Integrand size = 24, antiderivative size = 412 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\frac {5 (b d-a e)^5 (2 A b e-B (b d+a e)) \sqrt {a+b x} \sqrt {d+e x}}{1024 b^4 e^4}-\frac {5 (b d-a e)^4 (2 A b e-B (b d+a e)) (a+b x)^{3/2} \sqrt {d+e x}}{1536 b^4 e^3}+\frac {(b d-a e)^3 (2 A b e-B (b d+a e)) (a+b x)^{5/2} \sqrt {d+e x}}{384 b^4 e^2}+\frac {(b d-a e)^2 (2 A b e-B (b d+a e)) (a+b x)^{7/2} \sqrt {d+e x}}{64 b^4 e}+\frac {(b d-a e) (2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{3/2}}{24 b^3 e}+\frac {(2 A b e-B (b d+a e)) (a+b x)^{7/2} (d+e x)^{5/2}}{12 b^2 e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}-\frac {5 (b d-a e)^6 (2 A b e-B (b d+a e)) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{1024 b^{9/2} e^{9/2}} \]
1/24*(-a*e+b*d)*(2*A*b*e-B*(a*e+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*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*d))*arctanh(e ^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(9/2)/e^(9/2)-5/1536*(-a*e+b *d)^4*(2*A*b*e-B*(a*e+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*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*d))*(b*x+a)^(7/2)*(e*x+d)^(1/2)/b^4/e+5/1024* (-a*e+b*d)^5*(2*A*b*e-B*(a*e+b*d))*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^4/e^4
Time = 1.26 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.25 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\frac {(b d-a e)^6 \left (-\frac {\sqrt {b} \sqrt {e} \sqrt {a+b x} \sqrt {d+e x} \left (105 b B d e^6 (a+b x)^6-210 A b e^7 (a+b x)^6+105 a B e^7 (a+b x)^6-700 b^2 B d e^5 (a+b x)^5 (d+e x)+1400 A b^2 e^6 (a+b x)^5 (d+e x)-700 a b B e^6 (a+b x)^5 (d+e x)+1981 b^3 B d e^4 (a+b x)^4 (d+e x)^2-3962 A b^3 e^5 (a+b x)^4 (d+e x)^2+1981 a b^2 B e^5 (a+b x)^4 (d+e x)^2+3072 b^4 B d e^3 (a+b x)^3 (d+e x)^3-3072 a b^3 B e^4 (a+b x)^3 (d+e x)^3-1981 b^5 B d e^2 (a+b x)^2 (d+e x)^4+3962 A b^5 e^3 (a+b x)^2 (d+e x)^4-1981 a b^4 B e^3 (a+b x)^2 (d+e x)^4+700 b^6 B d e (a+b x) (d+e x)^5-1400 A b^6 e^2 (a+b x) (d+e x)^5+700 a b^5 B e^2 (a+b x) (d+e x)^5-105 b^7 B d (d+e x)^6+210 A b^7 e (d+e x)^6-105 a b^6 B e (d+e x)^6\right )}{(-b d+a e)^7}+105 (b B d-2 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )\right )}{21504 b^{9/2} e^{9/2}} \]
((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))
Time = 0.33 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.81, 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 |
| \(\displaystyle \frac {(2 A b e-B (a e+b d)) \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \left (\frac {(a+b x)^{5/2} \sqrt {d+e x}}{3 e}-\frac {5 (b d-a e) \left (\frac {(a+b x)^{3/2} \sqrt {d+e x}}{2 e}-\frac {3 (b d-a e) \left (\frac {\sqrt {a+b x} \sqrt {d+e x}}{e}-\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\right )}{4 e}\right )}{6 e}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {d+e x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (d+e x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (d+e x)^{5/2}}{6 b}\right )}{2 b e}+\frac {B (a+b x)^{7/2} (d+e x)^{7/2}}{7 b e}\) |
(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)
3.23.23.3.1 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. \(2395\) vs. \(2(356)=712\).
Time = 1.09 (sec) , antiderivative size = 2396, normalized size of antiderivative = 5.82
-1/43008*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-1016*B*((b*x+a)*(e*x+d))^(1/2)*(b*e )^(1/2)*a^2*b^4*d^3*e^3*x+644*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^5* d^4*e^2*x-1568*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b^3*d*e^5*x-33264 *A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^2*b^4*d^2*e^4*x-1568*A*((b*x+a)*( e*x+d))^(1/2)*(b*e)^(1/2)*a*b^5*d^3*e^3*x+644*B*((b*x+a)*(e*x+d))^(1/2)*(b *e)^(1/2)*a^4*b^2*d*e^5*x-1016*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^3*b ^3*d^2*e^4*x-96*B*b^6*d^3*e^3*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-224* A*a^3*b^3*e^6*x^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-224*A*b^6*d^3*e^3*x^ 2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+112*B*a^4*b^2*e^6*x^2*((b*x+a)*(e*x+ d))^(1/2)*(b*e)^(1/2)-420*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a^5*b*e^6- 420*A*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b^6*d^5*e-7168*A*b^6*e^6*x^5*((b *x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-6144*B*b^6*e^6*x^6*((b*x+a)*(e*x+d))^(1/2 )*(b*e)^(1/2)-14848*B*b^6*d*e^5*x^5*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-17 920*A*a*b^5*e^6*x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-17920*A*b^6*d*e^5* x^4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-9472*B*a^2*b^4*e^6*x^4*((b*x+a)*(e *x+d))^(1/2)*(b*e)^(1/2)-9472*B*b^6*d^2*e^4*x^4*((b*x+a)*(e*x+d))^(1/2)*(b *e)^(1/2)-12096*A*a^2*b^4*e^6*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-1209 6*A*b^6*d^2*e^4*x^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-96*B*a^3*b^3*e^6*x ^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-14848*B*a*b^5*e^6*x^5*((b*x+a)*(e*x +d))^(1/2)*(b*e)^(1/2)-980*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*a*b^5*...
Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (356) = 712\).
Time = 0.33 (sec) , antiderivative size = 1758, normalized size of antiderivative = 4.27 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \]
[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)*sq rt(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*(1 13*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 ...
\[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {5}{2}}\, dx \]
Exception generated. \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Exception raised: ValueError} \]
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. 7289 vs. \(2 (356) = 712\).
Time = 1.21 (sec) , antiderivative size = 7289, normalized size of antiderivative = 17.69 \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\text {Too large to display} \]
1/107520*(13440*(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^2*abs(b) + 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 + 13440*(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^2*abs(b)/b + 560*(sqr t(b^2*d + (b*x + a)*b*e - a*b*e)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)...
Timed out. \[ \int (a+b x)^{5/2} (A+B x) (d+e x)^{5/2} \, dx=\int \left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{5/2} \,d x \]