Integrand size = 24, antiderivative size = 358 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=-\frac {(b d-a e)^4 (7 b B d-12 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{512 b^3 e^4}+\frac {(b d-a e)^3 (7 b B d-12 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{768 b^3 e^3}-\frac {(b d-a e)^2 (7 b B d-12 A b e+5 a B e) (a+b x)^{5/2} \sqrt {d+e x}}{960 b^3 e^2}-\frac {(b d-a e) (7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} \sqrt {d+e x}}{160 b^3 e}-\frac {(7 b B d-12 A b e+5 a B e) (a+b x)^{7/2} (d+e x)^{3/2}}{60 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}+\frac {(b d-a e)^5 (7 b B d-12 A b e+5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{7/2} e^{9/2}} \]
-1/60*(-12*A*b*e+5*B*a*e+7*B*b*d)*(b*x+a)^(7/2)*(e*x+d)^(3/2)/b^2/e+1/6*B* (b*x+a)^(7/2)*(e*x+d)^(5/2)/b/e+1/512*(-a*e+b*d)^5*(-12*A*b*e+5*B*a*e+7*B* b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(7/2)/e^(9/2)+ 1/768*(-a*e+b*d)^3*(-12*A*b*e+5*B*a*e+7*B*b*d)*(b*x+a)^(3/2)*(e*x+d)^(1/2) /b^3/e^3-1/960*(-a*e+b*d)^2*(-12*A*b*e+5*B*a*e+7*B*b*d)*(b*x+a)^(5/2)*(e*x +d)^(1/2)/b^3/e^2-1/160*(-a*e+b*d)*(-12*A*b*e+5*B*a*e+7*B*b*d)*(b*x+a)^(7/ 2)*(e*x+d)^(1/2)/b^3/e-1/512*(-a*e+b*d)^4*(-12*A*b*e+5*B*a*e+7*B*b*d)*(b*x +a)^(1/2)*(e*x+d)^(1/2)/b^3/e^4
Time = 0.99 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.20 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (75 a^5 B e^5-5 a^4 b e^4 (49 B d+36 A e+10 B e x)+10 a^3 b^2 e^3 \left (12 A e (7 d+e x)+B \left (15 d^2+16 d e x+4 e^2 x^2\right )\right )+6 a^2 b^3 e^2 \left (4 A e \left (64 d^2+233 d e x+124 e^2 x^2\right )+B \left (-91 d^3+58 d^2 e x+564 d e^2 x^2+360 e^3 x^3\right )\right )+a b^4 e \left (24 A e \left (-35 d^3+23 d^2 e x+256 d e^2 x^2+168 e^3 x^3\right )+B \left (415 d^4-272 d^3 e x+216 d^2 e^2 x^2+4448 d e^3 x^3+3200 e^4 x^4\right )\right )+b^5 \left (12 A e \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )+B \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )\right )}{7680 b^3 e^4}+\frac {(b d-a e)^5 (7 b B d-12 A b e+5 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{512 b^{7/2} e^{9/2}} \]
(Sqrt[a + b*x]*Sqrt[d + e*x]*(75*a^5*B*e^5 - 5*a^4*b*e^4*(49*B*d + 36*A*e + 10*B*e*x) + 10*a^3*b^2*e^3*(12*A*e*(7*d + e*x) + B*(15*d^2 + 16*d*e*x + 4*e^2*x^2)) + 6*a^2*b^3*e^2*(4*A*e*(64*d^2 + 233*d*e*x + 124*e^2*x^2) + B* (-91*d^3 + 58*d^2*e*x + 564*d*e^2*x^2 + 360*e^3*x^3)) + a*b^4*e*(24*A*e*(- 35*d^3 + 23*d^2*e*x + 256*d*e^2*x^2 + 168*e^3*x^3) + B*(415*d^4 - 272*d^3* e*x + 216*d^2*e^2*x^2 + 4448*d*e^3*x^3 + 3200*e^4*x^4)) + b^5*(12*A*e*(15* d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 176*d*e^3*x^3 + 128*e^4*x^4) + B*(-105* d^5 + 70*d^4*e*x - 56*d^3*e^2*x^2 + 48*d^2*e^3*x^3 + 1664*d*e^4*x^4 + 1280 *e^5*x^5))))/(7680*b^3*e^4) + ((b*d - a*e)^5*(7*b*B*d - 12*A*b*e + 5*a*B*e )*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(512*b^(7/2)*e ^(9/2))
Time = 0.32 (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.
| \(\displaystyle \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \int (a+b x)^{5/2} (d+e x)^{3/2}dx}{12 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \int (a+b x)^{5/2} \sqrt {d+e x}dx}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {(a+b x)^{5/2}}{\sqrt {d+e x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {B (a+b x)^{7/2} (d+e x)^{5/2}}{6 b e}-\frac {(5 a B e-12 A b e+7 b B d) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b e}\) |
(B*(a + b*x)^(7/2)*(d + e*x)^(5/2))/(6*b*e) - ((7*b*B*d - 12*A*b*e + 5*a*B *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]*Sqr t[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/(4*e)))/(6*e)))/ (8*b)))/(10*b)))/(12*b*e)
3.23.24.3.1 Defintions of rubi rules used
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1847\) vs. \(2(308)=616\).
