Integrand size = 22, antiderivative size = 283 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\frac {(b c-a d)^3 (7 b c+3 a d) \sqrt {a+b x} \sqrt {c+d x}}{128 a^4 c^2 x}-\frac {(b c-a d)^2 (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{192 a^3 c^2 x^2}+\frac {(b c-a d) (7 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{240 a^2 c^2 x^3}+\frac {(7 b c+3 a d) \sqrt {a+b x} (c+d x)^{7/2}}{40 a c^2 x^4}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}-\frac {(b c-a d)^4 (7 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}} \]
-1/5*(b*x+a)^(3/2)*(d*x+c)^(7/2)/a/c/x^5-1/128*(-a*d+b*c)^4*(3*a*d+7*b*c)* arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(5/2)-1/192 *(-a*d+b*c)^2*(3*a*d+7*b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a^3/c^2/x^2+1/240* (-a*d+b*c)*(3*a*d+7*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/a^2/c^2/x^3+1/40*(3*a *d+7*b*c)*(d*x+c)^(7/2)*(b*x+a)^(1/2)/a/c^2/x^4+1/128*(-a*d+b*c)^3*(3*a*d+ 7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2/x
Time = 0.54 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^4 c^4 x^4-10 a b^3 c^3 x^3 (7 c+34 d x)+2 a^2 b^2 c^2 x^2 \left (28 c^2+111 c d x+173 d^2 x^2\right )-2 a^3 b c x \left (24 c^3+88 c^2 d x+109 c d^2 x^2+30 d^3 x^3\right )-3 a^4 \left (128 c^4+336 c^3 d x+248 c^2 d^2 x^2+10 c d^3 x^3-15 d^4 x^4\right )\right )}{1920 a^4 c^2 x^5}-\frac {(b c-a d)^4 (7 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{5/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^4*c^4*x^4 - 10*a*b^3*c^3*x^3*(7*c + 34 *d*x) + 2*a^2*b^2*c^2*x^2*(28*c^2 + 111*c*d*x + 173*d^2*x^2) - 2*a^3*b*c*x *(24*c^3 + 88*c^2*d*x + 109*c*d^2*x^2 + 30*d^3*x^3) - 3*a^4*(128*c^4 + 336 *c^3*d*x + 248*c^2*d^2*x^2 + 10*c*d^3*x^3 - 15*d^4*x^4)))/(1920*a^4*c^2*x^ 5) - ((b*c - a*d)^4*(7*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[ a]*Sqrt[c + d*x])])/(128*a^(9/2)*c^(5/2))
Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle -\frac {(3 a d+7 b c) \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^5}dx}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+7 b c) \left (\frac {(b c-a d) \int \frac {(c+d x)^{5/2}}{x^4 \sqrt {a+b x}}dx}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+7 b c) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+7 b c) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {(3 a d+7 b c) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {(3 a d+7 b c) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(3 a d+7 b c) \left (\frac {(b c-a d) \left (-\frac {5 (b c-a d) \left (-\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x} (c+d x)^{5/2}}{3 a x^3}\right )}{8 c}-\frac {\sqrt {a+b x} (c+d x)^{7/2}}{4 c x^4}\right )}{10 a c}-\frac {(a+b x)^{3/2} (c+d x)^{7/2}}{5 a c x^5}\) |
-1/5*((a + b*x)^(3/2)*(c + d*x)^(7/2))/(a*c*x^5) - ((7*b*c + 3*a*d)*(-1/4* (Sqrt[a + b*x]*(c + d*x)^(7/2))/(c*x^4) + ((b*c - a*d)*(-1/3*(Sqrt[a + b*x ]*(c + d*x)^(5/2))/(a*x^3) - (5*(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*a)))/(8*c)))/(10*a*c)
3.6.74.3.1 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. \(2(239)=478\).
