Integrand size = 22, antiderivative size = 437 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {(b c-a d)^4 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^5}-\frac {(b c-a d)^3 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^4}+\frac {(b c-a d)^2 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{1920 b^4 d^3}+\frac {(b c-a d) \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} \sqrt {c+d x}}{320 b^4 d^2}+\frac {\left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{120 b^3 d^2}-\frac {(9 b c+7 a d) (a+b x)^{7/2} (c+d x)^{5/2}}{84 b^2 d^2}+\frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {(b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{11/2}} \]
1/120*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^3/d^2 -1/84*(7*a*d+9*b*c)*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b^2/d^2+1/7*x*(b*x+a)^(7/2 )*(d*x+c)^(5/2)/b/d-1/1024*(-a*d+b*c)^5*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2)*a rctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(9/2)/d^(11/2)-1/153 6*(-a*d+b*c)^3*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(1/2 )/b^4/d^4+1/1920*(-a*d+b*c)^2*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2)*(b*x+a)^(5/ 2)*(d*x+c)^(1/2)/b^4/d^3+1/320*(-a*d+b*c)*(5*a^2*d^2+10*a*b*c*d+9*b^2*c^2) *(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^4/d^2+1/1024*(-a*d+b*c)^4*(5*a^2*d^2+10*a*b *c*d+9*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^5
Time = 0.88 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.87 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-525 a^6 d^6+350 a^5 b d^5 (4 c+d x)-35 a^4 b^2 d^4 \left (15 c^2+26 c d x+8 d^2 x^2\right )+60 a^3 b^3 d^3 \left (-10 c^3+5 c^2 d x+12 c d^2 x^2+4 d^3 x^3\right )+a^2 b^4 d^2 \left (3689 c^4-2332 c^3 d x+1824 c^2 d^2 x^2+33520 c d^3 x^3+23680 d^4 x^4\right )+2 a b^5 d \left (-1680 c^5+1099 c^4 d x-872 c^3 d^2 x^2+744 c^2 d^3 x^3+24320 c d^4 x^4+18560 d^5 x^5\right )+3 b^6 \left (315 c^6-210 c^5 d x+168 c^4 d^2 x^2-144 c^3 d^3 x^3+128 c^2 d^4 x^4+6400 c d^5 x^5+5120 d^6 x^6\right )\right )}{107520 b^4 d^5}-\frac {(b c-a d)^5 \left (9 b^2 c^2+10 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{11/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-525*a^6*d^6 + 350*a^5*b*d^5*(4*c + d*x) - 3 5*a^4*b^2*d^4*(15*c^2 + 26*c*d*x + 8*d^2*x^2) + 60*a^3*b^3*d^3*(-10*c^3 + 5*c^2*d*x + 12*c*d^2*x^2 + 4*d^3*x^3) + a^2*b^4*d^2*(3689*c^4 - 2332*c^3*d *x + 1824*c^2*d^2*x^2 + 33520*c*d^3*x^3 + 23680*d^4*x^4) + 2*a*b^5*d*(-168 0*c^5 + 1099*c^4*d*x - 872*c^3*d^2*x^2 + 744*c^2*d^3*x^3 + 24320*c*d^4*x^4 + 18560*d^5*x^5) + 3*b^6*(315*c^6 - 210*c^5*d*x + 168*c^4*d^2*x^2 - 144*c ^3*d^3*x^3 + 128*c^2*d^4*x^4 + 6400*c*d^5*x^5 + 5120*d^6*x^6)))/(107520*b^ 4*d^5) - ((b*c - a*d)^5*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt [d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(1024*b^(9/2)*d^(11/2))
Time = 0.38 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {101, 27, 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.
| \(\displaystyle \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\int -\frac {1}{2} (a+b x)^{5/2} (c+d x)^{3/2} (2 a c+(9 b c+7 a d) x)dx}{7 b d}+\frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\int (a+b x)^{5/2} (c+d x)^{3/2} (2 a c+(9 b c+7 a d) x)dx}{14 b d}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \int (a+b x)^{5/2} (c+d x)^{3/2}dx}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x (a+b x)^{7/2} (c+d x)^{5/2}}{7 b d}-\frac {\frac {(a+b x)^{7/2} (c+d x)^{5/2} (7 a d+9 b c)}{6 b d}-\frac {7 \left (5 a^2 d^2+10 a b c d+9 b^2 c^2\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
(x*(a + b*x)^(7/2)*(c + d*x)^(5/2))/(7*b*d) - (((9*b*c + 7*a*d)*(a + b*x)^ (7/2)*(c + d*x)^(5/2))/(6*b*d) - (7*(9*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*( ((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*d))/(14*b*d)
3.7.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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. \(2(381)=762\).
