Integrand size = 20, antiderivative size = 348 \[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=-\frac {5 (b c-a d)^5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^4}+\frac {5 (b c-a d)^4 (b c+a d) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^3}-\frac {(b c-a d)^3 (b c+a d) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^4 d^2}-\frac {(b c-a d)^2 (b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{64 b^4 d}-\frac {(b c-a d) (b c+a d) (a+b x)^{7/2} (c+d x)^{3/2}}{24 b^3 d}-\frac {(b c+a d) (a+b x)^{7/2} (c+d x)^{5/2}}{12 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}+\frac {5 (b c-a d)^6 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{9/2}} \]
-1/24*(-a*d+b*c)*(a*d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^3/d-1/12*(a*d+b*c )*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b^2/d+1/7*(b*x+a)^(7/2)*(d*x+c)^(7/2)/b/d+5/ 1024*(-a*d+b*c)^6*(a*d+b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^ (1/2))/b^(9/2)/d^(9/2)+5/1536*(-a*d+b*c)^4*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c) ^(1/2)/b^4/d^3-1/384*(-a*d+b*c)^3*(a*d+b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^ 4/d^2-1/64*(-a*d+b*c)^2*(a*d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^4/d-5/1024 *(-a*d+b*c)^5*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^4
Time = 0.76 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.04 \[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^6 d^6+70 a^5 b d^5 (7 c+d x)-7 a^4 b^2 d^4 \left (113 c^2+46 c d x+8 d^2 x^2\right )+4 a^3 b^3 d^3 \left (75 c^3+127 c^2 d x+64 c d^2 x^2+12 d^3 x^3\right )+a^2 b^4 d^2 \left (-791 c^4+508 c^3 d x+9840 c^2 d^2 x^2+12752 c d^3 x^3+4736 d^4 x^4\right )+2 a b^5 d \left (245 c^5-161 c^4 d x+128 c^3 d^2 x^2+6376 c^2 d^3 x^3+9344 c d^4 x^4+3712 d^5 x^5\right )+b^6 \left (-105 c^6+70 c^5 d x-56 c^4 d^2 x^2+48 c^3 d^3 x^3+4736 c^2 d^4 x^4+7424 c d^5 x^5+3072 d^6 x^6\right )\right )}{21504 b^4 d^4}+\frac {5 (b c-a d)^6 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{1024 b^{9/2} d^{9/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*a^6*d^6 + 70*a^5*b*d^5*(7*c + d*x) - 7* a^4*b^2*d^4*(113*c^2 + 46*c*d*x + 8*d^2*x^2) + 4*a^3*b^3*d^3*(75*c^3 + 127 *c^2*d*x + 64*c*d^2*x^2 + 12*d^3*x^3) + a^2*b^4*d^2*(-791*c^4 + 508*c^3*d* x + 9840*c^2*d^2*x^2 + 12752*c*d^3*x^3 + 4736*d^4*x^4) + 2*a*b^5*d*(245*c^ 5 - 161*c^4*d*x + 128*c^3*d^2*x^2 + 6376*c^2*d^3*x^3 + 9344*c*d^4*x^4 + 37 12*d^5*x^5) + b^6*(-105*c^6 + 70*c^5*d*x - 56*c^4*d^2*x^2 + 48*c^3*d^3*x^3 + 4736*c^2*d^4*x^4 + 7424*c*d^5*x^5 + 3072*d^6*x^6)))/(21504*b^4*d^4) + ( 5*(b*c - a*d)^6*(b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[ a + b*x])])/(1024*b^(9/2)*d^(9/2))
Time = 0.32 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, 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.
| \(\displaystyle \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \int (a+b x)^{5/2} (c+d x)^{5/2}dx}{2 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \int (a+b x)^{5/2} (c+d x)^{3/2}dx}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{7/2}}{7 b d}-\frac {(a d+b c) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\right )}{2 b d}\) |
((a + b*x)^(7/2)*(c + d*x)^(7/2))/(7*b*d) - ((b*c + a*d)*(((a + b*x)^(7/2) *(c + d*x)^(5/2))/(6*b) + (5*(b*c - a*d)*(((a + b*x)^(7/2)*(c + d*x)^(3/2) )/(5*b) + (3*(b*c - a*d)*(((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b) + ((b*c - a*d)*(((a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d) - (5*(b*c - a*d)*(((a + b*x)^( 3/2)*Sqrt[c + d*x])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/ d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]) /(Sqrt[b]*d^(3/2))))/(4*d)))/(6*d)))/(8*b)))/(10*b)))/(12*b)))/(2*b*d)
3.7.65.3.1 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. \(1320\) vs. \(2(292)=584\).