Time = 0.53 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.87
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (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^{5} d^{5} x^{5}-75 \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}-150 \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}+450 \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}-375 \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}+105 \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}-90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} d^{4} x^{4}+120 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}-692 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}+680 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}-210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}+436 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}-444 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}+1488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+352 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+2016 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x +96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{4} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{5} \sqrt {a c}}\) | \(813\) |
-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^2*(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^5*d^5*x^5-75*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^5-150*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+4 50*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^5-375*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+105*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^5-90*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*d^ 4*x^4+120*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c*d^3*x^4-692*((b*x+a) *(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^2*c^2*d^2*x^4+680*((b*x+a)*(d*x+c))^(1/2 )*(a*c)^(1/2)*a*b^3*c^3*d*x^4-210*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^4* c^4*x^4+60*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3+436*((b*x+a)* (d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3-444*((b*x+a)*(d*x+c))^(1/2)*( a*c)^(1/2)*a^2*b^2*c^3*d*x^3+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^3 *c^4*x^3+1488*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2+352*((b* x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2-112*((b*x+a)*(d*x+c))^(1/2 )*(a*c)^(1/2)*a^2*b^2*c^4*x^2+2016*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4 *c^3*d*x+96*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^4*x+768*((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 = 2.73 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.59 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{3} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 25 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 3 \, 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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 340 \, a^{2} b^{3} c^{4} d + 346 \, a^{3} b^{2} c^{3} d^{2} - 60 \, a^{4} b c^{2} d^{3} + 45 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 111 \, a^{3} b^{2} c^{4} d + 109 \, a^{4} b c^{3} d^{2} + 15 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 22 \, a^{4} b c^{4} d - 93 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + 21 \, a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{3} x^{5}}\right ] \]
[1/7680*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 10*a^3*b^2* c^2*d^3 - 5*a^4*b*c*d^4 + 3*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*(384*a^5*c^5 - ( 105*a*b^4*c^5 - 340*a^2*b^3*c^4*d + 346*a^3*b^2*c^3*d^2 - 60*a^4*b*c^2*d^3 + 45*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 111*a^3*b^2*c^4*d + 109*a^4*b*c ^3*d^2 + 15*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 22*a^4*b*c^4*d - 93*a^5* c^3*d^2)*x^2 + 48*(a^4*b*c^5 + 21*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c ))/(a^5*c^3*x^5), 1/3840*(15*(7*b^5*c^5 - 25*a*b^4*c^4*d + 30*a^2*b^3*c^3* d^2 - 10*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 + 3*a^5*d^5)*sqrt(-a*c)*x^5*arcta n(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*(384*a^5*c^5 - (105*a*b^4* c^5 - 340*a^2*b^3*c^4*d + 346*a^3*b^2*c^3*d^2 - 60*a^4*b*c^2*d^3 + 45*a^5* c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 111*a^3*b^2*c^4*d + 109*a^4*b*c^3*d^2 + 1 5*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 22*a^4*b*c^4*d - 93*a^5*c^3*d^2)*x ^2 + 48*(a^4*b*c^5 + 21*a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^ 3*x^5)]
\[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\int \frac {\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\text {Exception raised: ValueError} \]
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. 5928 vs. \(2 (239) = 478\).
Time = 15.10 (sec) , antiderivative size = 5928, normalized size of antiderivative = 20.95 \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\text {Too large to display} \]
-1/1920*(15*(7*sqrt(b*d)*b^6*c^5*abs(b) - 25*sqrt(b*d)*a*b^5*c^4*d*abs(b) + 30*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) - 10*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b ) - 5*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) + 3*sqrt(b*d)*a^5*b*d^5*abs(b))*arcta n(-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^4*b*c^2) - 2*(105*s qrt(b*d)*b^24*c^14*abs(b) - 1390*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 8471*sqr t(b*d)*a^2*b^22*c^12*d^2*abs(b) - 31420*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 79065*sqrt(b*d)*a^4*b^20*c^10*d^4*abs(b) - 142530*sqrt(b*d)*a^5*b^19*c^ 9*d^5*abs(b) + 189615*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) - 189192*sqrt(b*d) *a^7*b^17*c^7*d^7*abs(b) + 142755*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) - 8181 0*sqrt(b*d)*a^9*b^15*c^5*d^9*abs(b) + 35725*sqrt(b*d)*a^10*b^14*c^4*d^10*a bs(b) - 11900*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) + 2971*sqrt(b*d)*a^12*b^ 12*c^2*d^12*abs(b) - 510*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) + 45*sqrt(b*d)* a^14*b^10*d^14*abs(b) - 945*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) + 9535*sqrt(b*d)*(sqrt(b*d) *sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^21*c^12*d*abs( b) - 41870*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^20*c^11*d^2*abs(b) + 102090*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) - 141675*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*...
Timed out. \[ \int \frac {\sqrt {a+b x} (c+d x)^{5/2}}{x^6} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}}{x^6} \,d x \]