Integrand size = 22, antiderivative size = 319 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} (3 b c+a d) (b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} \sqrt {d}} \]
-5/4*(a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(5/2)/c/x-1/2*(b*x+a)^(5/2)*(d*x+c)^( 5/2)/x^2-5/4*(a*d+3*b*c)*(3*a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2) /(d*x+c)^(1/2))*a^(1/2)*c^(1/2)+5/8*(a*d+b*c)*(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^(1/2)/d^(1/2)+5/12 *(3*a^2*d^2+8*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/c+5/12*b*(3*a*d +5*b*c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/c+5/8*(5*a^2*d^2+10*a*b*c*d+b^2*c^2)*( b*x+a)^(1/2)*(d*x+c)^(1/2)
Time = 0.95 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\frac {1}{24} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^2 \left (4 c^2+18 c d x-11 d^2 x^2\right )+b^2 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )+2 a b x \left (-27 c^2+61 c d x+13 d^2 x^2\right )\right )}{x^2}-30 \sqrt {a} \sqrt {c} \left (3 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {15 \left (b^3 c^3+15 a b^2 c^2 d+15 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}}\right ) \]
((Sqrt[a + b*x]*Sqrt[c + d*x]*(-3*a^2*(4*c^2 + 18*c*d*x - 11*d^2*x^2) + b^ 2*x^2*(33*c^2 + 26*c*d*x + 8*d^2*x^2) + 2*a*b*x*(-27*c^2 + 61*c*d*x + 13*d ^2*x^2)))/x^2 - 30*Sqrt[a]*Sqrt[c]*(3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Ar cTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])] + (15*(b^3*c^3 + 15 *a*b^2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/( Sqrt[d]*Sqrt[a + b*x])])/(Sqrt[b]*Sqrt[d]))/24
Time = 0.52 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {108, 27, 166, 27, 171, 27, 171, 27, 171, 27, 175, 66, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{2} \int \frac {5 (a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{2 x^2}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{x^2}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {5}{4} \left (\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} ((3 b c+a d) (b c+3 a d)+2 b d (5 b c+3 a d) x)}{2 x}dx}{c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} \left (\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} ((3 b c+a d) (b c+3 a d)+2 b d (5 b c+3 a d) x)}{x}dx}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {\int \frac {d (c+d x)^{3/2} \left (3 a (3 b c+a d) (b c+3 a d)+4 b \left (b^2 c^2+8 a b d c+3 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx}{3 d}+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \int \frac {(c+d x)^{3/2} \left (3 a (3 b c+a d) (b c+3 a d)+4 b \left (b^2 c^2+8 a b d c+3 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (\frac {\int \frac {6 b c \sqrt {c+d x} \left (a (3 b c+a d) (b c+3 a d)+b \left (b^2 c^2+10 a b d c+5 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx}{2 b}+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \int \frac {\sqrt {c+d x} \left (a (3 b c+a d) (b c+3 a d)+b \left (b^2 c^2+10 a b d c+5 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \left (\frac {\int \frac {b \left (2 a c (3 b c+a d) (b c+3 a d)+(b c+a d) \left (b^2 c^2+14 a b d c+a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}+\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )\right )+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \left (\frac {1}{2} \int \frac {2 a c (3 b c+a d) (b c+3 a d)+(b c+a d) \left (b^2 c^2+14 a b d c+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx+\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )\right )+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \left (\frac {1}{2} \left ((a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+2 a c (a d+3 b c) (3 a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\right )+\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )\right )+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \left (\frac {1}{2} \left (2 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 a c (a d+3 b c) (3 a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\right )+\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )\right )+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \left (\frac {1}{2} \left (2 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+4 a c (a d+3 b c) (3 a d+b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )+\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )\right )+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5}{4} \left (\frac {\frac {1}{3} \left (3 c \left (\frac {1}{2} \left (\frac {2 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}-4 \sqrt {a} \sqrt {c} (a d+3 b c) (3 a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )+\sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )\right )+2 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )\right )+\frac {2}{3} b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{c x}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}\) |
-1/2*((a + b*x)^(5/2)*(c + d*x)^(5/2))/x^2 + (5*(-(((b*c + a*d)*(a + b*x)^ (3/2)*(c + d*x)^(5/2))/(c*x)) + ((2*b*(5*b*c + 3*a*d)*Sqrt[a + b*x]*(c + d *x)^(5/2))/3 + (2*(b^2*c^2 + 8*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*(c + d*x )^(3/2) + 3*c*((b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d *x] + (-4*Sqrt[a]*Sqrt[c]*(3*b*c + a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqr t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + (2*(b*c + a*d)*(b^2*c^2 + 14*a*b*c* d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sq rt[b]*Sqrt[d]))/2))/3)/(2*c)))/4
3.7.69.3.1 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[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_))^(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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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. \(730\) vs. \(2(261)=522\).
