\(\int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx\) [193]

Optimal result

Integrand size = 24, antiderivative size = 398 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=-\frac {5 (b d-a e)^5 (b B d-2 A b e+a B e) \sqrt {a+b x} \sqrt {d+e x}}{1024 b^4 e^4}+\frac {5 (b d-a e)^4 (b B d-2 A b e+a B e) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}-\frac {(b d-a e)^3 (b B d-2 A b e+a B e) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}-\frac {(b d-a e)^2 (b B d-2 A b e+a B e) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}-\frac {(b d-a e) (b B d-2 A b e+a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}-\frac {(b B d-2 A b e+a B e) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}+\frac {5 (b d-a e)^6 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{1024 b^{9/2} e^{9/2}} \] Output:

-5/1024*(-a*e+b*d)^5*(-2*A*b*e+B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^ 
4/e^4+5/1536*(-a*e+b*d)^4*(-2*A*b*e+B*a*e+B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/ 
2)/b^4/e^3-1/384*(-a*e+b*d)^3*(-2*A*b*e+B*a*e+B*b*d)*(b*x+a)^(5/2)*(e*x+d) 
^(1/2)/b^4/e^2-1/64*(-a*e+b*d)^2*(-2*A*b*e+B*a*e+B*b*d)*(b*x+a)^(7/2)*(e*x 
+d)^(1/2)/b^4/e-1/24*(-a*e+b*d)*(-2*A*b*e+B*a*e+B*b*d)*(b*x+a)^(7/2)*(e*x+ 
d)^(3/2)/b^3/e-1/12*(-2*A*b*e+B*a*e+B*b*d)*(b*x+a)^(7/2)*(e*x+d)^(5/2)/b^2 
/e+1/7*B*(b*x+a)^(7/2)*(e*x+d)^(7/2)/b/e+5/1024*(-a*e+b*d)^6*(-2*A*b*e+B*a 
*e+B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^( 
9/2)
 

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.29 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\frac {(b d-a e)^6 \left (-\frac {\sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {d+e x} \left (105 b B d e^6 (a+b x)^6-210 A b e^7 (a+b x)^6+105 a B e^7 (a+b x)^6-700 b^2 B d e^5 (a+b x)^5 (d+e x)+1400 A b^2 e^6 (a+b x)^5 (d+e x)-700 a b B e^6 (a+b x)^5 (d+e x)+1981 b^3 B d e^4 (a+b x)^4 (d+e x)^2-3962 A b^3 e^5 (a+b x)^4 (d+e x)^2+1981 a b^2 B e^5 (a+b x)^4 (d+e x)^2+3072 b^4 B d e^3 (a+b x)^3 (d+e x)^3-3072 a b^3 B e^4 (a+b x)^3 (d+e x)^3-1981 b^5 B d e^2 (a+b x)^2 (d+e x)^4+3962 A b^5 e^3 (a+b x)^2 (d+e x)^4-1981 a b^4 B e^3 (a+b x)^2 (d+e x)^4+700 b^6 B d e (a+b x) (d+e x)^5-1400 A b^6 e^2 (a+b x) (d+e x)^5+700 a b^5 B e^2 (a+b x) (d+e x)^5-105 b^7 B d (d+e x)^6+210 A b^7 e (d+e x)^6-105 a b^6 B e (d+e x)^6\right )}{(-b d+a e)^7}+105 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )\right )}{21504 b^{9/2} e^{9/2}} \] Input:

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

Output:

((b*d - a*e)^6*(-((Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sqrt[d + e*x]*(105*b*B*d* 
e^6*(a + b*x)^6 - 210*A*b*e^7*(a + b*x)^6 + 105*a*B*e^7*(a + b*x)^6 - 700* 
b^2*B*d*e^5*(a + b*x)^5*(d + e*x) + 1400*A*b^2*e^6*(a + b*x)^5*(d + e*x) - 
 700*a*b*B*e^6*(a + b*x)^5*(d + e*x) + 1981*b^3*B*d*e^4*(a + b*x)^4*(d + e 
*x)^2 - 3962*A*b^3*e^5*(a + b*x)^4*(d + e*x)^2 + 1981*a*b^2*B*e^5*(a + b*x 
)^4*(d + e*x)^2 + 3072*b^4*B*d*e^3*(a + b*x)^3*(d + e*x)^3 - 3072*a*b^3*B* 
e^4*(a + b*x)^3*(d + e*x)^3 - 1981*b^5*B*d*e^2*(a + b*x)^2*(d + e*x)^4 + 3 
962*A*b^5*e^3*(a + b*x)^2*(d + e*x)^4 - 1981*a*b^4*B*e^3*(a + b*x)^2*(d + 
e*x)^4 + 700*b^6*B*d*e*(a + b*x)*(d + e*x)^5 - 1400*A*b^6*e^2*(a + b*x)*(d 
 + e*x)^5 + 700*a*b^5*B*e^2*(a + b*x)*(d + e*x)^5 - 105*b^7*B*d*(d + e*x)^ 
6 + 210*A*b^7*e*(d + e*x)^6 - 105*a*b^6*B*e*(d + e*x)^6))/(-(b*d) + a*e)^7 
) + 105*(b*B*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e] 
*Sqrt[a + b*x])]))/(21504*b^(9/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.84, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {90, 60, 60, 60, 60, 60, 60, 66, 221}

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.

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \int (a+b x)^{5/2} (d+e x)^{3/2}dx}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \int (a+b x)^{5/2} \sqrt {d+e x}dx}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\)

Input:

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

Output:

(B*(a + b*x)^(7/2)*(d + e*x)^(7/2))/(7*b*e) + ((2*A*b*e - B*(b*d + a*e))*( 
((a + b*x)^(7/2)*(d + e*x)^(5/2))/(6*b) + (5*(b*d - a*e)*(((a + b*x)^(7/2) 
*(d + e*x)^(3/2))/(5*b) + (3*(b*d - a*e)*(((a + b*x)^(7/2)*Sqrt[d + e*x])/ 
(4*b) + ((b*d - a*e)*(((a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e) - (5*(b*d - a* 
e)*(((a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e) - (3*(b*d - a*e)*((Sqrt[a + b*x] 
*Sqrt[d + e*x])/e - ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]* 
Sqrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/(6*e)))/(8*b)))/(10*b)))/(12* 
b)))/(2*b*e)
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2395\) vs. \(2(342)=684\).

Time = 0.29 (sec) , antiderivative size = 2396, normalized size of antiderivative = 6.02

method	result	size
default	\(\text {Expression too large to display}\)	\(2396\)

Input:

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

Output:

-1/43008*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(210*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b* 
x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*a^6*b*e^7+210*A*ln(1/2*(2*b* 
e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2))*b^7*d^6*e- 
5544*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^2*b^4*d^3*e^3-19680*B*a^2*b^4 
*d^2*e^4*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-512*B*a*b^5*d^3*e^3*x^2*( 
(e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-37376*B*a*b^5*d*e^5*x^4*((e*x+d)*(b*x+a 
))^(1/2)*(b*e)^(1/2)-47488*A*a*b^5*d*e^5*x^3*((e*x+d)*(b*x+a))^(1/2)*(b*e) 
^(1/2)-25504*B*a^2*b^4*d*e^5*x^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-25504 
*B*a*b^5*d^2*e^4*x^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-35616*A*a^2*b^4*d 
*e^5*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-35616*A*a*b^5*d^2*e^4*x^2*((e 
*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-512*B*a^3*b^3*d*e^5*x^2*((e*x+d)*(b*x+a)) 
^(1/2)*(b*e)^(1/2)-1016*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^2*b^4*d^3* 
e^3*x+644*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a*b^5*d^4*e^2*x-1568*A*((e 
*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^3*b^3*d*e^5*x-33264*A*((e*x+d)*(b*x+a)) 
^(1/2)*(b*e)^(1/2)*a^2*b^4*d^2*e^4*x-1568*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^ 
(1/2)*a*b^5*d^3*e^3*x+644*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^4*b^2*d* 
e^5*x-1016*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^3*b^3*d^2*e^4*x-1260*A* 
ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+a*e+d*b)/(b*e)^(1/2) 
)*a^5*b^2*d*e^6+3150*A*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/ 
2)+a*e+d*b)/(b*e)^(1/2))*a^4*b^3*d^2*e^5-4200*A*ln(1/2*(2*b*e*x+2*((e*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (342) = 684\).

