Integrand size = 22, antiderivative size = 334 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=-\frac {5 \left (b^3 c^3-19 a b^2 c^2 d-45 a^2 b c d^2-a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b d}-\frac {5 \left (b^2 c^2-18 a b c d-31 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 d}+\frac {5 b (b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {5}{4} b (a+b x)^{3/2} (c+d x)^{5/2}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}-5 a^{3/2} c^{3/2} (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{3/2} d^{3/2}} \]
5/4*b*(b*x+a)^(3/2)*(d*x+c)^(5/2)-(b*x+a)^(5/2)*(d*x+c)^(5/2)/x-5*a^(3/2)* c^(3/2)*(a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))-5/6 4*(a^4*d^4-20*a^3*b*c*d^3-90*a^2*b^2*c^2*d^2-20*a*b^3*c^3*d+b^4*c^4)*arcta nh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/d^(3/2)-5/96*(-31* a^2*d^2-18*a*b*c*d+b^2*c^2)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/d+5/24*b*(7*a*d+b* c)*(d*x+c)^(5/2)*(b*x+a)^(1/2)/d-5/64*(-a^3*d^3-45*a^2*b*c*d^2-19*a*b^2*c^ 2*d+b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b/d
Time = 1.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^3 d^3 x+a^2 b d \left (-192 c^2+601 c d x+118 d^2 x^2\right )+a b^2 d x \left (601 c^2+452 c d x+136 d^2 x^2\right )+b^3 x \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b d x}-5 a^{3/2} c^{3/2} (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )-\frac {5 \left (b^4 c^4-20 a b^3 c^3 d-90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{3/2} d^{3/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^3*d^3*x + a^2*b*d*(-192*c^2 + 601*c*d*x + 118*d^2*x^2) + a*b^2*d*x*(601*c^2 + 452*c*d*x + 136*d^2*x^2) + b^3*x*(1 5*c^3 + 118*c^2*d*x + 136*c*d^2*x^2 + 48*d^3*x^3)))/(192*b*d*x) - 5*a^(3/2 )*c^(3/2)*(b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b* x])] - (5*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*b ^(3/2)*d^(3/2))
Time = 0.54 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {108, 27, 171, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \int \frac {5 (a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{2 x}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} (b c+a d+2 b d x)}{x}dx-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{2} \left (\frac {\int \frac {d \sqrt {a+b x} (c+d x)^{3/2} (4 a (b c+a d)+b (b c+7 a d) x)}{x}dx}{4 d}+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} (4 a (b c+a d)+b (b c+7 a d) x)}{x}dx+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\int \frac {(c+d x)^{3/2} \left (24 a^2 d (b c+a d)-b \left (b^2 c^2-18 a b d c-31 a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x}}dx}{3 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\int \frac {(c+d x)^{3/2} \left (24 a^2 d (b c+a d)-b \left (b^2 c^2-18 a b d c-31 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {3 b \sqrt {c+d x} \left (32 a^2 c d (b c+a d)-\left (b^3 c^3-19 a b^2 d c^2-45 a^2 b d^2 c-a^3 d^3\right ) x\right )}{2 x \sqrt {a+b x}}dx}{2 b}-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \int \frac {\sqrt {c+d x} \left (32 a^2 c d (b c+a d)-\left (b^3 c^3-19 a b^2 d c^2-45 a^2 b d^2 c-a^3 d^3\right ) x\right )}{x \sqrt {a+b x}}dx-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \left (\frac {\int \frac {64 a^2 b c^2 d (b c+a d)-\left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+a^4 d^4\right ) x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{b}\right )-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \left (\frac {\int \frac {64 a^2 b c^2 d (b c+a d)-\left (b^4 c^4-20 a b^3 d c^3-90 a^2 b^2 d^2 c^2-20 a^3 b d^3 c+a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{b}\right )-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \left (\frac {64 a^2 b c^2 d (a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-\left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{b}\right )-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \left (\frac {64 a^2 b c^2 d (a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{b}\right )-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \left (\frac {128 a^2 b c^2 d (a d+b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-2 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{b}\right )-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {5}{2} \left (\frac {1}{4} \left (\frac {\frac {3}{4} \left (\frac {-128 a^{3/2} b c^{3/2} d (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {2 \left (a^4 d^4-20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^3 d^3-45 a^2 b c d^2-19 a b^2 c^2 d+b^3 c^3\right )}{b}\right )-\frac {1}{2} \sqrt {a+b x} (c+d x)^{3/2} \left (-31 a^2 d^2-18 a b c d+b^2 c^2\right )}{6 d}+\frac {b \sqrt {a+b x} (c+d x)^{5/2} (7 a d+b c)}{3 d}\right )+\frac {1}{2} b (a+b x)^{3/2} (c+d x)^{5/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x}\) |