Time = 0.56 (sec) , antiderivative size = 1321, normalized size of antiderivative = 3.02
1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(97280*a*b^5*c*d^5*x^4*((b*x+a)*(d*x+ c))^(1/2)*(b*d)^(1/2)+67040*a^2*b^4*c*d^5*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d )^(1/2)+2976*a*b^5*c^2*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1440*a^ 3*b^3*c*d^5*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3648*a^2*b^4*c^2*d^4*x ^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-3488*a*b^5*c^3*d^3*x^2*((b*x+a)*(d* x+c))^(1/2)*(b*d)^(1/2)+2800*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*c*d ^5-1050*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c^2*d^4-1200*((b*x+a)* (d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d^3+7378*((b*x+a)*(d*x+c))^(1/2)*(b *d)^(1/2)*a^2*b^4*c^4*d^2+700*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*d^ 6*x-1260*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^6*c^5*d*x-1050*((b*x+a)*(d* x+c))^(1/2)*(b*d)^(1/2)*a^6*d^6+1890*((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+1575*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^3-945*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))*b^7*c^7+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^ 7*d^7+30720*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^5-1820*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c*d^5*x+600*((b*x +a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^2*d^4*x-4725*ln(1/2*(2*b*d*x+2...
Time = 0.29 (sec) , antiderivative size = 1110, normalized size of antiderivative = 2.54 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\left [-\frac {105 \, {\left (9 \, b^{7} c^{7} - 35 \, a b^{6} c^{6} d + 45 \, a^{2} b^{5} c^{5} d^{2} - 15 \, 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} + 15 \, a^{6} b c d^{6} - 5 \, 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 (15360 \, b^{7} d^{7} x^{6} + 945 \, b^{7} c^{6} d - 3360 \, a b^{6} c^{5} d^{2} + 3689 \, a^{2} b^{5} c^{4} d^{3} - 600 \, a^{3} b^{4} c^{3} d^{4} - 525 \, a^{4} b^{3} c^{2} d^{5} + 1400 \, a^{5} b^{2} c d^{6} - 525 \, a^{6} b d^{7} + 1280 \, {\left (15 \, b^{7} c d^{6} + 29 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 185 \, a^{2} b^{5} d^{7}\right )} x^{4} - 16 \, {\left (27 \, b^{7} c^{3} d^{4} - 93 \, a b^{6} c^{2} d^{5} - 2095 \, a^{2} b^{5} c d^{6} - 15 \, a^{3} b^{4} d^{7}\right )} x^{3} + 8 \, {\left (63 \, b^{7} c^{4} d^{3} - 218 \, a b^{6} c^{3} d^{4} + 228 \, a^{2} b^{5} c^{2} d^{5} + 90 \, a^{3} b^{4} c d^{6} - 35 \, a^{4} b^{3} d^{7}\right )} x^{2} - 2 \, {\left (315 \, b^{7} c^{5} d^{2} - 1099 \, a b^{6} c^{4} d^{3} + 1166 \, a^{2} b^{5} c^{3} d^{4} - 150 \, a^{3} b^{4} c^{2} d^{5} + 455 \, a^{4} b^{3} c d^{6} - 175 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{430080 \, b^{5} d^{6}}, \frac {105 \, {\left (9 \, b^{7} c^{7} - 35 \, a b^{6} c^{6} d + 45 \, a^{2} b^{5} c^{5} d^{2} - 15 \, 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} + 15 \, a^{6} b c d^{6} - 5 \, 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 (15360 \, b^{7} d^{7} x^{6} + 