Integrand size = 22, antiderivative size = 273 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=-\frac {3 (b c-a d)^3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^3 c^3 x}+\frac {(b c-a d)^2 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{64 a^2 c^3 x^2}+\frac {(b c-a d) (b c+a d) \sqrt {a+b x} (c+d x)^{5/2}}{16 a c^3 x^3}+\frac {(b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{8 a c^2 x^4}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}+\frac {3 (b c-a d)^4 (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{7/2}} \]
1/8*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/a/c^2/x^4-1/5*(b*x+a)^(5/2)*(d*x +c)^(5/2)/a/c/x^5+3/128*(-a*d+b*c)^4*(a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/ 2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(7/2)+1/64*(-a*d+b*c)^2*(a*d+b*c)*(d*x +c)^(3/2)*(b*x+a)^(1/2)/a^2/c^3/x^2+1/16*(-a*d+b*c)*(a*d+b*c)*(d*x+c)^(5/2 )*(b*x+a)^(1/2)/a/c^3/x^3-3/128*(-a*d+b*c)^3*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+ c)^(1/2)/a^3/c^3/x
Time = 0.55 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^4 c^4 x^4-10 a b^3 c^3 x^3 (c+4 d x)+2 a^2 b^2 c^2 x^2 \left (4 c^2+13 c d x+9 d^2 x^2\right )+2 a^3 b c x \left (88 c^3+136 c^2 d x+13 c d^2 x^2-20 d^3 x^3\right )+a^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )\right )}{640 a^3 c^3 x^5}+\frac {3 (b c-a d)^4 (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{7/2} c^{7/2}} \]
-1/640*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^4*c^4*x^4 - 10*a*b^3*c^3*x^3*(c + 4*d*x) + 2*a^2*b^2*c^2*x^2*(4*c^2 + 13*c*d*x + 9*d^2*x^2) + 2*a^3*b*c*x* (88*c^3 + 136*c^2*d*x + 13*c*d^2*x^2 - 20*d^3*x^3) + a^4*(128*c^4 + 176*c^ 3*d*x + 8*c^2*d^2*x^2 - 10*c*d^3*x^3 + 15*d^4*x^4)))/(a^3*c^3*x^5) + (3*(b *c - a*d)^4*(b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(7/2)*c^(7/2))
Time = 0.30 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {107, 105, 105, 105, 105, 104, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(a d+b c) \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^5}dx}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {3 (b c-a d) \int \frac {\sqrt {a+b x} (c+d x)^{3/2}}{x^4}dx}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4}\right )}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4}\right )}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4}\right )}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4}\right )}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4}\right )}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(a d+b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 c}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 c x^3}\right )}{8 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 c x^4}\right )}{2 a c}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 a c x^5}\) |
-1/5*((a + b*x)^(5/2)*(c + d*x)^(5/2))/(a*c*x^5) - ((b*c + a*d)*(-1/4*((a + b*x)^(3/2)*(c + d*x)^(5/2))/(c*x^4) + (3*(b*c - a*d)*(-1/3*(Sqrt[a + b*x ]*(c + d*x)^(5/2))/(c*x^3) + ((b*c - a*d)*(-1/2*(Sqrt[a + b*x]*(c + d*x)^( 3/2))/(a*x^2) - (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + ( (b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^( 3/2)*Sqrt[c])))/(4*a)))/(6*c)))/(8*c)))/(2*a*c)
3.7.15.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 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. \(812\) vs. \(2(229)=458\).
