Integrand size = 24, antiderivative size = 304 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=-\frac {3 (b d-a e)^3 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{128 b^3 e^3}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{64 b^3 e^2}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{16 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{5/2} (d+e x)^{3/2}}{8 b^2 e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}+\frac {3 (b d-a e)^4 (2 A b e-B (b d+a e)) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{128 b^{7/2} e^{7/2}} \]
1/8*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(5/2)*(e*x+d)^(3/2)/b^2/e+1/5*B*(b*x+a)^ (5/2)*(e*x+d)^(5/2)/b/e+3/128*(-a*e+b*d)^4*(2*A*b*e-B*(a*e+b*d))*arctanh(e ^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(7/2)/e^(7/2)+1/64*(-a*e+b*d )^2*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b^3/e^2+1/16*(-a*e+b *d)*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(5/2)*(e*x+d)^(1/2)/b^3/e-3/128*(-a*e+b* d)^3*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^3/e^3
Time = 0.84 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.05 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} \left (15 a^4 B e^4-10 a^3 b e^3 (4 B d+3 A e+B e x)+2 a^2 b^2 e^2 \left (5 A e (11 d+2 e x)+B \left (9 d^2+13 d e x+4 e^2 x^2\right )\right )+2 a b^3 e \left (5 A e \left (11 d^2+44 d e x+24 e^2 x^2\right )+B \left (-20 d^3+13 d^2 e x+136 d e^2 x^2+88 e^3 x^3\right )\right )+b^4 \left (10 A e \left (-3 d^3+2 d^2 e x+24 d e^2 x^2+16 e^3 x^3\right )+B \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )\right )\right )}{640 b^3 e^3}-\frac {3 (b d-a e)^4 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{128 b^{7/2} e^{7/2}} \]
(Sqrt[a + b*x]*Sqrt[d + e*x]*(15*a^4*B*e^4 - 10*a^3*b*e^3*(4*B*d + 3*A*e + B*e*x) + 2*a^2*b^2*e^2*(5*A*e*(11*d + 2*e*x) + B*(9*d^2 + 13*d*e*x + 4*e^ 2*x^2)) + 2*a*b^3*e*(5*A*e*(11*d^2 + 44*d*e*x + 24*e^2*x^2) + B*(-20*d^3 + 13*d^2*e*x + 136*d*e^2*x^2 + 88*e^3*x^3)) + b^4*(10*A*e*(-3*d^3 + 2*d^2*e *x + 24*d*e^2*x^2 + 16*e^3*x^3) + B*(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))))/(640*b^3*e^3) - (3*(b*d - a*e)^4*(b*B*d - 2*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])] )/(128*b^(7/2)*e^(7/2))
Time = 0.29 (sec) , antiderivative size = 252, 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 = {90, 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.
| \(\displaystyle \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \int (a+b x)^{3/2} (d+e x)^{3/2}dx}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {3 (b d-a e) \int (a+b x)^{3/2} \sqrt {d+e x}dx}{8 b}+\frac {(a+b x)^{5/2} (d+e x)^{3/2}}{4 b}\right )}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}}dx}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (d+e x)^{3/2}}{4 b}\right )}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}}dx}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (d+e x)^{3/2}}{4 b}\right )}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 e}\right )}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (d+e x)^{3/2}}{4 b}\right )}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{e}\right )}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (d+e x)^{3/2}}{4 b}\right )}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (d+e x)^{3/2}}{4 b}\right )}{2 b e}+\frac {B (a+b x)^{5/2} (d+e x)^{5/2}}{5 b e}\) |
(B*(a + b*x)^(5/2)*(d + e*x)^(5/2))/(5*b*e) + ((2*A*b*e - 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]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sqrt[b]*e^(3/2))))/(4* e)))/(6*b)))/(8*b)))/(2*b*e)
3.23.12.3.1 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. \(1371\) vs. \(2(260)=520\).
Time = 1.08 (sec) , antiderivative size = 1372, normalized size of antiderivative = 4.51
1/1280*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(544*B*a*b^3*d*e^3*x^2*((b*x+a)*(e*x+d) )^(1/2)*(b*e)^(1/2)+880*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^3*d*e^3* x+52*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b^2*d*e^3*x+52*B*((b*x+a)*( e*x+d))^(1/2)*(b*e)^(1/2)*a*b^3*d^2*e^2*x+180*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^2*b^3*d^2*e^3+45*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 ^4*b*d*e^4-30*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^3*b^2*d^2*e^3-30*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^2*b^3*d^3*e^2-120*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^3*b^2*d *e^4-60*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^4*d^3*e+45*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*b^4*d^4*e -120*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*b^4*d^3*e^2-60*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b*e^4+ 320*A*b^4*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+30*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*e^5+30* 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))*b^5*d^4*e+30*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^4*e^4+30*B*((b*x+ a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^4*d^4-15*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^5*d^5+352*B*a*b^3*e^4*x^...
