Integrand size = 22, antiderivative size = 157 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=-\frac {2 (b c-a d) (a+b x)^{3/2}}{3 c d (c+d x)^{3/2}}+\frac {2 \left (\frac {a^2}{c^2}-\frac {b^2}{d^2}\right ) \sqrt {a+b x}}{\sqrt {c+d x}}-\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{5/2}} \]
-2/3*(-a*d+b*c)*(b*x+a)^(3/2)/c/d/(d*x+c)^(3/2)-2*a^(5/2)*arctanh(c^(1/2)* (b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(5/2)+2*b^(5/2)*arctanh(d^(1/2)*(b* x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(5/2)+2*(a^2/c^2-b^2/d^2)*(b*x+a)^(1/2 )/(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 7.17 (sec) , antiderivative size = 1440, normalized size of antiderivative = 9.17 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\frac {8 a^3 b d^2 \left (2 c^2+9 c d x+3 d^2 x^2\right )+8 b^3 c^2 (3 c+4 d x) \left (b x (3 c-d x)+\sqrt {a-\frac {b c}{d}} \sqrt {a+b x} (-c+3 d x)\right )+2 a^2 b d \left (-4 \sqrt {a-\frac {b c}{d}} d \sqrt {a+b x} \left (2 c^2+9 c d x+3 d^2 x^2\right )+b \left (-51 c^3-84 c^2 d x+21 c d^2 x^2+6 d^3 x^3\right )\right )+a b^2 c \left (2 \sqrt {a-\frac {b c}{d}} d \sqrt {a+b x} \left (47 c^2+70 c d x-9 d^2 x^2\right )+b \left (73 c^3-15 c^2 d x-177 c d^2 x^2+7 d^3 x^3\right )\right )}{12 c^2 d^2 (c+d x)^{3/2} \left (b c \left (\sqrt {a-\frac {b c}{d}}-3 \sqrt {a+b x}\right )+b d x \left (-3 \sqrt {a-\frac {b c}{d}}+\sqrt {a+b x}\right )+a d \left (-4 \sqrt {a-\frac {b c}{d}}+4 \sqrt {a+b x}\right )\right )}+\frac {2 a^3 \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {2 i a^{7/2} d \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {2 a^3 \sqrt {d} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {2 i a^{7/2} d \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{5/2} \sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}+\frac {a \left (\frac {b \left (\sqrt {a-\frac {b c}{d}}-3 \sqrt {a+b x}\right )}{d}-\frac {4 a (4 c+3 d x) \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}{c^2}+\frac {b x \left (-3 \sqrt {a-\frac {b c}{d}}+\sqrt {a+b x}\right )}{c}+\frac {24 i a^{3/2} b (c+d x)^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{3/2} \sqrt {b c-a d} \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}-\frac {24 i a^{3/2} b (c+d x)^{3/2} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{c^{3/2} \sqrt {b c-a d} \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}}}\right )}{12 (c+d x)^{3/2}}-\frac {4 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{d^{5/2}} \]
(8*a^3*b*d^2*(2*c^2 + 9*c*d*x + 3*d^2*x^2) + 8*b^3*c^2*(3*c + 4*d*x)*(b*x* (3*c - d*x) + Sqrt[a - (b*c)/d]*Sqrt[a + b*x]*(-c + 3*d*x)) + 2*a^2*b*d*(- 4*Sqrt[a - (b*c)/d]*d*Sqrt[a + b*x]*(2*c^2 + 9*c*d*x + 3*d^2*x^2) + b*(-51 *c^3 - 84*c^2*d*x + 21*c*d^2*x^2 + 6*d^3*x^3)) + a*b^2*c*(2*Sqrt[a - (b*c) /d]*d*Sqrt[a + b*x]*(47*c^2 + 70*c*d*x - 9*d^2*x^2) + b*(73*c^3 - 15*c^2*d *x - 177*c*d^2*x^2 + 7*d^3*x^3)))/(12*c^2*d^2*(c + d*x)^(3/2)*(b*c*(Sqrt[a - (b*c)/d] - 3*Sqrt[a + b*x]) + b*d*x*(-3*Sqrt[a - (b*c)/d] + Sqrt[a + b* x]) + a*d*(-4*Sqrt[a - (b*c)/d] + 4*Sqrt[a + b*x]))) + (2*a^3*Sqrt[d]*ArcT an[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x ])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(5/2)*Sqrt[b *c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) - ((2*I)*a^(7/2)*d*Ar cTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d *x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(c^(5/2)*Sqrt [b*c - a*d]*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + ( 2*a^3*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x])) ])/(c^(5/2)*Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]) + ( (2*I)*a^(7/2)*d*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x] ))])/(c^(5/2)*Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*...
