d^2\right ) x \left (a+b x^2\right )^{5/2}}{48 b}+\frac {B (c+d x)^2 \left (a+b x^2\right )^{7/2}}{9 b}-\frac {\left (16 \left (a B d^2-b c (B c+9 A d)\right )-7 b d (2 B c+9 A d) x\right ) \left (a+b x^2\right )^{7/2}}{504 b^2}+\frac {5 a^3 \left (8 A b c^2-2 a B c d-a A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \] Output:

5/128*a^2*(-A*a*d^2+8*A*b*c^2-2*B*a*c*d)*x*(b*x^2+a)^(1/2)/b+5/192*a*(-A*a 
*d^2+8*A*b*c^2-2*B*a*c*d)*x*(b*x^2+a)^(3/2)/b+1/48*(-A*a*d^2+8*A*b*c^2-2*B 
*a*c*d)*x*(b*x^2+a)^(5/2)/b+1/9*B*(d*x+c)^2*(b*x^2+a)^(7/2)/b-1/504*(16*a* 
B*d^2-16*b*c*(9*A*d+B*c)-7*b*d*(9*A*d+2*B*c)*x)*(b*x^2+a)^(7/2)/b^2+5/128* 
a^3*(-A*a*d^2+8*A*b*c^2-2*B*a*c*d)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(3 
/2)
 

Mathematica [A] (verified)

Time = 2.46 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.10 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx=\frac {\sqrt {a+b x^2} \left (-256 a^4 B d^2+16 b^4 x^5 \left (3 A \left (28 c^2+48 c d x+21 d^2 x^2\right )+2 B x \left (36 c^2+63 c d x+28 d^2 x^2\right )\right )+a^3 b \left (9 A d (256 c+35 d x)+2 B \left (576 c^2+315 c d x+64 d^2 x^2\right )\right )+8 a b^3 x^3 \left (2 B x \left (216 c^2+357 c d x+152 d^2 x^2\right )+A \left (546 c^2+864 c d x+357 d^2 x^2\right )\right )+6 a^2 b^2 x \left (2 B x \left (288 c^2+413 c d x+160 d^2 x^2\right )+A \left (924 c^2+1152 c d x+413 d^2 x^2\right )\right )\right )+315 a^3 \sqrt {b} \left (-8 A b c^2+2 a B c d+a A d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8064 b^2} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(-256*a^4*B*d^2 + 16*b^4*x^5*(3*A*(28*c^2 + 48*c*d*x + 21 
*d^2*x^2) + 2*B*x*(36*c^2 + 63*c*d*x + 28*d^2*x^2)) + a^3*b*(9*A*d*(256*c 
+ 35*d*x) + 2*B*(576*c^2 + 315*c*d*x + 64*d^2*x^2)) + 8*a*b^3*x^3*(2*B*x*( 
216*c^2 + 357*c*d*x + 152*d^2*x^2) + A*(546*c^2 + 864*c*d*x + 357*d^2*x^2) 
) + 6*a^2*b^2*x*(2*B*x*(288*c^2 + 413*c*d*x + 160*d^2*x^2) + A*(924*c^2 + 
1152*c*d*x + 413*d^2*x^2))) + 315*a^3*Sqrt[b]*(-8*A*b*c^2 + 2*a*B*c*d + a* 
A*d^2)*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(8064*b^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {687, 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.

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {\int (c+d x) (9 A b c-2 a B d+b (2 B c+9 A d) x) \left (b x^2+a\right )^{5/2}dx}{9 b}+\frac {B \left (a+b x^2\right )^{7/2} (c+d x)^2}{9 b}\)

\(\Big \downarrow \) 676

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(B*(c + d*x)^2*(a + b*x^2)^(7/2))/(9*b) + ((-2*(a*B*d^2 - b*c*(B*c + 9*A*d 
))*(a + b*x^2)^(7/2))/(7*b) + (d*(2*B*c + 9*A*d)*x*(a + b*x^2)^(7/2))/8 + 
(9*(8*A*b*c^2 - 2*a*B*c*d - a*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)/(9*b)
 

