Integrand size = 22, antiderivative size = 314 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{192 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{240 b^2 d^3}-\frac {3 (3 b c+a d) (a+b x)^{7/2} \sqrt {c+d x}}{40 b^2 d^2}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{5/2} d^{11/2}} \]
-1/128*(-a*d+b*c)^3*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*arctanh(d^(1/2)*(b*x +a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(5/2)/d^(11/2)-1/192*(-a*d+b*c)*(3*a^2* d^2+14*a*b*c*d+63*b^2*c^2)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^2/d^4+1/240*(3*a^ 2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^2/d^3-3/40*(a*d +3*b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^2/d^2+1/5*x*(b*x+a)^(7/2)*(d*x+c)^(1 /2)/b/d+1/128*(-a*d+b*c)^2*(3*a^2*d^2+14*a*b*c*d+63*b^2*c^2)*(b*x+a)^(1/2) *(d*x+c)^(1/2)/b^2/d^5
Time = 0.63 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-45 a^4 d^4+30 a^3 b d^3 (-3 c+d x)+2 a^2 b^2 d^2 \left (782 c^2-481 c d x+372 d^2 x^2\right )+2 a b^3 d \left (-1155 c^3+749 c^2 d x-592 c d^2 x^2+504 d^3 x^3\right )+b^4 \left (945 c^4-630 c^3 d x+504 c^2 d^2 x^2-432 c d^3 x^3+384 d^4 x^4\right )\right )}{1920 b^2 d^5}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{5/2} d^{11/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-45*a^4*d^4 + 30*a^3*b*d^3*(-3*c + d*x) + 2* a^2*b^2*d^2*(782*c^2 - 481*c*d*x + 372*d^2*x^2) + 2*a*b^3*d*(-1155*c^3 + 7 49*c^2*d*x - 592*c*d^2*x^2 + 504*d^3*x^3) + b^4*(945*c^4 - 630*c^3*d*x + 5 04*c^2*d^2*x^2 - 432*c*d^3*x^3 + 384*d^4*x^4)))/(1920*b^2*d^5) - ((b*c - a *d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x] )/(Sqrt[d]*Sqrt[a + b*x])])/(128*b^(5/2)*d^(11/2))
Time = 0.32 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {101, 27, 90, 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.
| \(\displaystyle \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\int -\frac {(a+b x)^{5/2} (2 a c+3 (3 b c+a d) x)}{2 \sqrt {c+d x}}dx}{5 b d}+\frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\int \frac {(a+b x)^{5/2} (2 a c+3 (3 b c+a d) x)}{\sqrt {c+d x}}dx}{10 b d}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b d}}{10 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b d}}{10 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x (a+b x)^{7/2} \sqrt {c+d x}}{5 b d}-\frac {\frac {3 (a+b x)^{7/2} \sqrt {c+d x} (a d+3 b c)}{4 b d}-\frac {\left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b d}}{10 b d}\) |
(x*(a + b*x)^(7/2)*Sqrt[c + d*x])/(5*b*d) - ((3*(3*b*c + a*d)*(a + b*x)^(7 /2)*Sqrt[c + d*x])/(4*b*d) - ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(((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*d))/(10*b*d)
3.7.74.3.1 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. \(787\) vs. \(2(270)=540\).
Time = 1.78 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.51
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (768 b^{4} d^{4} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+2016 a \,b^{3} d^{4} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-864 b^{4} c \,d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1488 a^{2} b^{2} d^{4} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-2368 a \,b^{3} c \,d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1008 b^{4} c^{2} d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{5} d^{5}+75 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} b c \,d^{4}+450 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b^{2} c^{2} d^{3}-2250 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{3} c^{3} d^{2}+2625 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{4} c^{4} d -945 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{5} c^{5}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x -1924 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x +2996 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x -1260 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{3} d x -90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{4} d^{4}-180 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}+3128 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-4620 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d +1890 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{3840 b^{2} d^{5} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}\) | \(788\) |
1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(768*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1/2 )*(b*d)^(1/2)+2016*a*b^3*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-864*b ^4*c*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+1488*a^2*b^2*d^4*x^2*((b* x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2368*a*b^3*c*d^3*x^2*((b*x+a)*(d*x+c))^(1/ 2)*(b*d)^(1/2)+1008*b^4*c^2*d^2*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+45 *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*d^5+75*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*c*d^4+450*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^2*c^2*d^3-2250*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^2*b^3*c^3*d^2+2 625*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*b^4*c^4*d-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^5*c^5+60*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3 *b*d^4*x-1924*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^2*c*d^3*x+2996*((b *x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^3*c^2*d^2*x-1260*((b*x+a)*(d*x+c))^(1 /2)*(b*d)^(1/2)*b^4*c^3*d*x-90*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*d^4 -180*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b*c*d^3+3128*((b*x+a)*(d*x+c) )^(1/2)*(b*d)^(1/2)*a^2*b^2*c^2*d^2-4620*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/ 2)*a*b^3*c^3*d+1890*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^4*c^4)/b^2/d^5/( (b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)
Time = 0.30 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{3} d^{6}}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (384 \, b^{5} d^{5} x^{4} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{3} d^{6}}\right ] \]
[-1/7680*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3* b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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*(384*b^5*d^5*x^4 + 945*b^5 *c^4*d - 2310*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45 *a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^3 + 8*(63*b^5*c^2*d^3 - 148 *a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^ 3 + 481*a^2*b^3*c*d^4 - 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b ^3*d^6), 1/3840*(15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*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*(384*b^5*d^5*x^4 + 945*b^5*c^4*d - 231 0*a*b^4*c^3*d^2 + 1564*a^2*b^3*c^2*d^3 - 90*a^3*b^2*c*d^4 - 45*a^4*b*d^5 - 144*(3*b^5*c*d^4 - 7*a*b^4*d^5)*x^3 + 8*(63*b^5*c^2*d^3 - 148*a*b^4*c*d^4 + 93*a^2*b^3*d^5)*x^2 - 2*(315*b^5*c^3*d^2 - 749*a*b^4*c^2*d^3 + 481*a^2* b^3*c*d^4 - 15*a^3*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d^6)]
\[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\int \frac {x^{2} \left (a + b x\right )^{\frac {5}{2}}}{\sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.32 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.22 \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (6 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )}}{b^{3} d} - \frac {9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac {63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac {5 \, {\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} \sqrt {b x + a} + \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{5}}\right )} b}{1920 \, {\left | b \right |}} \]
1/1920*(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^3*d) - (9*b^7*c*d^7 + 11*a*b^6*d^8)/(b^9*d^9)) + (63*b^8*c^ 2*d^6 + 14*a*b^7*c*d^7 + 3*a^2*b^6*d^8)/(b^9*d^9)) - 5*(63*b^9*c^3*d^5 - 4 9*a*b^8*c^2*d^6 - 11*a^2*b^7*c*d^7 - 3*a^3*b^6*d^8)/(b^9*d^9))*(b*x + a) + 15*(63*b^10*c^4*d^4 - 112*a*b^9*c^3*d^5 + 38*a^2*b^8*c^2*d^6 + 8*a^3*b^7* c*d^7 + 3*a^4*b^6*d^8)/(b^9*d^9))*sqrt(b*x + a) + 15*(63*b^5*c^5 - 175*a*b ^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*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^2*d^5))*b/abs(b)
Timed out. \[ \int \frac {x^2 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^{5/2}}{\sqrt {c+d\,x}} \,d x \]