Time = 1.09 (sec) , antiderivative size = 1848, normalized size of antiderivative = 5.16
1/15360*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(11184*A*((b*x+a)*(e*x+d))^(1/2)*(b*e) ^(1/2)*a^2*b^3*d*e^4*x+1104*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^4*d^ 2*e^3*x+320*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b^2*d*e^4*x+696*B*(( b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b^3*d^2*e^3*x-544*B*((b*x+a)*(e*x+d) )^(1/2)*(b*e)^(1/2)*a*b^4*d^3*e^2*x+8064*A*a*b^4*e^5*x^3*((b*x+a)*(e*x+d)) ^(1/2)*(b*e)^(1/2)+4224*A*b^5*d*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2 )+4320*B*a^2*b^3*e^5*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+96*B*b^5*d^2* e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+5952*A*a^2*b^3*e^5*x^2*((b*x+a )*(e*x+d))^(1/2)*(b*e)^(1/2)+192*A*b^5*d^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2) *(b*e)^(1/2)+80*B*a^3*b^2*e^5*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-112* B*b^5*d^3*e^2*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+3072*A*a^2*b^3*d^2*e ^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+2560*B*b^5*e^5*x^5*((b*x+a)*(e*x+d) )^(1/2)*(b*e)^(1/2)+3072*A*b^5*e^5*x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2) -900*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e )^(1/2))*a^4*b^2*d*e^5+105*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b* e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^6*d^6-75*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e* x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^6*e^6+8896*B*a*b^4*d*e^4*x ^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12288*A*a*b^4*d*e^4*x^2*((b*x+a)*(e *x+d))^(1/2)*(b*e)^(1/2)+6768*B*a^2*b^3*d*e^4*x^2*((b*x+a)*(e*x+d))^(1/2)* (b*e)^(1/2)+432*B*a*b^4*d^2*e^3*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)...
Leaf count of result is larger than twice the leaf count of optimal. 687 vs. \(2 (308) = 616\).
Time = 0.31 (sec) , antiderivative size = 1388, normalized size of antiderivative = 3.88 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \]
[1/30720*(15*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*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*(B*a^4* b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*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 - 105*B*b^6*d^5*e + 5*(83*B *a*b^5 + 36*A*b^6)*d^4*e^2 - 42*(13*B*a^2*b^4 + 20*A*a*b^5)*d^3*e^3 + 6*(2 5*B*a^3*b^3 + 256*A*a^2*b^4)*d^2*e^4 - 35*(7*B*a^4*b^2 - 24*A*a^3*b^3)*d*e ^5 + 15*(5*B*a^5*b - 12*A*a^4*b^2)*e^6 + 128*(13*B*b^6*d*e^5 + (25*B*a*b^5 + 12*A*b^6)*e^6)*x^4 + 16*(3*B*b^6*d^2*e^4 + 2*(139*B*a*b^5 + 66*A*b^6)*d *e^5 + 9*(15*B*a^2*b^4 + 28*A*a*b^5)*e^6)*x^3 - 8*(7*B*b^6*d^3*e^3 - 3*(9* B*a*b^5 + 4*A*b^6)*d^2*e^4 - 3*(141*B*a^2*b^4 + 256*A*a*b^5)*d*e^5 - (5*B* a^3*b^3 + 372*A*a^2*b^4)*e^6)*x^2 + 2*(35*B*b^6*d^4*e^2 - 4*(34*B*a*b^5 + 15*A*b^6)*d^3*e^3 + 6*(29*B*a^2*b^4 + 46*A*a*b^5)*d^2*e^4 + 4*(20*B*a^3*b^ 3 + 699*A*a^2*b^4)*d*e^5 - 5*(5*B*a^4*b^2 - 12*A*a^3*b^3)*e^6)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^4*e^5), -1/15360*(15*(7*B*b^6*d^6 - 6*(5*B*a*b^5 + 2*A*b^6)*d^5*e + 15*(3*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*(B*a^4*b^2 - 8*A*a^3*b^3)*d^2*e^4 + 6*(3*B*a^5*b - 10*A*a^4*b^2)*d*e^5 - (5*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...
\[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Exception generated. \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 4126 vs. \(2 (308) = 616\).
Time = 0.83 (sec) , antiderivative size = 4126, normalized size of antiderivative = 11.53 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \]
1/7680*(960*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a )*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^ 2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*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*e^2))*A*a*d*abs(b) + 120*(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)*l og(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(s qrt(b*e)*b^2*e^3))*B*a*d*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^3*d*abs(b)/b^2 + 960*(sqrt (b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^ 2 + (b^6*d*e^3 - 13*a*b^5*e^4)/(b^7*e^4)) - 3*(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)/(b^7*e^4)) - 3*(b^3*d^3 + a*b^2*d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*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*e^2))*B*a^2*d*abs(b)/b + 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...
Timed out. \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]