Integrand size = 22, antiderivative size = 313 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=-\frac {\left (5 b^3 c^3+73 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 a c^2 x}-\frac {(5 b c-a d) (b c+3 a d) \sqrt {a+b x} (c+d x)^{3/2}}{32 c^2 x^2}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{24 c x^3}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}+\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
-1/24*(3*a*d+5*b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/2)/c/x^3-1/4*(b*x+a)^(5/2)*(d *x+c)^(3/2)/x^4+1/64*(-3*a^4*d^4+20*a^3*b*c*d^3-90*a^2*b^2*c^2*d^2-60*a*b^ 3*c^3*d+5*b^4*c^4)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^ (3/2)/c^(5/2)+2*b^(5/2)*d^(3/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x +c)^(1/2))-1/32*(-a*d+5*b*c)*(3*a*d+b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/c^2/x ^2-1/64*(3*a^3*d^3-17*a^2*b*c*d^2+73*a*b^2*c^2*d+5*b^3*c^3)*(b*x+a)^(1/2)* (d*x+c)^(1/2)/a/c^2/x
Time = 0.76 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3+a b^2 c^2 x^2 (118 c+337 d x)+a^2 b c x \left (136 c^2+244 c d x+57 d^2 x^2\right )+a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )\right )}{192 a c^2 x^4}-\frac {\left (-5 b^4 c^4+60 a b^3 c^3 d+90 a^2 b^2 c^2 d^2-20 a^3 b c d^3+3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 + a*b^2*c^2*x^2*(118*c + 337*d*x) + a^2*b*c*x*(136*c^2 + 244*c*d*x + 57*d^2*x^2) + a^3*(48*c^3 + 72*c^2*d*x + 6*c*d^2*x^2 - 9*d^3*x^3)))/(a*c^2*x^4) - ((-5*b^4*c^4 + 60*a *b^3*c^3*d + 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqr t[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(5/2)) + 2*b^( 5/2)*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]
Time = 0.49 (sec) , antiderivative size = 339, 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, 166, 27, 166, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{4} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} (5 b c+3 a d+8 b d x)}{2 x^4}dx-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} (5 b c+3 a d+8 b d x)}{x^4}dx-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{8} \left (\frac {\int \frac {3 \sqrt {a+b x} \sqrt {c+d x} \left (16 c d x b^2+(5 b c-a d) (b c+3 a d)\right )}{2 x^3}dx}{3 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (16 c d x b^2+(5 b c-a d) (b c+3 a d)\right )}{x^3}dx}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\int \frac {\sqrt {c+d x} \left (5 c^3 b^3+64 c^2 d x b^3+73 a c^2 d b^2-17 a^2 c d^2 b+3 a^3 d^3\right )}{2 x^2 \sqrt {a+b x}}dx}{2 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\int \frac {\sqrt {c+d x} \left (5 c^3 b^3+64 c^2 d x b^3+73 a c^2 d b^2-17 a^2 c d^2 b+3 a^3 d^3\right )}{x^2 \sqrt {a+b x}}dx}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {\frac {\int -\frac {5 b^4 c^4-60 a b^3 d c^3-90 a^2 b^2 d^2 c^2-128 a b^3 d^2 x c^2+20 a^3 b d^3 c-3 a^4 d^4}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {-\frac {\int \frac {5 b^4 c^4-60 a b^3 d c^3-90 a^2 b^2 d^2 c^2-128 a b^3 d^2 x c^2+20 a^3 b d^3 c-3 a^4 d^4}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {-\frac {\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-128 a b^3 c^2 d^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {-\frac {\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-256 a b^3 c^2 d^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {-\frac {2 \left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-256 a b^3 c^2 d^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{a x}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{8} \left (\frac {\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{a x}-\frac {-\frac {2 \left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}-256 a b^{5/2} c^2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{2 a}}{4 c}-\frac {\sqrt {a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{2 c x^2}}{2 c}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{3 c x^3}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}\) |
-1/4*((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^4 + (-1/3*((5*b*c + 3*a*d)*(a + b *x)^(3/2)*(c + d*x)^(3/2))/(c*x^3) + (-1/2*((5*b*c - a*d)*(b*c + 3*a*d)*Sq rt[a + b*x]*(c + d*x)^(3/2))/(c*x^2) + (-(((5*b^3*c^3 + 73*a*b^2*c^2*d - 1 7*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - ((-2*(5*b ^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4) *ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*Sqrt[c ]) - 256*a*b^(5/2)*c^2*d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sq rt[c + d*x])])/(2*a))/(4*c))/(2*c))/8
3.7.63.3.1 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[(((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. \(732\) vs. \(2(263)=526\).