-(((a + b*x)^(5/2)*(c + d*x)^(5/2))/x) + (5*((b*(a + b*x)^(3/2)*(c + d*x)^ (5/2))/2 + ((b*(b*c + 7*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(3*d) + (-1/2* ((b^2*c^2 - 18*a*b*c*d - 31*a^2*d^2)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + (3*( -(((b^3*c^3 - 19*a*b^2*c^2*d - 45*a^2*b*c*d^2 - a^3*d^3)*Sqrt[a + b*x]*Sqr t[c + d*x])/b) + (-128*a^(3/2)*b*c^(3/2)*d*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sq rt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - (2*(b^4*c^4 - 20*a*b^3*c^3*d - 90* a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x] )/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[d]))/(2*b)))/4)/(6*d))/4))/2
3.7.68.3.1 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[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[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. \(816\) vs. \(2(280)=560\).
Time = 0.57 (sec) , antiderivative size = 817, normalized size of antiderivative = 2.45
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-96 b^{3} d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-272 a \,b^{2} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}-272 b^{3} c \,d^{2} x^{3} \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^{4} d^{4} x -300 \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} b 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 ) \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2} x -300 \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^{3} 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 ) \sqrt {a c}\, b^{4} c^{4} x +960 \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} b \,c^{2} d^{2} x +960 \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^{2} c^{3} d x -236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,d^{3} x^{2}-904 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c \,d^{2} x^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{2} d \,x^{2}-30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} d^{3} x -1202 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b c \,d^{2} x -1202 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{2} d x -30 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, b^{3} c^{3} x +384 a^{2} b \,c^{2} d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{384 b d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, x \sqrt {a c}}\) | \(817\) |
-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-96*b^3*d^3*x^4*((b*x+a)*(d*x+c))^(1/2 )*(b*d)^(1/2)*(a*c)^(1/2)-272*a*b^2*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^ (1/2)*(a*c)^(1/2)-272*b^3*c*d^2*x^3*((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^4*d^4*x-300*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*b*c*d^3*x-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*c)^(1/2)*a^2*b^2*c^2*d^2*x-300*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^3*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))*(a*c) ^(1/2)*b^4*c^4*x+960*(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*b*c^2*d^2*x+960*(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^2*c^3*d*x-236*((b*x+a)* (d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*d^3*x^2-904*((b*x+a)*(d*x+c)) ^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2*c*d^2*x^2-236*((b*x+a)*(d*x+c))^(1/2) *(b*d)^(1/2)*(a*c)^(1/2)*b^3*c^2*d*x^2-30*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1 /2)*(a*c)^(1/2)*a^3*d^3*x-1202*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^( 1/2)*a^2*b*c*d^2*x-1202*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a* b^2*c^2*d*x-30*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*b^3*c^3*x+3 84*a^2*b*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2))/b/d/((b...