Time = 0.57 (sec) , antiderivative size = 1321, normalized size of antiderivative = 3.80
1/43008*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(37376*a*b^5*c*d^5*x^4*((b*x+a)*(d*x+c ))^(1/2)*(b*d)^(1/2)+25504*a^2*b^4*c*d^5*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d) ^(1/2)+25504*a*b^5*c^2*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+512*a^3 *b^3*c*d^5*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+19680*a^2*b^4*c^2*d^4*x ^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+512*a*b^5*c^3*d^3*x^2*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(1/2)+980*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*c*d^5 -1582*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c^2*d^4+600*((b*x+a)*(d* x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d^3-1582*((b*x+a)*(d*x+c))^(1/2)*(b*d) ^(1/2)*a^2*b^4*c^4*d^2+140*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*d^6*x +140*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^6*c^5*d*x-210*((b*x+a)*(d*x+c)) ^(1/2)*(b*d)^(1/2)*a^6*d^6-210*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^6*c^6 -525*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^ (1/2))*a^4*b^3*c^3*d^4-525*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d) ^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^4*c^4*d^3+105*ln(1/2*(2*b*d*x+2*((b*x+a )*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b^7*c^7+105*ln(1/2*(2*b *d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^7*d^7+6 144*b^6*d^6*x^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+945*ln(1/2*(2*b*d*x+2* ((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b^2*c^2*d^5- 644*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c*d^5*x+1016*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^2*d^4*x+945*ln(1/2*(2*b*d*x+2*((b*x+a)...
Time = 0.30 (sec) , antiderivative size = 1102, normalized size of antiderivative = 3.17 \[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\left [\frac {105 \, {\left (b^{7} c^{7} - 5 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{3} b^{4} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} - 5 \, a^{6} b c d^{6} + a^{7} d^{7}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (3072 \, b^{7} d^{7} x^{6} - 105 \, b^{7} c^{6} d + 490 \, a b^{6} c^{5} d^{2} - 791 \, a^{2} b^{5} c^{4} d^{3} + 300 \, a^{3} b^{4} c^{3} d^{4} - 791 \, a^{4} b^{3} c^{2} d^{5} + 490 \, a^{5} b^{2} c d^{6} - 105 \, a^{6} b d^{7} + 7424 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (37 \, b^{7} c^{2} d^{5} + 146 \, a b^{6} c d^{6} + 37 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (3 \, b^{7} c^{3} d^{4} + 797 \, a b^{6} c^{2} d^{5} + 797 \, a^{2} b^{5} c d^{6} + 3 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (7 \, b^{7} c^{4} d^{3} - 32 \, a b^{6} c^{3} d^{4} - 1230 \, a^{2} b^{5} c^{2} d^{5} - 32 \, a^{3} b^{4} c d^{6} + 7 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (35 \, b^{7} c^{5} d^{2} - 161 \, a b^{6} c^{4} d^{3} + 254 \, a^{2} b^{5} c^{3} d^{4} + 254 \, a^{3} b^{4} c^{2} d^{5} - 161 \, a^{4} b^{3} c d^{6} + 35 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{86016 \, b^{5} d^{5}}, -\frac {105 \, {\left (b^{7} c^{7} - 5 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{3} b^{4} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} - 5 \, a^{6} b c d^{6} + a^{7} d^{7}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (3072 \, b^{7} d^{7} x^{6} - 105 \, b^{7} c^{6} d + 490 \, a b^{6} c^{5} d^{2} - 791 \, a^{2} b^{5} c^{4} d^{3} + 300 \, a^{3} b^{4} c^{3} d^{4} - 791 \, a^{4} b^{3} c^{2} d^{5} + 490 \, a^{5} b^{2} c d^{6} - 105 \, a^{6} b d^{7} + 7424 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (37 \, b^{7} c^{2} d^{5} + 146 \, a b^{6} c d^{6} + 37 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (3 \, b^{7} c^{3} d^{4} + 797 \, a b^{6} c^{2} d^{5} + 797 \, a^{2} b^{5} c d^{6} + 3 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (7 \, b^{7} c^{4} d^{3} - 32 \, a b^{6} c^{3} d^{4} - 1230 \, a^{2} b^{5} c^{2} d^{5} - 32 \, a^{3} b^{4} c d^{6} + 7 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (35 \, b^{7} c^{5} d^{2} - 161 \, a b^{6} c^{4} d^{3} + 254 \, a^{2} b^{5} c^{3} d^{4} + 254 \, a^{3} b^{4} c^{2} d^{5} - 161 \, a^{4} b^{3} c d^{6} + 35 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{43008 \, b^{5} d^{5}}\right ] \]
[1/86016*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4 *d^3 - 5*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*sq rt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b *c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(3072*b^7*d^7*x^6 - 105*b^7*c^6*d + 490*a*b^6*c^5*d^2 - 791*a^2*b^5*c ^4*d^3 + 300*a^3*b^4*c^3*d^4 - 791*a^4*b^3*c^2*d^5 + 490*a^5*b^2*c*d^6 - 1 05*a^6*b*d^7 + 7424*(b^7*c*d^6 + a*b^6*d^7)*x^5 + 128*(37*b^7*c^2*d^5 + 14 6*a*b^6*c*d^6 + 37*a^2*b^5*d^7)*x^4 + 16*(3*b^7*c^3*d^4 + 797*a*b^6*c^2*d^ 5 + 797*a^2*b^5*c*d^6 + 3*a^3*b^4*d^7)*x^3 - 8*(7*b^7*c^4*d^3 - 32*a*b^6*c ^3*d^4 - 1230*a^2*b^5*c^2*d^5 - 32*a^3*b^4*c*d^6 + 7*a^4*b^3*d^7)*x^2 + 2* (35*b^7*c^5*d^2 - 161*a*b^6*c^4*d^3 + 254*a^2*b^5*c^3*d^4 + 254*a^3*b^4*c^ 2*d^5 - 161*a^4*b^3*c*d^6 + 35*a^5*b^2*d^7)*x)*sqrt(b*x + a)*sqrt(d*x + c) )/(b^5*d^5), -1/43008*(105*(b^7*c^7 - 5*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 - 5*a^3*b^4*c^4*d^3 - 5*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 - 5*a^6*b*c*d^6 + a^7*d^7)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 2*( 3072*b^7*d^7*x^6 - 105*b^7*c^6*d + 490*a*b^6*c^5*d^2 - 791*a^2*b^5*c^4*d^3 + 300*a^3*b^4*c^3*d^4 - 791*a^4*b^3*c^2*d^5 + 490*a^5*b^2*c*d^6 - 105*a^6 *b*d^7 + 7424*(b^7*c*d^6 + a*b^6*d^7)*x^5 + 128*(37*b^7*c^2*d^5 + 146*a*b^ 6*c*d^6 + 37*a^2*b^5*d^7)*x^4 + 16*(3*b^7*c^3*d^4 + 797*a*b^6*c^2*d^5 +...
\[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\int x \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \]
Exception generated. \[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 4150 vs. \(2 (292) = 584\).
Time = 0.81 (sec) , antiderivative size = 4150, normalized size of antiderivative = 11.93 \[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\text {Too large to display} \]
1/107520*(1680*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13* c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3* d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))* sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3* b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*a*c^2*abs(b) + 13440*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b ^6*c*d^3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11* a^2*b^5*d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3 *d^3)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b* d)))/(sqrt(b*d)*b*d^2))*a^2*c^2*abs(b)/b + 56*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41* a*b^19*d^8)/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19 *d^8)/(b^23*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c* d^7 - 447*a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a* b^22*c^3*d^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8) /(b^23*d^8))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3 *d^2 + 10*a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*log(abs(-sqrt(b*d )*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*...
Timed out. \[ \int x (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\int x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2} \,d x \]