\(\int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx\) [182]

Optimal result

Integrand size = 24, antiderivative size = 358 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx=\frac {(b d-a e)^4 (5 b B d-12 A b e+7 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^4 e^3}-\frac {(b d-a e)^3 (5 b B d-12 A b e+7 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^4 e^2}-\frac {(b d-a e)^2 (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{192 b^4 e}-\frac {(b d-a e) (5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{3/2}}{96 b^3 e}-\frac {(5 b B d-12 A b e+7 a B e) (a+b x)^{5/2} (d+e x)^{5/2}}{60 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{7/2}}{6 b e}-\frac {(b d-a e)^5 (5 b B d-12 A b e+7 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{9/2} e^{7/2}} \] Output:

1/512*(-a*e+b*d)^4*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2) 
/b^4/e^3-1/768*(-a*e+b*d)^3*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^(3/2)*(e*x 
+d)^(1/2)/b^4/e^2-1/192*(-a*e+b*d)^2*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^( 
5/2)*(e*x+d)^(1/2)/b^4/e-1/96*(-a*e+b*d)*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+ 
a)^(5/2)*(e*x+d)^(3/2)/b^3/e-1/60*(-12*A*b*e+7*B*a*e+5*B*b*d)*(b*x+a)^(5/2 
)*(e*x+d)^(5/2)/b^2/e+1/6*B*(b*x+a)^(5/2)*(e*x+d)^(7/2)/b/e-1/512*(-a*e+b* 
d)^5*(-12*A*b*e+7*B*a*e+5*B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e* 
x+d)^(1/2))/b^(9/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.21 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (-105 a^5 B e^5+5 a^4 b e^4 (83 B d+36 A e+14 B e x)-2 a^3 b^2 e^3 \left (60 A e (7 d+e x)+B \left (273 d^2+136 d e x+28 e^2 x^2\right )\right )+6 a^2 b^3 e^2 \left (4 A e \left (64 d^2+23 d e x+4 e^2 x^2\right )+B \left (25 d^3+58 d^2 e x+36 d e^2 x^2+8 e^3 x^3\right )\right )+a b^4 e \left (24 A e \left (35 d^3+233 d^2 e x+256 d e^2 x^2+88 e^3 x^3\right )+B \left (-245 d^4+160 d^3 e x+3384 d^2 e^2 x^2+4448 d e^3 x^3+1664 e^4 x^4\right )\right )+b^5 \left (12 A e \left (-15 d^4+10 d^3 e x+248 d^2 e^2 x^2+336 d e^3 x^3+128 e^4 x^4\right )+5 B \left (15 d^5-10 d^4 e x+8 d^3 e^2 x^2+432 d^2 e^3 x^3+640 d e^4 x^4+256 e^5 x^5\right )\right )\right )}{7680 b^4 e^3}+\frac {(b d-a e)^5 (-5 b B d+12 A b e-7 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{512 b^{9/2} e^{7/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-105*a^5*B*e^5 + 5*a^4*b*e^4*(83*B*d + 36*A* 
e + 14*B*e*x) - 2*a^3*b^2*e^3*(60*A*e*(7*d + e*x) + B*(273*d^2 + 136*d*e*x 
 + 28*e^2*x^2)) + 6*a^2*b^3*e^2*(4*A*e*(64*d^2 + 23*d*e*x + 4*e^2*x^2) + B 
*(25*d^3 + 58*d^2*e*x + 36*d*e^2*x^2 + 8*e^3*x^3)) + a*b^4*e*(24*A*e*(35*d 
^3 + 233*d^2*e*x + 256*d*e^2*x^2 + 88*e^3*x^3) + B*(-245*d^4 + 160*d^3*e*x 
 + 3384*d^2*e^2*x^2 + 4448*d*e^3*x^3 + 1664*e^4*x^4)) + b^5*(12*A*e*(-15*d 
^4 + 10*d^3*e*x + 248*d^2*e^2*x^2 + 336*d*e^3*x^3 + 128*e^4*x^4) + 5*B*(15 
*d^5 - 10*d^4*e*x + 8*d^3*e^2*x^2 + 432*d^2*e^3*x^3 + 640*d*e^4*x^4 + 256* 
e^5*x^5))))/(7680*b^4*e^3) + ((b*d - a*e)^5*(-5*b*B*d + 12*A*b*e - 7*a*B*e 
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])])/(512*b^(9/2)*e 
^(7/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.82, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {90, 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.