Time = 0.18 (sec) , antiderivative size = 1758, normalized size of antiderivative = 4.42 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/86016*(105*(B*b^7*d^7 - (5*B*a*b^6 + 2*A*b^7)*d^6*e + 3*(3*B*a^2*b^5 + 
 4*A*a*b^6)*d^5*e^2 - 5*(B*a^3*b^4 + 6*A*a^2*b^5)*d^4*e^3 - 5*(B*a^4*b^3 - 
 8*A*a^3*b^4)*d^3*e^4 + 3*(3*B*a^5*b^2 - 10*A*a^4*b^3)*d^2*e^5 - (5*B*a^6* 
b - 12*A*a^5*b^2)*d*e^6 + (B*a^7 - 2*A*a^6*b)*e^7)*sqrt(b*e)*log(8*b^2*e^2 
*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*s 
qrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(3072*B*b^7*e^7* 
x^6 - 105*B*b^7*d^6*e + 70*(7*B*a*b^6 + 3*A*b^7)*d^5*e^2 - 7*(113*B*a^2*b^ 
5 + 170*A*a*b^6)*d^4*e^3 + 12*(25*B*a^3*b^4 + 231*A*a^2*b^5)*d^3*e^4 - 7*( 
113*B*a^4*b^3 - 396*A*a^3*b^4)*d^2*e^5 + 70*(7*B*a^5*b^2 - 17*A*a^4*b^3)*d 
*e^6 - 105*(B*a^6*b - 2*A*a^5*b^2)*e^7 + 256*(29*B*b^7*d*e^6 + (29*B*a*b^6 
 + 14*A*b^7)*e^7)*x^5 + 128*(37*B*b^7*d^2*e^5 + 2*(73*B*a*b^6 + 35*A*b^7)* 
d*e^6 + (37*B*a^2*b^5 + 70*A*a*b^6)*e^7)*x^4 + 16*(3*B*b^7*d^3*e^4 + (797* 
B*a*b^6 + 378*A*b^7)*d^2*e^5 + (797*B*a^2*b^5 + 1484*A*a*b^6)*d*e^6 + 3*(B 
*a^3*b^4 + 126*A*a^2*b^5)*e^7)*x^3 - 8*(7*B*b^7*d^4*e^3 - 2*(16*B*a*b^6 + 
7*A*b^7)*d^3*e^4 - 6*(205*B*a^2*b^5 + 371*A*a*b^6)*d^2*e^5 - 2*(16*B*a^3*b 
^4 + 1113*A*a^2*b^5)*d*e^6 + 7*(B*a^4*b^3 - 2*A*a^3*b^4)*e^7)*x^2 + 2*(35* 
B*b^7*d^5*e^2 - 7*(23*B*a*b^6 + 10*A*b^7)*d^4*e^3 + 2*(127*B*a^2*b^5 + 196 
*A*a*b^6)*d^3*e^4 + 2*(127*B*a^3*b^4 + 4158*A*a^2*b^5)*d^2*e^5 - 7*(23*B*a 
^4*b^3 - 56*A*a^3*b^4)*d*e^6 + 35*(B*a^5*b^2 - 2*A*a^4*b^3)*e^7)*x)*sqrt(b 
*x + a)*sqrt(e*x + d))/(b^5*e^5), -1/43008*(105*(B*b^7*d^7 - (5*B*a*b^6...
 

Sympy [F]

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

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

Output:

Integral((A + B*x)*(a + b*x)**(5/2)*(d + e*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6983 vs. \(2 (342) = 684\).