Defintions of rubi rules used

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 = 1.11 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.91

method	result	size
default	\(A \,c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+d \left (A d +2 B c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )+\frac {c \left (2 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+B \,d^{2} \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )\)	\(234\)
risch	\(\frac {\left (896 B \,b^{4} d^{2} x^{8}+1008 A \,b^{4} d^{2} x^{7}+2016 B \,b^{4} c d \,x^{7}+2304 A \,b^{4} c d \,x^{6}+2432 B a \,b^{3} d^{2} x^{6}+1152 B \,b^{4} c^{2} x^{6}+2856 A a \,b^{3} d^{2} x^{5}+1344 A \,b^{4} c^{2} x^{5}+5712 B a \,b^{3} c d \,x^{5}+6912 A a \,b^{3} c d \,x^{4}+1920 B \,a^{2} b^{2} d^{2} x^{4}+3456 B a \,b^{3} c^{2} x^{4}+2478 A \,a^{2} b^{2} d^{2} x^{3}+4368 A a \,b^{3} c^{2} x^{3}+4956 B \,a^{2} b^{2} c d \,x^{3}+6912 A \,a^{2} b^{2} c d \,x^{2}+128 B \,a^{3} b \,d^{2} x^{2}+3456 B \,a^{2} b^{2} c^{2} x^{2}+315 A \,a^{3} b \,d^{2} x +5544 A \,a^{2} b^{2} c^{2} x +630 B \,a^{3} b c d x +2304 A \,a^{3} b c d -256 B \,a^{4} d^{2}+1152 B \,a^{3} b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{8064 b^{2}}-\frac {5 a^{3} \left (A a \,d^{2}-8 A b \,c^{2}+2 B a c d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\)	\(355\)

Input:

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

Output:

A*c^2*(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*(A*d+2*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*A*d+B*c)*(b*x^2+a)^(7/2)/b+B*d^2*(1/9*x^2*(b*x^2+a) 
^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.74 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx=\left [-\frac {315 \, {\left (8 \, A a^{3} b c^{2} - 2 \, B a^{4} c d - A a^{4} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (896 \, B b^{4} d^{2} x^{8} + 1008 \, {\left (2 \, B b^{4} c d + A b^{4} d^{2}\right )} x^{7} + 1152 \, B a^{3} b c^{2} + 2304 \, A a^{3} b c d - 256 \, B a^{4} d^{2} + 128 \, {\left (9 \, B b^{4} c^{2} + 18 \, A b^{4} c d + 19 \, B a b^{3} d^{2}\right )} x^{6} + 168 \, {\left (8 \, A b^{4} c^{2} + 34 \, B a b^{3} c d + 17 \, A a b^{3} d^{2}\right )} x^{5} + 384 \, {\left (9 \, B a b^{3} c^{2} + 18 \, A a b^{3} c d + 5 \, B a^{2} b^{2} d^{2}\right )} x^{4} + 42 \, {\left (104 \, A a b^{3} c^{2} + 118 \, B a^{2} b^{2} c d + 59 \, A a^{2} b^{2} d^{2}\right )} x^{3} + 128 \, {\left (27 \, B a^{2} b^{2} c^{2} + 54 \, A a^{2} b^{2} c d + B a^{3} b d^{2}\right )} x^{2} + 63 \, {\left (88 \, A a^{2} b^{2} c^{2} + 10 \, B a^{3} b c d + 5 \, A a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16128 \, b^{2}}, -\frac {315 \, {\left (8 \, A a^{3} b c^{2} - 2 \, B a^{4} c d - A a^{4} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (896 \, B b^{4} d^{2} x^{8} + 1008 \, {\left (2 \, B b^{4} c d + A b^{4} d^{2}\right )} x^{7} + 1152 \, B a^{3} b c^{2} + 2304 \, A a^{3} b c d - 256 \, B a^{4} d^{2} + 128 \, {\left (9 \, B b^{4} c^{2} + 18 \, A b^{4} c d + 19 \, B a b^{3} d^{2}\right )} x^{6} + 168 \, {\left (8 \, A b^{4} c^{2} + 34 \, B a b^{3} c d + 17 \, A a b^{3} d^{2}\right )} x^{5} + 384 \, {\left (9 \, B a b^{3} c^{2} + 18 \, A a b^{3} c d + 5 \, B a^{2} b^{2} d^{2}\right )} x^{4} + 42 \, {\left (104 \, A a b^{3} c^{2} + 118 \, B a^{2} b^{2} c d + 59 \, A a^{2} b^{2} d^{2}\right )} x^{3} + 128 \, {\left (27 \, B a^{2} b^{2} c^{2} + 54 \, A a^{2} b^{2} c d + B a^{3} b d^{2}\right )} x^{2} + 63 \, {\left (88 \, A a^{2} b^{2} c^{2} + 10 \, B a^{3} b c d + 5 \, A a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8064 \, b^{2}}\right ] \] Input:

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

Output:

[-1/16128*(315*(8*A*a^3*b*c^2 - 2*B*a^4*c*d - A*a^4*d^2)*sqrt(b)*log(-2*b* 
x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(896*B*b^4*d^2*x^8 + 1008*(2*B* 
b^4*c*d + A*b^4*d^2)*x^7 + 1152*B*a^3*b*c^2 + 2304*A*a^3*b*c*d - 256*B*a^4 
*d^2 + 128*(9*B*b^4*c^2 + 18*A*b^4*c*d + 19*B*a*b^3*d^2)*x^6 + 168*(8*A*b^ 
4*c^2 + 34*B*a*b^3*c*d + 17*A*a*b^3*d^2)*x^5 + 384*(9*B*a*b^3*c^2 + 18*A*a 
*b^3*c*d + 5*B*a^2*b^2*d^2)*x^4 + 42*(104*A*a*b^3*c^2 + 118*B*a^2*b^2*c*d 
+ 59*A*a^2*b^2*d^2)*x^3 + 128*(27*B*a^2*b^2*c^2 + 54*A*a^2*b^2*c*d + B*a^3 
*b*d^2)*x^2 + 63*(88*A*a^2*b^2*c^2 + 10*B*a^3*b*c*d + 5*A*a^3*b*d^2)*x)*sq 
rt(b*x^2 + a))/b^2, -1/8064*(315*(8*A*a^3*b*c^2 - 2*B*a^4*c*d - A*a^4*d^2) 
*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (896*B*b^4*d^2*x^8 + 1008*( 
2*B*b^4*c*d + A*b^4*d^2)*x^7 + 1152*B*a^3*b*c^2 + 2304*A*a^3*b*c*d - 256*B 
*a^4*d^2 + 128*(9*B*b^4*c^2 + 18*A*b^4*c*d + 19*B*a*b^3*d^2)*x^6 + 168*(8* 
A*b^4*c^2 + 34*B*a*b^3*c*d + 17*A*a*b^3*d^2)*x^5 + 384*(9*B*a*b^3*c^2 + 18 
*A*a*b^3*c*d + 5*B*a^2*b^2*d^2)*x^4 + 42*(104*A*a*b^3*c^2 + 118*B*a^2*b^2* 
c*d + 59*A*a^2*b^2*d^2)*x^3 + 128*(27*B*a^2*b^2*c^2 + 54*A*a^2*b^2*c*d + B 
*a^3*b*d^2)*x^2 + 63*(88*A*a^2*b^2*c^2 + 10*B*a^3*b*c*d + 5*A*a^3*b*d^2)*x 
)*sqrt(b*x^2 + a))/b^2]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (257) = 514\).