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(B*(a + b*x)^(5/2)*(d + e*x)^(7/2))/(6*b*e) - ((5*b*B*d - 12*A*b*e + 7*a*B 
*e)*(((a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b) + ((b*d - a*e)*(((a + b*x)^(5 
/2)*(d + e*x)^(3/2))/(4*b) + (3*(b*d - a*e)*(((a + b*x)^(5/2)*Sqrt[d + e*x 
])/(3*b) + ((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]*Sqr 
t[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/(6*b)))/ 
(8*b)))/(2*b)))/(12*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. \(1847\) vs. \(2(308)=616\).

Time = 0.27 (sec) , antiderivative size = 1848, normalized size of antiderivative = 5.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1848\)

Input:

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

Output:

-1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(240*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^ 
(1/2)*a^3*b^2*e^5*x-240*A*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*b^5*d^3*e^2* 
x-140*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^4*b*e^5*x+100*B*((e*x+d)*(b* 
x+a))^(1/2)*(b*e)^(1/2)*b^5*d^4*e*x-8896*B*a*b^4*d*e^4*x^3*((e*x+d)*(b*x+a 
))^(1/2)*(b*e)^(1/2)-12288*A*a*b^4*d*e^4*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e) 
^(1/2)-432*B*a^2*b^3*d*e^4*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-6768*B* 
a*b^4*d^2*e^3*x^2*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-1104*A*((e*x+d)*(b*x 
+a))^(1/2)*(b*e)^(1/2)*a^2*b^3*d*e^4*x-11184*A*((e*x+d)*(b*x+a))^(1/2)*(b* 
e)^(1/2)*a*b^4*d^2*e^3*x+544*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^3*b^2 
*d*e^4*x-696*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^2*b^3*d^2*e^3*x-320*B 
*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a*b^4*d^3*e^2*x+900*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*b^5*d^4*e^2 
+450*B*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*d*e^5-2560*B*b^5*e^5*x^5*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2 
)-3072*A*b^5*e^5*x^4*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)-4320*B*b^5*d^2*e^ 
3*x^3*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)+180*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*e^6-180*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^6*d^ 
5*e+210*B*((e*x+d)*(b*x+a))^(1/2)*(b*e)^(1/2)*a^5*e^5-150*B*((e*x+d)*(b*x+ 
a))^(1/2)*(b*e)^(1/2)*b^5*d^5+300*B*ln(1/2*(2*b*e*x+2*((e*x+d)*(b*x+a))...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (308) = 616\).

Time = 0.16 (sec) , antiderivative size = 1384, normalized size of antiderivative = 3.87 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/30720*(15*(5*B*b^6*d^6 - 6*(3*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(B*a^2*b^4 
+ 4*A*a*b^5)*d^4*e^2 + 20*(B*a^3*b^3 - 6*A*a^2*b^4)*d^3*e^3 - 15*(3*B*a^4* 
b^2 - 8*A*a^3*b^3)*d^2*e^4 + 30*(B*a^5*b - 2*A*a^4*b^2)*d*e^5 - (7*B*a^6 - 
 12*A*a^5*b)*e^6)*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)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b 
^2*d*e + a*b*e^2)*x) + 4*(1280*B*b^6*e^6*x^5 + 75*B*b^6*d^5*e - 5*(49*B*a* 
b^5 + 36*A*b^6)*d^4*e^2 + 30*(5*B*a^2*b^4 + 28*A*a*b^5)*d^3*e^3 - 6*(91*B* 
a^3*b^3 - 256*A*a^2*b^4)*d^2*e^4 + 5*(83*B*a^4*b^2 - 168*A*a^3*b^3)*d*e^5 
- 15*(7*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*(25*B*b^6*d*e^5 + (13*B*a*b^5 + 
12*A*b^6)*e^6)*x^4 + 16*(135*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 126*A*b^6)*d 
*e^5 + 3*(B*a^2*b^4 + 44*A*a*b^5)*e^6)*x^3 + 8*(5*B*b^6*d^3*e^3 + 3*(141*B 
*a*b^5 + 124*A*b^6)*d^2*e^4 + 3*(9*B*a^2*b^4 + 256*A*a*b^5)*d*e^5 - (7*B*a 
^3*b^3 - 12*A*a^2*b^4)*e^6)*x^2 - 2*(25*B*b^6*d^4*e^2 - 20*(4*B*a*b^5 + 3* 
A*b^6)*d^3*e^3 - 6*(29*B*a^2*b^4 + 466*A*a*b^5)*d^2*e^4 + 4*(34*B*a^3*b^3 
- 69*A*a^2*b^4)*d*e^5 - 5*(7*B*a^4*b^2 - 12*A*a^3*b^3)*e^6)*x)*sqrt(b*x + 
a)*sqrt(e*x + d))/(b^5*e^4), 1/15360*(15*(5*B*b^6*d^6 - 6*(3*B*a*b^5 + 2*A 
*b^6)*d^5*e + 15*(B*a^2*b^4 + 4*A*a*b^5)*d^4*e^2 + 20*(B*a^3*b^3 - 6*A*a^2 
*b^4)*d^3*e^3 - 15*(3*B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 30*(B*a^5*b - 2*A 
*a^4*b^2)*d*e^5 - (7*B*a^6 - 12*A*a^5*b)*e^6)*sqrt(-b*e)*arctan(1/2*(2*b*e 
*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a...
 

