Integrand size = 22, antiderivative size = 340 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=-\frac {(3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 c x^4}-\frac {\left (\frac {3 b^2 c}{a}+12 b d-\frac {7 a d^2}{c}\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 c x^3}+\frac {\left (15 b^3 c^3-9 a b^2 c^2 d+61 a^2 b c d^2-35 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^2 c^3 x^2}-\frac {\left (45 b^4 c^4-30 a b^3 c^3 d-36 a^2 b^2 c^2 d^2+190 a^3 b c d^3-105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^3 c^4 x}-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{7/2} c^{9/2}} \]
1/128*(-a*d+b*c)^3*(7*a^2*d^2+6*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2)*(b*x+a) ^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(9/2)-1/5*(b*x+a)^(3/2)*(d*x+c)^(1 /2)/x^5-1/40*(a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^4-1/240*(3*b^2*c/ a+12*b*d-7*a*d^2/c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c/x^3+1/960*(-35*a^3*d^3+6 1*a^2*b*c*d^2-9*a*b^2*c^2*d+15*b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^ 3/x^2-1/1920*(-105*a^4*d^4+190*a^3*b*c*d^3-36*a^2*b^2*c^2*d^2-30*a*b^3*c^3 *d+45*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^4/x
Time = 0.54 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (45 b^4 c^4 x^4-30 a b^3 c^3 x^3 (c+d x)+6 a^2 b^2 c^2 x^2 \left (4 c^2+3 c d x-6 d^2 x^2\right )+2 a^3 b c x \left (264 c^3+48 c^2 d x-61 c d^2 x^2+95 d^3 x^3\right )+a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )\right )}{1920 a^3 c^4 x^5}+\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{7/2} c^{9/2}} \]
-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(45*b^4*c^4*x^4 - 30*a*b^3*c^3*x^3*(c + d*x) + 6*a^2*b^2*c^2*x^2*(4*c^2 + 3*c*d*x - 6*d^2*x^2) + 2*a^3*b*c*x*(2 64*c^3 + 48*c^2*d*x - 61*c*d^2*x^2 + 95*d^3*x^3) + a^4*(384*c^4 + 48*c^3*d *x - 56*c^2*d^2*x^2 + 70*c*d^3*x^3 - 105*d^4*x^4)))/(a^3*c^4*x^5) + ((b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x ])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(7/2)*c^(9/2))
Time = 0.48 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {108, 27, 166, 27, 168, 27, 168, 27, 168, 27, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{5} \int \frac {\sqrt {a+b x} (3 b c+a d+4 b d x)}{2 x^5 \sqrt {c+d x}}dx-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \frac {\sqrt {a+b x} (3 b c+a d+4 b d x)}{x^5 \sqrt {c+d x}}dx-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{10} \left (\frac {\int \frac {3 b^2 c^2+12 a b d c-7 a^2 d^2+2 b d (7 b c-3 a d) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {\int \frac {3 b^2 c^2+12 a b d c-7 a^2 d^2+2 b d (7 b c-3 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3+4 b d \left (3 b^2 c^2+12 a b d c-7 a^2 d^2\right ) x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {\int \frac {15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3+4 b d \left (3 b^2 c^2+12 a b d c-7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {\int \frac {45 b^4 c^4-30 a b^3 d c^3-36 a^2 b^2 d^2 c^2+190 a^3 b d^3 c-105 a^4 d^4+2 b d \left (15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {\int \frac {45 b^4 c^4-30 a b^3 d c^3-36 a^2 b^2 d^2 c^2+190 a^3 b d^3 c-105 a^4 d^4+2 b d \left (15 b^3 c^3-9 a b^2 d c^2+61 a^2 b d^2 c-35 a^3 d^3\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {-\frac {\int \frac {15 (b c-a d)^3 \left (3 b^2 c^2+6 a b d c+7 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {-\frac {15 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {-\frac {15 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{10} \left (\frac {-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-35 a^3 d^3+61 a^2 b c d^2-9 a b^2 c^2 d+15 b^3 c^3\right )}{2 a c x^2}-\frac {\frac {15 (b c-a d)^3 \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 a^4 d^4+190 a^3 b c d^3-36 a^2 b^2 c^2 d^2-30 a b^3 c^3 d+45 b^4 c^4\right )}{a c x}}{4 a c}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3 b^2 c}{a}-\frac {7 a d^2}{c}+12 b d\right )}{3 x^3}}{8 c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c x^4}\right )-\frac {(a+b x)^{3/2} \sqrt {c+d x}}{5 x^5}\) |
-1/5*((a + b*x)^(3/2)*Sqrt[c + d*x])/x^5 + (-1/4*((3*b*c + a*d)*Sqrt[a + b *x]*Sqrt[c + d*x])/(c*x^4) + (-1/3*(((3*b^2*c)/a + 12*b*d - (7*a*d^2)/c)*S qrt[a + b*x]*Sqrt[c + d*x])/x^3 - (-1/2*((15*b^3*c^3 - 9*a*b^2*c^2*d + 61* a^2*b*c*d^2 - 35*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^2) - (-(((45 *b^4*c^4 - 30*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 190*a^3*b*c*d^3 - 105*a^4 *d^4)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (15*(b*c - a*d)^3*(3*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*c^(3/2)))/(4*a*c))/(6*a*c))/(8*c))/10
3.7.6.3.1 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_))/((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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[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. \(2(296)=592\).