Time = 0.57 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.34
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (384 \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 ) a \,b^{3} c^{2} d^{2} x^{4} \sqrt {a c}-9 \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} d^{4} x^{4} \sqrt {b d}+60 \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 \,d^{3} x^{4} \sqrt {b d}-270 \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^{2} d^{2} x^{4} \sqrt {b d}-180 \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^{3} c^{3} d \,x^{4} \sqrt {b d}+15 \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^{4} c^{4} x^{4} \sqrt {b d}+18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{3} x^{3}-114 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c \,d^{2} x^{3}-674 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{3} x^{3}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c \,d^{2} x^{2}-488 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{2} d \,x^{2}-236 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a \,b^{2} c^{3} x^{2}-144 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{2} d x -272 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{2} b \,c^{3} x -96 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, a^{3} c^{3}\right )}{384 a \,c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{4} \sqrt {b d}\, \sqrt {a c}}\) | \(733\) |
1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^2*(384*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*b^3*c^2*d^2*x^4*(a*c)^(1/ 2)-9*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*d ^4*x^4*(b*d)^(1/2)+60*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*d^3*x^4*(b*d)^(1/2)-270*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^2*d^2*x^4*(b*d)^(1/2)-180*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^3*c^3*d*x^ 4*(b*d)^(1/2)+15*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^4*c^4*x^4*(b*d)^(1/2)+18*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c) )^(1/2)*a^3*d^3*x^3-114*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^ 2*b*c*d^2*x^3-674*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^ 2*d*x^3-30*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^3*x^3-12* ((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c*d^2*x^2-488*((b*x+a) *(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*b*c^2*d*x^2-236*((b*x+a)*(d*x+ c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a*b^2*c^3*x^2-144*((b*x+a)*(d*x+c))^(1/2 )*(b*d)^(1/2)*(a*c)^(1/2)*a^3*c^2*d*x-272*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1 /2)*(a*c)^(1/2)*a^2*b*c^3*x-96*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^( 1/2)*a^3*c^3)/((b*x+a)*(d*x+c))^(1/2)/x^4/(b*d)^(1/2)/(a*c)^(1/2)
Time = 4.42 (sec) , antiderivative size = 1525, normalized size of antiderivative = 4.87 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\text {Too large to display} \]
[1/768*(384*sqrt(b*d)*a^2*b^2*c^3*d*x^4*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*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2* b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*sqrt(a*c)*x^4*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)*s qrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(48*a^4*c^4 + (15*a*b^3*c^4 + 337*a^2*b^2*c^3*d + 57*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 122*a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b* c^4 + 9*a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*c^3*x^4), -1/768*( 768*sqrt(-b*d)*a^2*b^2*c^3*d*x^4*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)) + 3*(5*b^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^ 3 - 3*a^4*d^4)*sqrt(a*c)*x^4*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*a^4*c^4 + (15*a*b^3*c^4 + 337*a^2* b^2*c^3*d + 57*a^3*b*c^2*d^2 - 9*a^4*c*d^3)*x^3 + 2*(59*a^2*b^2*c^4 + 122* a^3*b*c^3*d + 3*a^4*c^2*d^2)*x^2 + 8*(17*a^3*b*c^4 + 9*a^4*c^3*d)*x)*sqrt( b*x + a)*sqrt(d*x + c))/(a^2*c^3*x^4), 1/384*(192*sqrt(b*d)*a^2*b^2*c^3*d* x^4*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) ...
\[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\text {Exception raised: ValueError} \]
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. 3887 vs. \(2 (263) = 526\).
Time = 1.32 (sec) , antiderivative size = 3887, normalized size of antiderivative = 12.42 \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\text {Too large to display} \]
-1/192*(192*sqrt(b*d)*b^2*d*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2 *c + (b*x + a)*b*d - a*b*d))^2) - 3*(5*sqrt(b*d)*b^5*c^4*abs(b) - 60*sqrt( b*d)*a*b^4*c^3*d*abs(b) - 90*sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) + 20*sqrt(b* d)*a^3*b^2*c*d^3*abs(b) - 3*sqrt(b*d)*a^4*b*d^4*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*b*c^2) + 2*(15*sqrt(b*d)*b^19*c^ 11*abs(b) + 217*sqrt(b*d)*a*b^18*c^10*d*abs(b) - 2219*sqrt(b*d)*a^2*b^17*c ^9*d^2*abs(b) + 8131*sqrt(b*d)*a^3*b^16*c^8*d^3*abs(b) - 16154*sqrt(b*d)*a ^4*b^15*c^7*d^4*abs(b) + 19306*sqrt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 13958*s qrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 5494*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 581*sqrt(b*d)*a^8*b^11*c^3*d^8*abs(b) - 371*sqrt(b*d)*a^9*b^10*c^2*d^9*a bs(b) + 129*sqrt(b*d)*a^10*b^9*c*d^10*abs(b) - 9*sqrt(b*d)*a^11*b^8*d^11*a bs(b) - 105*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^2*b^17*c^10*abs(b) - 1598*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^16*c^9*d*abs(b) + 9523*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 ^15*c^8*d^2*abs(b) - 18024*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^14*c^7*d^3*abs(b) + 10942*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^13*c^ 6*d^4*abs(b) + 7372*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (...
Timed out. \[ \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^5} \,d x \]