Sympy [F]

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

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

Output:

Integral((A + B*x)*(a + b*x)**(3/2)*(d + e*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(3/2)*(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. 4461 vs. \(2 (308) = 616\).

Time = 0.68 (sec) , antiderivative size = 4461, normalized size of antiderivative = 12.46 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/7680*(40*(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*d^2*abs(b) - 7680*((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^2*d^2*a 
bs(b)/b^2 + 80*(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*d*e*abs(b) + 160*(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*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.40 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*sqrt(d + e*x)*sqrt(a + b*x)*a**5*b*e**6 - 85*sqrt(d + e*x)*sqrt(a + b* 
x)*a**4*b**2*d*e**5 - 10*sqrt(d + e*x)*sqrt(a + b*x)*a**4*b**2*e**6*x + 19 
8*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**3*d**2*e**4 + 56*sqrt(d + e*x)*sqrt( 
a + b*x)*a**3*b**3*d*e**5*x + 8*sqrt(d + e*x)*sqrt(a + b*x)*a**3*b**3*e**6 
*x**2 + 198*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**4*d**3*e**3 + 1188*sqrt(d 
+ e*x)*sqrt(a + b*x)*a**2*b**4*d**2*e**4*x + 1272*sqrt(d + e*x)*sqrt(a + b 
*x)*a**2*b**4*d*e**5*x**2 + 432*sqrt(d + e*x)*sqrt(a + b*x)*a**2*b**4*e**6 
*x**3 - 85*sqrt(d + e*x)*sqrt(a + b*x)*a*b**5*d**4*e**2 + 56*sqrt(d + e*x) 
*sqrt(a + b*x)*a*b**5*d**3*e**3*x + 1272*sqrt(d + e*x)*sqrt(a + b*x)*a*b** 
5*d**2*e**4*x**2 + 1696*sqrt(d + e*x)*sqrt(a + b*x)*a*b**5*d*e**5*x**3 + 6 
40*sqrt(d + e*x)*sqrt(a + b*x)*a*b**5*e**6*x**4 + 15*sqrt(d + e*x)*sqrt(a 
+ b*x)*b**6*d**5*e - 10*sqrt(d + e*x)*sqrt(a + b*x)*b**6*d**4*e**2*x + 8*s 
qrt(d + e*x)*sqrt(a + b*x)*b**6*d**3*e**3*x**2 + 432*sqrt(d + e*x)*sqrt(a 
+ b*x)*b**6*d**2*e**4*x**3 + 640*sqrt(d + e*x)*sqrt(a + b*x)*b**6*d*e**5*x 
**4 + 256*sqrt(d + e*x)*sqrt(a + b*x)*b**6*e**6*x**5 - 15*sqrt(e)*sqrt(b)* 
log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a**6* 
e**6 + 90*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e* 
x))/sqrt(a*e - b*d))*a**5*b*d*e**5 - 225*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt 
(a + b*x) + sqrt(b)*sqrt(d + e*x))/sqrt(a*e - b*d))*a**4*b**2*d**2*e**4 + 
300*sqrt(e)*sqrt(b)*log((sqrt(e)*sqrt(a + b*x) + sqrt(b)*sqrt(d + e*x))...