Time = 0.28 (sec) , antiderivative size = 1036, normalized size of antiderivative = 3.41 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\left [\frac {15 \, {\left (B b^{5} d^{5} - {\left (3 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 2 \, {\left (B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} + 2 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - {\left (3 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} - 2 \, A a^{4} b\right )} e^{5}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (128 \, B b^{5} e^{5} x^{4} + 15 \, B b^{5} d^{4} e - 10 \, {\left (4 \, B a b^{4} + 3 \, A b^{5}\right )} d^{3} e^{2} + 2 \, {\left (9 \, B a^{2} b^{3} + 55 \, A a b^{4}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} d e^{4} + 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{5} + 16 \, {\left (11 \, B b^{5} d e^{4} + {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} e^{5}\right )} x^{3} + 8 \, {\left (B b^{5} d^{2} e^{3} + 2 \, {\left (17 \, B a b^{4} + 15 \, A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} e^{5}\right )} x^{2} - 2 \, {\left (5 \, B b^{5} d^{3} e^{2} - {\left (13 \, B a b^{4} + 10 \, A b^{5}\right )} d^{2} e^{3} - {\left (13 \, B a^{2} b^{3} + 220 \, A a b^{4}\right )} d e^{4} + 5 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{2560 \, b^{4} e^{4}}, \frac {15 \, {\left (B b^{5} d^{5} - {\left (3 \, B a b^{4} + 2 \, A b^{5}\right )} d^{4} e + 2 \, {\left (B a^{2} b^{3} + 4 \, A a b^{4}\right )} d^{3} e^{2} + 2 \, {\left (B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} d^{2} e^{3} - {\left (3 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} - 2 \, A a^{4} b\right )} e^{5}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (128 \, B b^{5} e^{5} x^{4} + 15 \, B b^{5} d^{4} e - 10 \, {\left (4 \, B a b^{4} + 3 \, A b^{5}\right )} d^{3} e^{2} + 2 \, {\left (9 \, B a^{2} b^{3} + 55 \, A a b^{4}\right )} d^{2} e^{3} - 10 \, {\left (4 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} d e^{4} + 15 \, {\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{5} + 16 \, {\left (11 \, B b^{5} d e^{4} + {\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} e^{5}\right )} x^{3} + 8 \, {\left (B b^{5} d^{2} e^{3} + 2 \, {\left (17 \, B a b^{4} + 15 \, A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} e^{5}\right )} x^{2} - 2 \, {\left (5 \, B b^{5} d^{3} e^{2} - {\left (13 \, B a b^{4} + 10 \, A b^{5}\right )} d^{2} e^{3} - {\left (13 \, B a^{2} b^{3} + 220 \, A a b^{4}\right )} d e^{4} + 5 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{1280 \, b^{4} e^{4}}\right ] \]
[1/2560*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*A*b^5)*d^4*e + 2*(B*a^2*b^3 + 4*A* a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*a^2*b^3)*d^2*e^3 - (3*B*a^4*b - 8*A*a^ 3*b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*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*(128*B*b^5*e^5*x^4 + 15*B*b ^5*d^4*e - 10*(4*B*a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A*a*b^4) *d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(B*a^4*b - 2*A*a^3*b ^2)*e^5 + 16*(11*B*b^5*d*e^4 + (11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8*(B*b^5 *d^2*e^3 + 2*(17*B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a*b^4)*e^5) *x^2 - 2*(5*B*b^5*d^3*e^2 - (13*B*a*b^4 + 10*A*b^5)*d^2*e^3 - (13*B*a^2*b^ 3 + 220*A*a*b^4)*d*e^4 + 5*(B*a^3*b^2 - 2*A*a^2*b^3)*e^5)*x)*sqrt(b*x + a) *sqrt(e*x + d))/(b^4*e^4), 1/1280*(15*(B*b^5*d^5 - (3*B*a*b^4 + 2*A*b^5)*d ^4*e + 2*(B*a^2*b^3 + 4*A*a*b^4)*d^3*e^2 + 2*(B*a^3*b^2 - 6*A*a^2*b^3)*d^2 *e^3 - (3*B*a^4*b - 8*A*a^3*b^2)*d*e^4 + (B*a^5 - 2*A*a^4*b)*e^5)*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*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(128*B*b^5*e^5*x^4 + 15*B*b^5*d^4*e - 10*(4*B*a*b^4 + 3*A*b^5)*d^3*e^2 + 2*(9*B*a^2*b^3 + 55*A *a*b^4)*d^2*e^3 - 10*(4*B*a^3*b^2 - 11*A*a^2*b^3)*d*e^4 + 15*(B*a^4*b - 2* A*a^3*b^2)*e^5 + 16*(11*B*b^5*d*e^4 + (11*B*a*b^4 + 10*A*b^5)*e^5)*x^3 + 8 *(B*b^5*d^2*e^3 + 2*(17*B*a*b^4 + 15*A*b^5)*d*e^4 + (B*a^2*b^3 + 30*A*a...
\[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
Exception generated. \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
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. 2589 vs. \(2 (260) = 520\).
Time = 0.67 (sec) , antiderivative size = 2589, normalized size of antiderivative = 8.52 \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\text {Too large to display} \]
1/1920*(80*(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*d*abs(b) + 10*(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^1 2*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(a bs(-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*abs(b) - 1920*((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*abs(b)/b^2 + 160*(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*d*abs(b)/b + 10*(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*...
Timed out. \[ \int (a+b x)^{3/2} (A+B x) (d+e x)^{3/2} \, dx=\int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]