Time = 0.77 (sec) , antiderivative size = 994, normalized size of antiderivative = 3.87 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(B*b**2*d**2*x**8/9 + x**7*(A*b**3*d**2 + 2*B* 
b**3*c*d)/(8*b) + x**6*(2*A*b**3*c*d + 19*B*a*b**2*d**2/9 + B*b**3*c**2)/( 
7*b) + x**5*(3*A*a*b**2*d**2 + A*b**3*c**2 + 6*B*a*b**2*c*d - 7*a*(A*b**3* 
d**2 + 2*B*b**3*c*d)/(8*b))/(6*b) + x**4*(6*A*a*b**2*c*d + 3*B*a**2*b*d**2 
 + 3*B*a*b**2*c**2 - 6*a*(2*A*b**3*c*d + 19*B*a*b**2*d**2/9 + B*b**3*c**2) 
/(7*b))/(5*b) + x**3*(3*A*a**2*b*d**2 + 3*A*a*b**2*c**2 + 6*B*a**2*b*c*d - 
 5*a*(3*A*a*b**2*d**2 + A*b**3*c**2 + 6*B*a*b**2*c*d - 7*a*(A*b**3*d**2 + 
2*B*b**3*c*d)/(8*b))/(6*b))/(4*b) + x**2*(6*A*a**2*b*c*d + B*a**3*d**2 + 3 
*B*a**2*b*c**2 - 4*a*(6*A*a*b**2*c*d + 3*B*a**2*b*d**2 + 3*B*a*b**2*c**2 - 
 6*a*(2*A*b**3*c*d + 19*B*a*b**2*d**2/9 + B*b**3*c**2)/(7*b))/(5*b))/(3*b) 
 + x*(A*a**3*d**2 + 3*A*a**2*b*c**2 + 2*B*a**3*c*d - 3*a*(3*A*a**2*b*d**2 
+ 3*A*a*b**2*c**2 + 6*B*a**2*b*c*d - 5*a*(3*A*a*b**2*d**2 + A*b**3*c**2 + 
6*B*a*b**2*c*d - 7*a*(A*b**3*d**2 + 2*B*b**3*c*d)/(8*b))/(6*b))/(4*b))/(2* 
b) + (2*A*a**3*c*d + B*a**3*c**2 - 2*a*(6*A*a**2*b*c*d + B*a**3*d**2 + 3*B 
*a**2*b*c**2 - 4*a*(6*A*a*b**2*c*d + 3*B*a**2*b*d**2 + 3*B*a*b**2*c**2 - 6 
*a*(2*A*b**3*c*d + 19*B*a*b**2*d**2/9 + B*b**3*c**2)/(7*b))/(5*b))/(3*b))/ 
b) + (A*a**3*c**2 - a*(A*a**3*d**2 + 3*A*a**2*b*c**2 + 2*B*a**3*c*d - 3*a* 
(3*A*a**2*b*d**2 + 3*A*a*b**2*c**2 + 6*B*a**2*b*c*d - 5*a*(3*A*a*b**2*d**2 
 + A*b**3*c**2 + 6*B*a*b**2*c*d - 7*a*(A*b**3*d**2 + 2*B*b**3*c*d)/(8*b))/ 
(6*b))/(4*b))/(2*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.12 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B d^{2} x^{2}}{9 \, b} + \frac {1}{6} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A c^{2} x + \frac {5}{24} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a c^{2} x + \frac {5}{16} \, \sqrt {b x^{2} + a} A a^{2} c^{2} x + \frac {5 \, A a^{3} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, \sqrt {b}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B c^{2}}{7 \, b} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A c d}{7 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a d^{2}}{63 \, b^{2}} + \frac {{\left (2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {7}{2}} x}{8 \, b} - \frac {{\left (2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a x}{48 \, b} - \frac {5 \, {\left (2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{192 \, b} - \frac {5 \, {\left (2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a} a^{3} x}{128 \, b} - \frac {5 \, {\left (2 \, B c d + A d^{2}\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} \] Input:

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

Output:

1/9*(b*x^2 + a)^(7/2)*B*d^2*x^2/b + 1/6*(b*x^2 + a)^(5/2)*A*c^2*x + 5/24*( 
b*x^2 + a)^(3/2)*A*a*c^2*x + 5/16*sqrt(b*x^2 + a)*A*a^2*c^2*x + 5/16*A*a^3 
*c^2*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 1/7*(b*x^2 + a)^(7/2)*B*c^2/b + 2/7* 
(b*x^2 + a)^(7/2)*A*c*d/b - 2/63*(b*x^2 + a)^(7/2)*B*a*d^2/b^2 + 1/8*(2*B* 
c*d + A*d^2)*(b*x^2 + a)^(7/2)*x/b - 1/48*(2*B*c*d + A*d^2)*(b*x^2 + a)^(5 
/2)*a*x/b - 5/192*(2*B*c*d + A*d^2)*(b*x^2 + a)^(3/2)*a^2*x/b - 5/128*(2*B 
*c*d + A*d^2)*sqrt(b*x^2 + a)*a^3*x/b - 5/128*(2*B*c*d + A*d^2)*a^4*arcsin 
h(b*x/sqrt(a*b))/b^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.50 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx=\frac {1}{8064} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, B b^{2} d^{2} x + \frac {9 \, {\left (2 \, B b^{9} c d + A b^{9} d^{2}\right )}}{b^{7}}\right )} x + \frac {8 \, {\left (9 \, B b^{9} c^{2} + 18 \, A b^{9} c d + 19 \, B a b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {21 \, {\left (8 \, A b^{9} c^{2} + 34 \, B a b^{8} c d + 17 \, A a b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {48 \, {\left (9 \, B a b^{8} c^{2} + 18 \, A a b^{8} c d + 5 \, B a^{2} b^{7} d^{2}\right )}}{b^{7}}\right )} x + \frac {21 \, {\left (104 \, A a b^{8} c^{2} + 118 \, B a^{2} b^{7} c d + 59 \, A a^{2} b^{7} d^{2}\right )}}{b^{7}}\right )} x + \frac {64 \, {\left (27 \, B a^{2} b^{7} c^{2} + 54 \, A a^{2} b^{7} c d + B a^{3} b^{6} d^{2}\right )}}{b^{7}}\right )} x + \frac {63 \, {\left (88 \, A a^{2} b^{7} c^{2} + 10 \, B a^{3} b^{6} c d + 5 \, A a^{3} b^{6} d^{2}\right )}}{b^{7}}\right )} x + \frac {128 \, {\left (9 \, B a^{3} b^{6} c^{2} + 18 \, A a^{3} b^{6} c d - 2 \, B a^{4} b^{5} d^{2}\right )}}{b^{7}}\right )} - \frac {5 \, {\left (8 \, A a^{3} b c^{2} - 2 \, B a^{4} c d - A a^{4} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} \] Input:

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

Output:

1/8064*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*B*b^2*d^2*x + 9*(2*B*b^9*c*d + A* 
b^9*d^2)/b^7)*x + 8*(9*B*b^9*c^2 + 18*A*b^9*c*d + 19*B*a*b^8*d^2)/b^7)*x + 
 21*(8*A*b^9*c^2 + 34*B*a*b^8*c*d + 17*A*a*b^8*d^2)/b^7)*x + 48*(9*B*a*b^8 
*c^2 + 18*A*a*b^8*c*d + 5*B*a^2*b^7*d^2)/b^7)*x + 21*(104*A*a*b^8*c^2 + 11 
8*B*a^2*b^7*c*d + 59*A*a^2*b^7*d^2)/b^7)*x + 64*(27*B*a^2*b^7*c^2 + 54*A*a 
^2*b^7*c*d + B*a^3*b^6*d^2)/b^7)*x + 63*(88*A*a^2*b^7*c^2 + 10*B*a^3*b^6*c 
*d + 5*A*a^3*b^6*d^2)/b^7)*x + 128*(9*B*a^3*b^6*c^2 + 18*A*a^3*b^6*c*d - 2 
*B*a^4*b^5*d^2)/b^7) - 5/128*(8*A*a^3*b*c^2 - 2*B*a^4*c*d - A*a^4*d^2)*log 
(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 5.02 (sec) , antiderivative size = 572, normalized size of antiderivative = 2.23 \[ \int (A+B x) (c+d x)^2 \left (a+b x^2\right )^{5/2} \, dx=\frac {315 \sqrt {b \,x^{2}+a}\, a^{4} b \,d^{2} x +5544 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c^{2} x +2478 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d^{2} x^{3}+128 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} d^{2} x^{2}+4368 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{2} x^{3}+3456 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{2} x^{2}+2856 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d^{2} x^{5}+1920 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} d^{2} x^{4}+1344 \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{2} x^{5}+3456 \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{2} x^{4}+1008 \sqrt {b \,x^{2}+a}\, a \,b^{4} d^{2} x^{7}+2432 \sqrt {b \,x^{2}+a}\, a \,b^{4} d^{2} x^{6}+2016 \sqrt {b \,x^{2}+a}\, b^{5} c d \,x^{7}+2520 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,c^{2}+2304 \sqrt {b \,x^{2}+a}\, a^{4} b c d +6912 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c d \,x^{2}+630 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c d x +6912 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c d \,x^{4}+4956 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c d \,x^{3}+2304 \sqrt {b \,x^{2}+a}\, a \,b^{4} c d \,x^{6}+5712 \sqrt {b \,x^{2}+a}\, a \,b^{4} c d \,x^{5}-630 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b c d -256 \sqrt {b \,x^{2}+a}\, a^{4} b \,d^{2}+1152 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c^{2}+1152 \sqrt {b \,x^{2}+a}\, b^{5} c^{2} x^{6}+896 \sqrt {b \,x^{2}+a}\, b^{5} d^{2} x^{8}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} d^{2}}{8064 b^{2}} \] Input:

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

Output:

(2304*sqrt(a + b*x**2)*a**4*b*c*d + 315*sqrt(a + b*x**2)*a**4*b*d**2*x - 2 
56*sqrt(a + b*x**2)*a**4*b*d**2 + 5544*sqrt(a + b*x**2)*a**3*b**2*c**2*x + 
 1152*sqrt(a + b*x**2)*a**3*b**2*c**2 + 6912*sqrt(a + b*x**2)*a**3*b**2*c* 
d*x**2 + 630*sqrt(a + b*x**2)*a**3*b**2*c*d*x + 2478*sqrt(a + b*x**2)*a**3 
*b**2*d**2*x**3 + 128*sqrt(a + b*x**2)*a**3*b**2*d**2*x**2 + 4368*sqrt(a + 
 b*x**2)*a**2*b**3*c**2*x**3 + 3456*sqrt(a + b*x**2)*a**2*b**3*c**2*x**2 + 
 6912*sqrt(a + b*x**2)*a**2*b**3*c*d*x**4 + 4956*sqrt(a + b*x**2)*a**2*b** 
3*c*d*x**3 + 2856*sqrt(a + b*x**2)*a**2*b**3*d**2*x**5 + 1920*sqrt(a + b*x 
**2)*a**2*b**3*d**2*x**4 + 1344*sqrt(a + b*x**2)*a*b**4*c**2*x**5 + 3456*s 
qrt(a + b*x**2)*a*b**4*c**2*x**4 + 2304*sqrt(a + b*x**2)*a*b**4*c*d*x**6 + 
 5712*sqrt(a + b*x**2)*a*b**4*c*d*x**5 + 1008*sqrt(a + b*x**2)*a*b**4*d**2 
*x**7 + 2432*sqrt(a + b*x**2)*a*b**4*d**2*x**6 + 1152*sqrt(a + b*x**2)*b** 
5*c**2*x**6 + 2016*sqrt(a + b*x**2)*b**5*c*d*x**7 + 896*sqrt(a + b*x**2)*b 
**5*d**2*x**8 - 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a* 
*5*d**2 + 2520*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b* 
c**2 - 630*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*c*d) 
/(8064*b**2)