Time = 0.95 (sec) , antiderivative size = 6983, normalized size of antiderivative = 17.55 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/107520*(1680*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + 
 a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13* 
d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5*b^14*d^3* 
e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))* 
sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 20*a^3* 
b*d*e^3 - 35*a^4*e^4)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x 
 + a)*b*e - a*b*e)))/(sqrt(b*e)*b^2*e^3))*B*a*d^2*abs(b) - 107520*((b^2*d 
- a*b*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a 
*b*e)))/sqrt(b*e) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*a 
^3*d^2*abs(b)/b^2 + 560*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)* 
(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - 
 (5*b^13*d^2*e^4 + 14*a*b^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5* 
b^14*d^3*e^3 + 9*a*b^13*d^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^ 
14*e^6))*sqrt(b*x + a) + 3*(5*b^4*d^4 + 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 
+ 20*a^3*b*d*e^3 - 35*a^4*e^4)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2 
*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*b^2*e^3))*A*b*d^2*abs(b) + 3360*( 
sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a) 
/b^3 + (b^12*d*e^5 - 25*a*b^11*e^6)/(b^14*e^6)) - (5*b^13*d^2*e^4 + 14*a*b 
^12*d*e^5 - 163*a^2*b^11*e^6)/(b^14*e^6)) + 3*(5*b^14*d^3*e^3 + 9*a*b^13*d 
^2*e^4 + 15*a^2*b^12*d*e^5 - 93*a^3*b^11*e^6)/(b^14*e^6))*sqrt(b*x + a)...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 1098, normalized size of antiderivative = 2.76 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(105*sqrt(d + e*x)*sqrt(a + b*x)*a**6*b*e**7 - 700*sqrt(d + e*x)*sqrt(a + 
b*x)*a**5*b**2*d*e**6 - 70*sqrt(d + e*x)*sqrt(a + b*x)*a**5*b**2*e**7*x + 
1981*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**3*d**2*e**5 + 462*sqrt(d + e*x)*s 
qrt(a + b*x)*a**4*b**3*d*e**6*x + 56*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**3 
*e**7*x**2 + 3072*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**4*d**3*e**4 + 17140* 
sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**4*d**2*e**5*x + 18064*sqrt(d + e*x)*sq 
rt(a + b*x)*a**3*b**4*d*e**6*x**2 + 6096*sqrt(d + e*x)*sqrt(a + b*x)*a**3* 
b**4*e**7*x**3 - 1981*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**5*d**4*e**3 + 12 
92*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**5*d**3*e**4*x + 27648*sqrt(d + e*x) 
*sqrt(a + b*x)*a**2*b**5*d**2*e**5*x**2 + 36496*sqrt(d + e*x)*sqrt(a + b*x 
)*a**2*b**5*d*e**6*x**3 + 13696*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**5*e**7 
*x**4 + 700*sqrt(d + e*x)*sqrt(a + b*x)*a*b**6*d**5*e**2 - 462*sqrt(d + e* 
x)*sqrt(a + b*x)*a*b**6*d**4*e**3*x + 368*sqrt(d + e*x)*sqrt(a + b*x)*a*b* 
*6*d**3*e**4*x**2 + 18800*sqrt(d + e*x)*sqrt(a + b*x)*a*b**6*d**2*e**5*x** 
3 + 27648*sqrt(d + e*x)*sqrt(a + b*x)*a*b**6*d*e**6*x**4 + 11008*sqrt(d + 
e*x)*sqrt(a + b*x)*a*b**6*e**7*x**5 - 105*sqrt(d + e*x)*sqrt(a + b*x)*b**7 
*d**6*e + 70*sqrt(d + e*x)*sqrt(a + b*x)*b**7*d**5*e**2*x - 56*sqrt(d + e* 
x)*sqrt(a + b*x)*b**7*d**4*e**3*x**2 + 48*sqrt(d + e*x)*sqrt(a + b*x)*b**7 
*d**3*e**4*x**3 + 4736*sqrt(d + e*x)*sqrt(a + b*x)*b**7*d**2*e**5*x**4 + 7 
424*sqrt(d + e*x)*sqrt(a + b*x)*b**7*d*e**6*x**5 + 3072*sqrt(d + e*x)*s...