Time = 1.54 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.39
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (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 ) a^{5} d^{5} x^{5}-225 \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}+90 \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}-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 ) b^{5} c^{5} x^{5}-210 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} d^{4} x^{4}+380 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b c \,d^{3} x^{4}-72 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x^{4}-60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{3} d \,x^{4}+90 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, b^{4} c^{4} x^{4}+140 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}-244 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}+36 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}-60 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+192 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}+48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x +1056 \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^{3} c^{4} \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^3/c^4*(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)*a^5*d^5*x^5-225*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+90*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^5-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 1/2)+2*a*c)/x)*b^5*c^5*x^5-210*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*d^4 *x^4+380*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c*d^3*x^4-72*((b*x+a)*( d*x+c))^(1/2)*(a*c)^(1/2)*a^2*b^2*c^2*d^2*x^4-60*((b*x+a)*(d*x+c))^(1/2)*( a*c)^(1/2)*a*b^3*c^3*d*x^4+90*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^4*c^4* x^4+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3-244*((b*x+a)*(d* x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3+36*((b*x+a)*(d*x+c))^(1/2)*(a*c) ^(1/2)*a^2*b^2*c^3*d*x^3-60*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^3*c^4* x^3-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2+192*((b*x+a)*( d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2+48*((b*x+a)*(d*x+c))^(1/2)*(a*c) ^(1/2)*a^2*b^2*c^4*x^2+96*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^3*d*x+ 1056*((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 = 3.17 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\left [-\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, 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 (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{4} c^{5} x^{5}}, -\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, 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 (45 \, a b^{4} c^{5} - 30 \, a^{2} b^{3} c^{4} d - 36 \, a^{3} b^{2} c^{3} d^{2} + 190 \, a^{4} b c^{2} d^{3} - 105 \, a^{5} c d^{4}\right )} x^{4} - 2 \, {\left (15 \, a^{2} b^{3} c^{5} - 9 \, a^{3} b^{2} c^{4} d + 61 \, a^{4} b c^{3} d^{2} - 35 \, a^{5} c^{2} d^{3}\right )} x^{3} + 8 \, {\left (3 \, a^{3} b^{2} c^{5} + 12 \, a^{4} b c^{4} d - 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (11 \, a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{4} c^{5} x^{5}}\right ] \]
[-1/7680*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^ 2*d^3 + 15*a^4*b*c*d^4 - 7*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 + (4 5*a*b^4*c^5 - 30*a^2*b^3*c^4*d - 36*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x^4 - 2*(15*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 61*a^4*b*c^3*d^ 2 - 35*a^5*c^2*d^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^ 2)*x^2 + 48*(11*a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^ 4*c^5*x^5), -1/3840*(15*(3*b^5*c^5 - 3*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 6 *a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 7*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*(384*a^5*c^5 + (45*a*b^4*c^5 - 30 *a^2*b^3*c^4*d - 36*a^3*b^2*c^3*d^2 + 190*a^4*b*c^2*d^3 - 105*a^5*c*d^4)*x ^4 - 2*(15*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 61*a^4*b*c^3*d^2 - 35*a^5*c^2*d ^3)*x^3 + 8*(3*a^3*b^2*c^5 + 12*a^4*b*c^4*d - 7*a^5*c^3*d^2)*x^2 + 48*(11* a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^5*x^5)]
\[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\text {Exception raised: ValueError} \]
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. 5872 vs. \(2 (296) = 592\).
Time = 14.87 (sec) , antiderivative size = 5872, normalized size of antiderivative = 17.27 \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\text {Too large to display} \]
1/1920*(15*(3*sqrt(b*d)*b^6*c^5*abs(b) - 3*sqrt(b*d)*a*b^5*c^4*d*abs(b) - 2*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) - 6*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) + 15*sqrt(b*d)*a^4*b^2*c*d^4*abs(b) - 7*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^4) - 2*(45*sqrt( b*d)*b^24*c^14*abs(b) - 480*sqrt(b*d)*a*b^23*c^13*d*abs(b) + 2289*sqrt(b*d )*a^2*b^22*c^12*d^2*abs(b) - 6200*sqrt(b*d)*a^3*b^21*c^11*d^3*abs(b) + 942 5*sqrt(b*d)*a^4*b^20*c^10*d^4*abs(b) - 3720*sqrt(b*d)*a^5*b^19*c^9*d^5*abs (b) - 18075*sqrt(b*d)*a^6*b^18*c^8*d^6*abs(b) + 49872*sqrt(b*d)*a^7*b^17*c ^7*d^7*abs(b) - 71865*sqrt(b*d)*a^8*b^16*c^6*d^8*abs(b) + 68880*sqrt(b*d)* a^9*b^15*c^5*d^9*abs(b) - 46125*sqrt(b*d)*a^10*b^14*c^4*d^10*abs(b) + 2148 0*sqrt(b*d)*a^11*b^13*c^3*d^11*abs(b) - 6661*sqrt(b*d)*a^12*b^12*c^2*d^12* abs(b) + 1240*sqrt(b*d)*a^13*b^11*c*d^13*abs(b) - 105*sqrt(b*d)*a^14*b^10* d^14*abs(b) - 405*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) + 3045*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) - 9150* 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) + 10550*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^3*b^19*c^10*d^3*abs(b) + 12625*sqrt (b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2...
Timed out. \[ \int \frac {(a+b x)^{3/2} \sqrt {c+d x}}{x^6} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}}{x^6} \,d x \]