Time = 0.58 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.29
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (16 b^{2} d^{2} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}+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 ) \sqrt {a c}\, a^{3} d^{3} x^{2}+225 \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 ) \sqrt {a c}\, a^{2} b c \,d^{2} x^{2}+225 \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 ) \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{2}+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 ) \sqrt {a c}\, b^{3} c^{3} x^{2}-90 \sqrt {b d}\, \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} c \,d^{2} x^{2}-300 \sqrt {b d}\, \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 \,c^{2} d \,x^{2}-90 \sqrt {b d}\, \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^{2} c^{3} x^{2}+52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x^{3}+52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d \,x^{3}+66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}+244 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}+66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}-108 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x -108 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x -24 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2}\right )}{48 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, x^{2} \sqrt {a c}}\) | \(731\) |
1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*b^2*d^2*x^4*((b*x+a)*(d*x+c))^(1/2)*( b*d)^(1/2)*(a*c)^(1/2)+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*c)^(1/2)*a^3*d^3*x^2+225*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*c)^(1/2)*a^2* b*c*d^2*x^2+225*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*c)^(1/2)*a*b^2*c^2*d*x^2+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*c)^(1/2)*b^3*c^3*x^2 -90*(b*d)^(1/2)*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*c*d^2*x^2-300*(b*d)^(1/2)*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*c^2*d*x^2-90*(b*d)^(1/2)*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^2*c^3*x^2+52*(b*d)^(1/2 )*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*d^2*x^3+52*(b*d)^(1/2)*(a*c)^(1/ 2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c*d*x^3+66*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a) *(d*x+c))^(1/2)*a^2*d^2*x^2+244*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^ (1/2)*a*b*c*d*x^2+66*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c ^2*x^2-108*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c*d*x-108*( b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^2*x-24*(b*d)^(1/2)*(a *c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^( 1/2)/x^2/(a*c)^(1/2)
Time = 4.52 (sec) , antiderivative size = 1469, normalized size of antiderivative = 4.61 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\text {Too large to display} \]
[1/96*(15*(b^3*c^3 + 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 + a^3*d^3)*sqrt(b*d)* x^2*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) + 3 0*(3*b^3*c^2*d + 10*a*b^2*c*d^2 + 3*a^2*b*d^3)*sqrt(a*c)*x^2*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*(8*b^3 *d^3*x^4 - 12*a^2*b*c^2*d + 26*(b^3*c*d^2 + a*b^2*d^3)*x^3 + (33*b^3*c^2*d + 122*a*b^2*c*d^2 + 33*a^2*b*d^3)*x^2 - 54*(a*b^2*c^2*d + a^2*b*c*d^2)*x) *sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x^2), -1/48*(15*(b^3*c^3 + 15*a*b^2*c^2 *d + 15*a^2*b*c*d^2 + a^3*d^3)*sqrt(-b*d)*x^2*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)) - 15*(3*b^3*c^2*d + 10*a*b^2*c*d^2 + 3*a^2*b*d^3)*sqrt( a*c)*x^2*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) - 2*(8*b^3*d^3*x^4 - 12*a^2*b*c^2*d + 26*(b^3*c*d^2 + a*b^2*d^ 3)*x^3 + (33*b^3*c^2*d + 122*a*b^2*c*d^2 + 33*a^2*b*d^3)*x^2 - 54*(a*b^2*c ^2*d + a^2*b*c*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x^2), 1/96*(60*(3 *b^3*c^2*d + 10*a*b^2*c*d^2 + 3*a^2*b*d^3)*sqrt(-a*c)*x^2*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)) + 15*(b^3*c^3 + 15*a*b^2*c^2*d + 15*a^...
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{3}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\text {Exception raised: ValueError} \]
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. 1315 vs. \(2 (261) = 522\).
Time = 1.52 (sec) , antiderivative size = 1315, normalized size of antiderivative = 4.12 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\text {Too large to display} \]
1/48*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*d^2*abs(b)/b + (13*b*c*d^5*abs(b) + 5*a*d^6*abs(b))/(b*d^4))*(b*x + a) + 3*(11*b^2*c^2*d ^4*abs(b) + 32*a*b*c*d^5*abs(b) + 5*a^2*d^6*abs(b))/(b*d^4))*sqrt(b*x + a) - 15*(b^3*c^3*abs(b) + 15*a*b^2*c^2*d*abs(b) + 15*a^2*b*c*d^2*abs(b) + a^ 3*d^3*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) - 60*(3*sqrt(b*d)*a*b^3*c^3*abs(b) + 10*sqrt(b*d)*a^2 *b^2*c^2*d*abs(b) + 3*sqrt(b*d)*a^3*b*c*d^2*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)*b) - 24*(9*sqrt(b*d)*a*b^9*c^6*abs(b) - 27*sqrt(b*d)*a^2*b^8*c^5*d*abs(b) + 18*sqrt(b*d)*a^3*b^7*c^4*d^2*abs(b) + 18*sqrt(b*d)*a^4*b^6*c^3*d^3*abs(b) - 27*sqrt(b*d)*a^5*b^5*c^2*d^4*abs(b ) + 9*sqrt(b*d)*a^6*b^4*c*d^5*abs(b) - 27*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^7*c^5*abs(b) + 4*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^6 *c^4*d*abs(b) + 46*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^5*c^3*d^2*abs(b) + 4*sqrt(b*d)*(sqrt(b*d)*sqrt( b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^4*c^2*d^3*abs(b) - 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* d))^2*a^5*b^3*c*d^4*abs(b) + 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt( b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^5*c^4*abs(b) + 45*sqrt(b*d)*(sqrt...
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^3} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^3} \,d x \]