Integrand size = 22, antiderivative size = 306 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=-\frac {5 (b c-a d) \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d}-\frac {5 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^4 d}-\frac {\left (14 a c+\frac {b c^2}{d}-\frac {63 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{5/2}}{24 b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 b^2 d}-\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{3/2}} \]
-5/64*(-a*d+b*c)^2*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b*x+a )^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(11/2)/d^(3/2)-2*a^2*(d*x+c)^(7/2)/b^2/(- a*d+b*c)/(b*x+a)^(1/2)-5/96*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2) *(b*x+a)^(1/2)/b^4/d-1/24*(14*a*c+b*c^2/d-63*a^2*d/b)*(d*x+c)^(5/2)*(b*x+a )^(1/2)/b^2/(-a*d+b*c)+1/4*(d*x+c)^(7/2)*(b*x+a)^(1/2)/b^2/d-5/64*(-a*d+b* c)*(-63*a^2*d^2+14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^5/d
Time = 0.65 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (-945 a^4 d^3+105 a^3 b d^2 (17 c-3 d x)+a^2 b^2 d \left (-839 c^2+637 c d x+126 d^2 x^2\right )+a b^3 \left (15 c^3-337 c^2 d x-244 c d^2 x^2-72 d^3 x^3\right )+b^4 x \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^5 d \sqrt {a+b x}}-\frac {5 (b c-a d)^2 \left (b^2 c^2+14 a b c d-63 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{11/2} d^{3/2}} \]
(Sqrt[c + d*x]*(-945*a^4*d^3 + 105*a^3*b*d^2*(17*c - 3*d*x) + a^2*b^2*d*(- 839*c^2 + 637*c*d*x + 126*d^2*x^2) + a*b^3*(15*c^3 - 337*c^2*d*x - 244*c*d ^2*x^2 - 72*d^3*x^3) + b^4*x*(15*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*d^ 3*x^3)))/(192*b^5*d*Sqrt[a + b*x]) - (5*(b*c - a*d)^2*(b^2*c^2 + 14*a*b*c* d - 63*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/ (64*b^(11/2)*d^(3/2))
Time = 0.34 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {100, 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.
| \(\displaystyle \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {2 \int -\frac {(c+d x)^{5/2} (a (b c-7 a d)-b (b c-a d) x)}{2 \sqrt {a+b x}}dx}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(c+d x)^{5/2} (a (b c-7 a d)-b (b c-a d) x)}{\sqrt {a+b x}}dx}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {\frac {\left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}}dx}{8 d}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)}{4 d}}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\frac {\left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)}{4 d}}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\frac {\left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)}{4 d}}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\frac {\left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)}{4 d}}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {\frac {\left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)}{4 d}}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\left (-63 a^2 d^2+14 a b c d+b^2 c^2\right ) \left (\frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 b}\right )}{8 d}-\frac {\sqrt {a+b x} (c+d x)^{7/2} (b c-a d)}{4 d}}{b^2 (b c-a d)}-\frac {2 a^2 (c+d x)^{7/2}}{b^2 \sqrt {a+b x} (b c-a d)}\) |
(-2*a^2*(c + d*x)^(7/2))/(b^2*(b*c - a*d)*Sqrt[a + b*x]) - (-1/4*((b*c - a *d)*Sqrt[a + b*x]*(c + d*x)^(7/2))/d + ((b^2*c^2 + 14*a*b*c*d - 63*a^2*d^2 )*((Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*b) + (5*(b*c - a*d)*((Sqrt[a + b*x]* (c + d*x)^(3/2))/(2*b) + (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/b + ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b ^(3/2)*Sqrt[d])))/(4*b)))/(6*b)))/(8*d))/(b^2*(b*c - a*d))
3.8.68.3.1 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*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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. \(960\) vs. \(2(264)=528\).