Time = 8.14 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.83 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\text {Too large to display} \]
[1/768*(15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*d)*x*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) + 960*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*sqrt(a*c)*x*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* (48*b^4*d^4*x^4 - 192*a^2*b^2*c^2*d^2 + 136*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 2*(59*b^4*c^2*d^2 + 226*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x^2 + (15*b^4*c^3*d + 601*a*b^3*c^2*d^2 + 601*a^2*b^2*c*d^3 + 15*a^3*b*d^4)*x)*sqrt(b*x + a)*s qrt(d*x + c))/(b^2*d^2*x), 1/384*(15*(b^4*c^4 - 20*a*b^3*c^3*d - 90*a^2*b^ 2*c^2*d^2 - 20*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b*d)*x*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)) + 480*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*sqrt(a*c)*x* 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*(48*b^4*d^4*x^4 - 192*a^2*b^2*c^2*d^2 + 136*(b^4*c*d^3 + a*b^3*d^4) *x^3 + 2*(59*b^4*c^2*d^2 + 226*a*b^3*c*d^3 + 59*a^2*b^2*d^4)*x^2 + (15*b^4 *c^3*d + 601*a*b^3*c^2*d^2 + 601*a^2*b^2*c*d^3 + 15*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^2*x), 1/768*(1920*(a*b^3*c^2*d^2 + a^2*b^2*c*d ^3)*sqrt(-a*c)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x...
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (280) = 560\).
Time = 1.00 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.27 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{2}} + \frac {17 \, b^{3} c d^{7} {\left | b \right |} - a b^{2} d^{8} {\left | b \right |}}{b^{4} d^{6}}\right )} + \frac {59 \, b^{4} c^{2} d^{6} {\left | b \right |} + 90 \, a b^{3} c d^{7} {\left | b \right |} - 5 \, a^{2} b^{2} d^{8} {\left | b \right |}}{b^{4} d^{6}}\right )} {\left (b x + a\right )} + \frac {3 \, {\left (5 \, b^{5} c^{3} d^{5} {\left | b \right |} + 161 \, a b^{4} c^{2} d^{6} {\left | b \right |} + 95 \, a^{2} b^{3} c d^{7} {\left | b \right |} - 5 \, a^{3} b^{2} d^{8} {\left | b \right |}\right )}}{b^{4} d^{6}}\right )} \sqrt {b x + a} - \frac {1920 \, {\left (\sqrt {b d} a^{2} b^{2} c^{3} {\left | b \right |} + \sqrt {b d} a^{3} b c^{2} d {\left | b \right |}\right )} \arctan \left (-\frac {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}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {768 \, {\left (\sqrt {b d} a^{2} b^{4} c^{4} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c^{3} d {\left | b \right |} + \sqrt {b d} a^{4} b^{2} c^{2} d^{2} {\left | b \right |} - \sqrt {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} a^{2} b^{2} c^{3} {\left | b \right |} - \sqrt {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} a^{3} b c^{2} d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} 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 )}^{4}} + \frac {15 \, {\left (b^{4} c^{4} {\left | b \right |} - 20 \, a b^{3} c^{3} d {\left | b \right |} - 90 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 20 \, a^{3} b c d^{3} {\left | b \right |} + 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 )}{\sqrt {b d} b d}}{384 \, b} \]
1/384*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(6*(b*x + a)* d^2*abs(b)/b^2 + (17*b^3*c*d^7*abs(b) - a*b^2*d^8*abs(b))/(b^4*d^6)) + (59 *b^4*c^2*d^6*abs(b) + 90*a*b^3*c*d^7*abs(b) - 5*a^2*b^2*d^8*abs(b))/(b^4*d ^6))*(b*x + a) + 3*(5*b^5*c^3*d^5*abs(b) + 161*a*b^4*c^2*d^6*abs(b) + 95*a ^2*b^3*c*d^7*abs(b) - 5*a^3*b^2*d^8*abs(b))/(b^4*d^6))*sqrt(b*x + a) - 192 0*(sqrt(b*d)*a^2*b^2*c^3*abs(b) + sqrt(b*d)*a^3*b*c^2*d*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) - 768*(sqrt(b*d)*a^2*b^ 4*c^4*abs(b) - 2*sqrt(b*d)*a^3*b^3*c^3*d*abs(b) + sqrt(b*d)*a^4*b^2*c^2*d^ 2*abs(b) - 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*c^3*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr t(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b*c^2*d*abs(b))/(b^4*c^2 - 2*a*b^3 *c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b *d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a) *b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a) *b*d - a*b*d))^4) + 15*(b^4*c^4*abs(b) - 20*a*b^3*c^3*d*abs(b) - 90*a^2*b^ 2*c^2*d^2*abs(b) - 20*a^3*b*c*d^3*abs(b) + 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*d))/b
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{5/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^2} \,d x \]