Time = 0.55 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.98
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{5} d^{5} x^{5}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{4} b c \,d^{4} x^{5}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} b^{2} c^{2} d^{3} x^{5}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b^{3} c^{3} d^{2} x^{5}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{4} c^{4} d \,x^{5}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{5} c^{5} x^{5}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} d^{4} x^{4}+80 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}-36 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+80 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}-52 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}-52 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}+20 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}-16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}-544 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}-16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}-352 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x -352 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x -256 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{1280 a^{3} c^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{5} \sqrt {a c}}\) | \(813\) |
1/1280*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/ 2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^5*d^5*x^5-45*ln((a*d*x+b*c*x+2*(a*c )^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b*c*d^4*x^5+30*ln((a*d*x+b*c *x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^2*c^2*d^3*x^5+30* ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^3*c^ 3*d^2*x^5-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/ x)*a*b^4*c^4*d*x^5+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2 )+2*a*c)/x)*b^5*c^5*x^5-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*d^4*x^4 +80*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c*d^3*x^4-36*((b*x+a)*(d*x+c ))^(1/2)*(a*c)^(1/2)*a^2*b^2*c^2*d^2*x^4+80*((b*x+a)*(d*x+c))^(1/2)*(a*c)^ (1/2)*a*b^3*c^3*d*x^4-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^4*c^4*x^4+2 0*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3-52*((b*x+a)*(d*x+c))^( 1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3-52*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)* a^2*b^2*c^3*d*x^3+20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^3*c^4*x^3-16* ((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2-544*((b*x+a)*(d*x+c))^ (1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2-16*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a ^2*b^2*c^4*x^2-352*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^3*d*x-352*((b *x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^4*x-256*((b*x+a)*(d*x+c))^(1/2)*a ^4*c^4*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^5/(a*c)^(1/2)
Time = 3.17 (sec) , antiderivative size = 720, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=\left [\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {a c} x^{5} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, a^{4} c^{4} x^{5}}, -\frac {15 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 2 \, a^{2} b^{3} c^{3} d^{2} + 2 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (128 \, a^{5} c^{5} + {\left (15 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d + 18 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 15 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{5} - 13 \, a^{3} b^{2} c^{4} d - 13 \, a^{4} b c^{3} d^{2} + 5 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (a^{3} b^{2} c^{5} + 34 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} x^{2} + 176 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, a^{4} c^{4} x^{5}}\right ] \]
[1/2560*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d ^3 - 3*a^4*b*c*d^4 + a^5*d^5)*sqrt(a*c)*x^5*log((8*a^2*c^2 + (b^2*c^2 + 6* a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a) *sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(128*a^5*c^5 + (15*a*b^ 4*c^5 - 40*a^2*b^3*c^4*d + 18*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 15*a^5* c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 13*a^3*b^2*c^4*d - 13*a^4*b*c^3*d^2 + 5*a^ 5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^5 + 34*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 176* (a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^5), -1/ 1280*(15*(b^5*c^5 - 3*a*b^4*c^4*d + 2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a^4*b*c*d^4 + a^5*d^5)*sqrt(-a*c)*x^5*arctan(1/2*(2*a*c + (b*c + a*d)* x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^ 2 + a^2*c*d)*x)) + 2*(128*a^5*c^5 + (15*a*b^4*c^5 - 40*a^2*b^3*c^4*d + 18* a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 15*a^5*c*d^4)*x^4 - 2*(5*a^2*b^3*c^5 - 13*a^3*b^2*c^4*d - 13*a^4*b*c^3*d^2 + 5*a^5*c^2*d^3)*x^3 + 8*(a^3*b^2*c^ 5 + 34*a^4*b*c^4*d + a^5*c^3*d^2)*x^2 + 176*(a^4*b*c^5 + a^5*c^4*d)*x)*sqr t(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^5)]
\[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=\text {Exception raised: ValueError} \]
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. 5926 vs. \(2 (229) = 458\).
Time = 15.19 (sec) , antiderivative size = 5926, normalized size of antiderivative = 21.71 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=\text {Too large to display} \]
1/640*(15*(sqrt(b*d)*b^6*c^5*abs(b) - 3*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 2*s qrt(b*d)*a^2*b^4*c^3*d^2*abs(b) + 2*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) - 3*s qrt(b*d)*a^4*b^2*c*d^4*abs(b) + sqrt(b*d)*a^5*b*d^5*abs(b))*arctan(-1/2*(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(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^3*b*c^3) - 2*(15*sqrt(b*d)*b ^24*c^14*abs(b) - 190*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 1093*sqrt(b*d)*a^2* b^22*c^12*d^2*abs(b) - 3820*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 9175*sqrt (b*d)*a^4*b^20*c^10*d^4*abs(b) - 16290*sqrt(b*d)*a^5*b^19*c^9*d^5*abs(b) + 22485*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) - 24936*sqrt(b*d)*a^7*b^17*c^7*d^ 7*abs(b) + 22485*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) - 16290*sqrt(b*d)*a^9*b ^15*c^5*d^9*abs(b) + 9175*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) - 3820*sqrt( b*d)*a^11*b^13*c^3*d^11*abs(b) + 1093*sqrt(b*d)*a^12*b^12*c^2*d^12*abs(b) - 190*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) + 15*sqrt(b*d)*a^14*b^10*d^14*abs( b) - 135*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^22*c^13*abs(b) + 1285*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^21*c^12*d*abs(b) - 5170*sqrt(b*d) *(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^2 0*c^11*d^2*abs(b) + 11550*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^19*c^10*d^3*abs(b) - 15725*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^18...
Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^6} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x^6} \,d x \]