Time = 1.69 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.14
| method | result | size |
| default | \(\frac {\sqrt {d x +c}\, \left (96 b^{4} d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-144 a \,b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+272 b^{4} c \,d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b 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 ) a^{4} b \,d^{4} x -2100 \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 \,d^{3} x +1350 \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^{2} d^{2} x -180 \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^{3} d x -15 \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^{4} x +252 a^{2} b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-488 a \,b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+236 b^{4} c^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b 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 ) a^{5} d^{4}-2100 \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^{3}+1350 \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^{2}-180 \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 -15 \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}-630 a^{3} b \,d^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+1274 a^{2} b^{2} c \,d^{2} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-674 a \,b^{3} c^{2} d x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+30 b^{4} c^{3} x \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-1890 a^{4} d^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+3570 a^{3} b c \,d^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-1678 a^{2} b^{2} c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+30 a \,b^{3} c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\right )}{384 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {b x +a}\, b^{5} d}\) | \(961\) |
1/384*(d*x+c)^(1/2)*(96*b^4*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-14 4*a*b^3*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+272*b^4*c*d^2*x^3*((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^4*b*d^4*x-2100*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*d^3*x+1 350*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^2*d^2*x-180*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^3*d*x-15*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^4*x+252*a^2*b^2*d^3 *x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-488*a*b^3*c*d^2*x^2*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(1/2)+236*b^4*c^2*d*x^2*((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*d^4-2100*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^3+1350*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^2-180*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-15*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^4-630*a^3*b*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+12 74*a^2*b^2*c*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-674*a*b^3*c^2*d*x*( (b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+30*b^4*c^3*x*((b*x+a)*(d*x+c))^(1/2)...
Time = 0.41 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.58 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (a b^{4} c^{4} + 12 \, a^{2} b^{3} c^{3} d - 90 \, a^{3} b^{2} c^{2} d^{2} + 140 \, a^{4} b c d^{3} - 63 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 12 \, a b^{4} c^{3} d - 90 \, a^{2} b^{3} c^{2} d^{2} + 140 \, a^{3} b^{2} c d^{3} - 63 \, a^{4} b d^{4}\right )} x\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 (48 \, b^{5} d^{4} x^{4} + 15 \, a b^{4} c^{3} d - 839 \, a^{2} b^{3} c^{2} d^{2} + 1785 \, a^{3} b^{2} c d^{3} - 945 \, a^{4} b d^{4} + 8 \, {\left (17 \, b^{5} c d^{3} - 9 \, a b^{4} d^{4}\right )} x^{3} + 2 \, {\left (59 \, b^{5} c^{2} d^{2} - 122 \, a b^{4} c d^{3} + 63 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (15 \, b^{5} c^{3} d - 337 \, a b^{4} c^{2} d^{2} + 637 \, a^{2} b^{3} c d^{3} - 315 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}, \frac {15 \, {\left (a b^{4} c^{4} + 12 \, a^{2} b^{3} c^{3} d - 90 \, a^{3} b^{2} c^{2} d^{2} + 140 \, a^{4} b c d^{3} - 63 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 12 \, a b^{4} c^{3} d - 90 \, a^{2} b^{3} c^{2} d^{2} + 140 \, a^{3} b^{2} c d^{3} - 63 \, a^{4} b d^{4}\right )} x\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 (48 \, b^{5} d^{4} x^{4} + 15 \, a b^{4} c^{3} d - 839 \, a^{2} b^{3} c^{2} d^{2} + 1785 \, a^{3} b^{2} c d^{3} - 945 \, a^{4} b d^{4} + 8 \, {\left (17 \, b^{5} c d^{3} - 9 \, a b^{4} d^{4}\right )} x^{3} + 2 \, {\left (59 \, b^{5} c^{2} d^{2} - 122 \, a b^{4} c d^{3} + 63 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (15 \, b^{5} c^{3} d - 337 \, a b^{4} c^{2} d^{2} + 637 \, a^{2} b^{3} c d^{3} - 315 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}\right ] \]
[-1/768*(15*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b *c*d^3 - 63*a^5*d^4 + (b^5*c^4 + 12*a*b^4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140 *a^3*b^2*c*d^3 - 63*a^4*b*d^4)*x)*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*(48*b^5*d^4*x^4 + 15*a*b^4*c^3*d - 839*a^2*b^3*c^2*d^2 + 1785*a^3*b^2*c*d^3 - 945*a^4*b*d^4 + 8*(17*b^5*c*d ^3 - 9*a*b^4*d^4)*x^3 + 2*(59*b^5*c^2*d^2 - 122*a*b^4*c*d^3 + 63*a^2*b^3*d ^4)*x^2 + (15*b^5*c^3*d - 337*a*b^4*c^2*d^2 + 637*a^2*b^3*c*d^3 - 315*a^3* b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^2*x + a*b^6*d^2), 1/384*(1 5*(a*b^4*c^4 + 12*a^2*b^3*c^3*d - 90*a^3*b^2*c^2*d^2 + 140*a^4*b*c*d^3 - 6 3*a^5*d^4 + (b^5*c^4 + 12*a*b^4*c^3*d - 90*a^2*b^3*c^2*d^2 + 140*a^3*b^2*c *d^3 - 63*a^4*b*d^4)*x)*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*(48*b^5*d^4*x^4 + 15*a*b^4*c^3*d - 839*a^2*b^3*c^2*d^2 + 1785* a^3*b^2*c*d^3 - 945*a^4*b*d^4 + 8*(17*b^5*c*d^3 - 9*a*b^4*d^4)*x^3 + 2*(59 *b^5*c^2*d^2 - 122*a*b^4*c*d^3 + 63*a^2*b^3*d^4)*x^2 + (15*b^5*c^3*d - 337 *a*b^4*c^2*d^2 + 637*a^2*b^3*c*d^3 - 315*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqr t(d*x + c))/(b^7*d^2*x + a*b^6*d^2)]
\[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.52 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.41 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{192} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{7}} + \frac {17 \, b^{28} c d^{7} {\left | b \right |} - 33 \, a b^{27} d^{8} {\left | b \right |}}{b^{34} d^{6}}\right )} + \frac {59 \, b^{29} c^{2} d^{6} {\left | b \right |} - 326 \, a b^{28} c d^{7} {\left | b \right |} + 315 \, a^{2} b^{27} d^{8} {\left | b \right |}}{b^{34} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{30} c^{3} d^{5} {\left | b \right |} - 191 \, a b^{29} c^{2} d^{6} {\left | b \right |} + 511 \, a^{2} b^{28} c d^{7} {\left | b \right |} - 325 \, a^{3} b^{27} d^{8} {\left | b \right |}\right )}}{b^{34} d^{6}}\right )} \sqrt {b x + a} - \frac {4 \, {\left (a^{2} b^{3} c^{3} d {\left | b \right |} - 3 \, a^{3} b^{2} c^{2} d^{2} {\left | b \right |} + 3 \, a^{4} b c d^{3} {\left | b \right |} - a^{5} d^{4} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{5}} + \frac {5 \, {\left (b^{4} c^{4} {\left | b \right |} + 12 \, a b^{3} c^{3} d {\left | b \right |} - 90 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + 140 \, a^{3} b c d^{3} {\left | b \right |} - 63 \, a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, \sqrt {b d} b^{6} d} \]
1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b* x + a)*d^2*abs(b)/b^7 + (17*b^28*c*d^7*abs(b) - 33*a*b^27*d^8*abs(b))/(b^3 4*d^6)) + (59*b^29*c^2*d^6*abs(b) - 326*a*b^28*c*d^7*abs(b) + 315*a^2*b^27 *d^8*abs(b))/(b^34*d^6)) + 3*(5*b^30*c^3*d^5*abs(b) - 191*a*b^29*c^2*d^6*a bs(b) + 511*a^2*b^28*c*d^7*abs(b) - 325*a^3*b^27*d^8*abs(b))/(b^34*d^6))*s qrt(b*x + a) - 4*(a^2*b^3*c^3*d*abs(b) - 3*a^3*b^2*c^2*d^2*abs(b) + 3*a^4* b*c*d^3*abs(b) - a^5*d^4*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a ) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*sqrt(b*d)*b^5) + 5/128*(b^4*c^ 4*abs(b) + 12*a*b^3*c^3*d*abs(b) - 90*a^2*b^2*c^2*d^2*abs(b) + 140*a^3*b*c *d^3*abs(b) - 63*a^4*d^4*abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^6*d)
Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]