Integrand size = 22, antiderivative size = 259 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=-\frac {\left (b^2 c^2-14 a b c d-19 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d}+\frac {b (b c+7 a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {4}{3} b (a+b x)^{3/2} (c+d x)^{3/2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}-a^{3/2} \sqrt {c} (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{3/2}} \]
4/3*b*(b*x+a)^(3/2)*(d*x+c)^(3/2)-(b*x+a)^(5/2)*(d*x+c)^(3/2)/x-1/8*(-5*a^ 3*d^3-45*a^2*b*c*d^2-15*a*b^2*c^2*d+b^3*c^3)*arctanh(d^(1/2)*(b*x+a)^(1/2) /b^(1/2)/(d*x+c)^(1/2))/d^(3/2)/b^(1/2)-a^(3/2)*(3*a*d+5*b*c)*arctanh(c^(1 /2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))*c^(1/2)+1/4*b*(7*a*d+b*c)*(d*x+c) ^(3/2)*(b*x+a)^(1/2)/d-1/8*(-19*a^2*d^2-14*a*b*c*d+b^2*c^2)*(b*x+a)^(1/2)* (d*x+c)^(1/2)/d
Time = 0.73 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d (-8 c+11 d x)+2 a b d x (34 c+13 d x)+b^2 x \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 d x}-a^{3/2} \sqrt {c} (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (b^3 c^3-15 a b^2 c^2 d-45 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{3/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(3*a^2*d*(-8*c + 11*d*x) + 2*a*b*d*x*(34*c + 13*d*x) + b^2*x*(3*c^2 + 14*c*d*x + 8*d^2*x^2)))/(24*d*x) - a^(3/2)*Sqrt[c ]*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - ((b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[ d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(8*Sqrt[b]*d^(3/2))
Time = 0.43 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {108, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \int \frac {(a+b x)^{3/2} \sqrt {c+d x} (5 b c+3 a d+8 b d x)}{2 x}dx-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} (5 b c+3 a d+8 b d x)}{x}dx-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {3 d \sqrt {a+b x} \sqrt {c+d x} (a (5 b c+3 a d)+b (b c+7 a d) x)}{x}dx}{3 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {\sqrt {a+b x} \sqrt {c+d x} (a (5 b c+3 a d)+b (b c+7 a d) x)}{x}dx+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {c+d x} \left (4 a^2 d (5 b c+3 a d)-b \left (b^2 c^2-14 a b d c-19 a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x}}dx}{2 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {c+d x} \left (4 a^2 d (5 b c+3 a d)-b \left (b^2 c^2-14 a b d c-19 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {b \left (8 a^2 c d (5 b c+3 a d)-\left (b^3 c^3-15 a b^2 d c^2-45 a^2 b d^2 c-5 a^3 d^3\right ) x\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}-\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \int \frac {8 a^2 c d (5 b c+3 a d)-\left (b^3 c^3-15 a b^2 d c^2-45 a^2 b d^2 c-5 a^3 d^3\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx-\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \left (8 a^2 c d (3 a d+5 b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx\right )-\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \left (8 a^2 c d (3 a d+5 b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 \left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \left (16 a^2 c d (3 a d+5 b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-2 \left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {1}{2} \left (-16 a^{3/2} \sqrt {c} d (3 a d+5 b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {2 \left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}\right )-\sqrt {a+b x} \sqrt {c+d x} \left (-19 a^2 d^2-14 a b c d+b^2 c^2\right )}{4 d}+\frac {8}{3} b (a+b x)^{3/2} (c+d x)^{3/2}+\frac {b \sqrt {a+b x} (c+d x)^{3/2} (7 a d+b c)}{2 d}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x}\) |
-(((a + b*x)^(5/2)*(c + d*x)^(3/2))/x) + ((b*(b*c + 7*a*d)*Sqrt[a + b*x]*( c + d*x)^(3/2))/(2*d) + (8*b*(a + b*x)^(3/2)*(c + d*x)^(3/2))/3 + (-((b^2* c^2 - 14*a*b*c*d - 19*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x]) + (-16*a^(3/2) *Sqrt[c]*d*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] - (2*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*Arc Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[d]))/ 2)/(4*d))/2
3.7.60.3.1 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. \(597\) vs. \(2(211)=422\).