945 \, b^{7} c^{6} d - 3360 \, a b^{6} c^{5} d^{2} + 3689 \, a^{2} b^{5} c^{4} d^{3} - 600 \, a^{3} b^{4} c^{3} d^{4} - 525 \, a^{4} b^{3} c^{2} d^{5} + 1400 \, a^{5} b^{2} c d^{6} - 525 \, a^{6} b d^{7} + 1280 \, {\left (15 \, b^{7} c d^{6} + 29 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (3 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 185 \, a^{2} b^{5} d^{7}\right )} x^{4} - 16 \, {\left (27 \, b^{7} c^{3} d^{4} - 93 \, a b^{6} c^{2} d^{5} - 2095 \, a^{2} b^{5} c d^{6} - 15 \, a^{3} b^{4} d^{7}\right )} x^{3} + 8 \, {\left (63 \, b^{7} c^{4} d^{3} - 218 \, a b^{6} c^{3} d^{4} + 228 \, a^{2} b^{5} c^{2} d^{5} + 90 \, a^{3} b^{4} c d^{6} - 35 \, a^{4} b^{3} d^{7}\right )} x^{2} - 2 \, {\left (315 \, b^{7} c^{5} d^{2} - 1099 \, a b^{6} c^{4} d^{3} + 1166 \, a^{2} b^{5} c^{3} d^{4} - 150 \, a^{3} b^{4} c^{2} d^{5} + 455 \, a^{4} b^{3} c d^{6} - 175 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{215040 \, b^{5} d^{6}}\right ] \]
[-1/430080*(105*(9*b^7*c^7 - 35*a*b^6*c^6*d + 45*a^2*b^5*c^5*d^2 - 15*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 + 15*a^6*b*c*d^6 - 5*a ^7*d^7)*sqrt(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*(15360*b^7*d^7*x^6 + 945*b^7*c^6*d - 3360*a*b^6*c^5*d^2 + 3 689*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 - 525*a^4*b^3*c^2*d^5 + 1400*a^5 *b^2*c*d^6 - 525*a^6*b*d^7 + 1280*(15*b^7*c*d^6 + 29*a*b^6*d^7)*x^5 + 128* (3*b^7*c^2*d^5 + 380*a*b^6*c*d^6 + 185*a^2*b^5*d^7)*x^4 - 16*(27*b^7*c^3*d ^4 - 93*a*b^6*c^2*d^5 - 2095*a^2*b^5*c*d^6 - 15*a^3*b^4*d^7)*x^3 + 8*(63*b ^7*c^4*d^3 - 218*a*b^6*c^3*d^4 + 228*a^2*b^5*c^2*d^5 + 90*a^3*b^4*c*d^6 - 35*a^4*b^3*d^7)*x^2 - 2*(315*b^7*c^5*d^2 - 1099*a*b^6*c^4*d^3 + 1166*a^2*b ^5*c^3*d^4 - 150*a^3*b^4*c^2*d^5 + 455*a^4*b^3*c*d^6 - 175*a^5*b^2*d^7)*x) *sqrt(b*x + a)*sqrt(d*x + c))/(b^5*d^6), 1/215040*(105*(9*b^7*c^7 - 35*a*b ^6*c^6*d + 45*a^2*b^5*c^5*d^2 - 15*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 + 15*a^6*b*c*d^6 - 5*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*(15360*b^7*d^7*x^6 + 945*b^7*c^6*d - 33 60*a*b^6*c^5*d^2 + 3689*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 - 525*a^4*b^ 3*c^2*d^5 + 1400*a^5*b^2*c*d^6 - 525*a^6*b*d^7 + 1280*(15*b^7*c*d^6 + 29*a *b^6*d^7)*x^5 + 128*(3*b^7*c^2*d^5 + 380*a*b^6*c*d^6 + 185*a^2*b^5*d^7)...
\[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x^{2} \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \]
Exception generated. \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 3120 vs. \(2 (381) = 762\).
Time = 0.65 (sec) , antiderivative size = 3120, normalized size of antiderivative = 7.14 \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \]
1/107520*(168*(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^2 1*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*d^4))*a*c*abs(b) + 4480*(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^3*c*abs(b)/b^2 + 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^...
Timed out. \[ \int x^2 (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x^2\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]