Time = 0.56 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.31
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (16 b^{2} d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} d^{3} x +135 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b c \,d^{2} x +45 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{2} c^{2} d x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{3} x -72 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x -120 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d x +52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x^{2}+28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d \,x^{2}+66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x +136 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d x +6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x -48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d \right )}{48 d \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, x \sqrt {a c}}\) | \(598\) |
1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*b^2*d^2*x^3*(a*c)^(1/2)*((b*x+a)*(d*x +c))^(1/2)*(b*d)^(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+135*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+45*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-3*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-72*(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-120*(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+52*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 1/2)*a*b*d^2*x^2+28*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c* d*x^2+66*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*x+136*(b* d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d*x+6*(b*d)^(1/2)*(a*c) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x-48*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+ a)*(d*x+c))^(1/2)*a^2*c*d)/d/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/x/(a*c)^( 1/2)
Time = 3.22 (sec) , antiderivative size = 1333, normalized size of antiderivative = 5.15 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\text {Too large to display} \]
[-1/96*(3*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*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) - 2 4*(5*a*b^2*c*d^2 + 3*a^2*b*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*(8*b^3*d^3*x^3 - 24*a^2 *b*c*d^2 + 2*(7*b^3*c*d^2 + 13*a*b^2*d^3)*x^2 + (3*b^3*c^2*d + 68*a*b^2*c* d^2 + 33*a^2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^2*x), 1/48*(3*(b^ 3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*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)) + 12*(5*a*b^2*c*d^2 + 3*a^2*b*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*(8*b^3*d^3*x^3 - 24*a^2*b*c*d^2 + 2*(7*b^3*c*d^2 + 13*a *b^2*d^3)*x^2 + (3*b^3*c^2*d + 68*a*b^2*c*d^2 + 33*a^2*b*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^2*x), 1/96*(48*(5*a*b^2*c*d^2 + 3*a^2*b*d^3)*sqrt (-a*c)*x*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)) - 3*(b^3*c^3 - 1 5*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*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)*sqr...
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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. 654 vs. \(2 (211) = 422\).
Time = 0.75 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} d {\left | b \right |}}{b} + \frac {7 \, b c d^{4} {\left | b \right |} + 5 \, a d^{5} {\left | b \right |}}{b d^{4}}\right )} + \frac {3 \, {\left (b^{2} c^{2} d^{3} {\left | b \right |} + 18 \, a b c d^{4} {\left | b \right |} + 5 \, a^{2} d^{5} {\left | b \right |}\right )}}{b d^{4}}\right )} \sqrt {b x + a} - \frac {48 \, {\left (5 \, \sqrt {b d} a^{2} b^{2} c^{2} {\left | b \right |} + 3 \, \sqrt {b d} a^{3} b c 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 {3 \, {\left (b^{3} c^{3} {\left | b \right |} - 15 \, a b^{2} c^{2} d {\left | b \right |} - 45 \, a^{2} b c d^{2} {\left | b \right |} - 5 \, a^{3} d^{3} {\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} d} - \frac {96 \, {\left (\sqrt {b d} a^{2} b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a^{3} b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{4} b^{2} c 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^{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^{3} b c 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}}}{48 \, b} \]
1/48*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*d*ab s(b)/b + (7*b*c*d^4*abs(b) + 5*a*d^5*abs(b))/(b*d^4)) + 3*(b^2*c^2*d^3*abs (b) + 18*a*b*c*d^4*abs(b) + 5*a^2*d^5*abs(b))/(b*d^4))*sqrt(b*x + a) - 48* (5*sqrt(b*d)*a^2*b^2*c^2*abs(b) + 3*sqrt(b*d)*a^3*b*c*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) + 3*(b^3*c^3*abs(b) - 1 5*a*b^2*c^2*d*abs(b) - 45*a^2*b*c*d^2*abs(b) - 5*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)*d ) - 96*(sqrt(b*d)*a^2*b^4*c^3*abs(b) - 2*sqrt(b*d)*a^3*b^3*c^2*d*abs(b) + sqrt(b*d)*a^4*b^2*c*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^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^3*b*c*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) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^4))/b
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^2} \,d x \]