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {109, 27, 167, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \int \frac {3 \sqrt {a+b x} \left (d a^2+b^2 c x\right )}{2 x (c+d x)^{3/2}}dx}{3 c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b x} \left (d a^2+b^2 c x\right )}{x (c+d x)^{3/2}}dx}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {d^2 a^3+b^3 c^2 x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c d}-\frac {2 \sqrt {a+b x} \left (\frac {b^2 c}{d}-\frac {a^2 d}{c}\right )}{\sqrt {c+d x}}}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {d^2 a^3+b^3 c^2 x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c d}-\frac {2 \sqrt {a+b x} \left (\frac {b^2 c}{d}-\frac {a^2 d}{c}\right )}{\sqrt {c+d x}}}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\frac {a^3 d^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+b^3 c^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{c d}-\frac {2 \sqrt {a+b x} \left (\frac {b^2 c}{d}-\frac {a^2 d}{c}\right )}{\sqrt {c+d x}}}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {a^3 d^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+2 b^3 c^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c d}-\frac {2 \sqrt {a+b x} \left (\frac {b^2 c}{d}-\frac {a^2 d}{c}\right )}{\sqrt {c+d x}}}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\frac {2 a^3 d^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 b^3 c^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c d}-\frac {2 \sqrt {a+b x} \left (\frac {b^2 c}{d}-\frac {a^2 d}{c}\right )}{\sqrt {c+d x}}}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {2 b^{5/2} c^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {2 a^{5/2} d^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}}{c d}-\frac {2 \sqrt {a+b x} \left (\frac {b^2 c}{d}-\frac {a^2 d}{c}\right )}{\sqrt {c+d x}}}{c d}-\frac {2 (a+b x)^{3/2} (b c-a d)}{3 c d (c+d x)^{3/2}}\) |
(-2*(b*c - a*d)*(a + b*x)^(3/2))/(3*c*d*(c + d*x)^(3/2)) + ((-2*((b^2*c)/d - (a^2*d)/c)*Sqrt[a + b*x])/Sqrt[c + d*x] + ((-2*a^(5/2)*d^2*ArcTanh[(Sqr t[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] + (2*b^(5/2)*c^2*Arc Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[d])/(c*d))/(c* d)
3.7.94.3.1 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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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] && LtQ[m, -1] && GtQ[n, 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. \(565\) vs. \(2(123)=246\).
Time = 0.57 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.61
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \left (3 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{4} x^{2}-3 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} d^{2} x^{2}+6 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{3} x -6 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3} d x +3 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d^{2}-3 \sqrt {a c}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{4}-6 a^{2} d^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}-2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{2} x +8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d x -8 a^{2} c \,d^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d +6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3}\right )}{3 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \left (d x +c \right )^{\frac {3}{2}} d^{2} c^{2}}\) | \(566\) |
-1/3*(b*x+a)^(1/2)*(3*(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*d^4*x^2-3*(a*c)^(1/2)*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))*b^3*c^2*d^2*x^2+6*(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^3*x-6*(a*c)^(1/2)*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))*b^3*c^3*d*x+3*(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^2*d^2-3*(a*c)^(1/2)*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))*b^3* c^4-6*a^2*d^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2*(b*d)^(1 /2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d^2*x+8*(b*d)^(1/2)*(a*c)^(1 /2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*d*x-8*a^2*c*d^2*(a*c)^(1/2)*((b*x+a)*( d*x+c))^(1/2)*(b*d)^(1/2)+2*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2 )*a*b*c^2*d+6*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^3)/((b *x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*c)^(1/2)/(d*x+c)^(3/2)/d^2/c^2
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (123) = 246\).
Time = 1.08 (sec) , antiderivative size = 1361, normalized size of antiderivative = 8.67 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
[1/6*(3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*sqrt(b/d)*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^2*x + b*c*d + a*d^2)*sqrt (b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) + 3*(a^2*d^4* x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(a/c)*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^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sq rt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(3*b^2*c^3 + a*b *c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^3*x + c^4*d^2), -1/6*(6*(b^2*c^2 *d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) - 3*(a^2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(a/c)*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^2 + (b*c^2 + a *c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/ x^2) + 4*(3*b^2*c^3 + a*b*c^2*d - 4*a^2*c*d^2 + (4*b^2*c^2*d - a*b*c*d^2 - 3*a^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*d^4*x^2 + 2*c^3*d^3*x + c ^4*d^2), 1/6*(6*(a^2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*sqrt(-a/c)*arc tan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-a/c)/(a* b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) + 3*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*sqrt(b/d)*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^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d) + 8...
\[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 347 vs. \(2 (123) = 246\).
Time = 0.43 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.21 \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b d} a^{3} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} - \frac {\sqrt {b d} b^{3} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{3} {\left | b \right |}} - \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (4 \, b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} - 2 \, a^{2} b^{6} c^{3} d^{4} + 3 \, a^{3} b^{5} c^{2} d^{5}\right )} {\left (b x + a\right )}}{b^{3} c^{5} d^{3} {\left | b \right |} - a b^{2} c^{4} d^{4} {\left | b \right |}} + \frac {3 \, {\left (b^{9} c^{6} d - 2 \, a b^{8} c^{5} d^{2} + 2 \, a^{3} b^{6} c^{3} d^{4} - a^{4} b^{5} c^{2} d^{5}\right )}}{b^{3} c^{5} d^{3} {\left | b \right |} - a b^{2} c^{4} d^{4} {\left | b \right |}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \]
-2*sqrt(b*d)*a^3*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)*c^2*abs(b)) - sqrt(b*d)*b^3*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(d^3*abs(b)) - 2/3*sqrt(b*x + a)*((4*b^8*c^5*d ^2 - 5*a*b^7*c^4*d^3 - 2*a^2*b^6*c^3*d^4 + 3*a^3*b^5*c^2*d^5)*(b*x + a)/(b ^3*c^5*d^3*abs(b) - a*b^2*c^4*d^4*abs(b)) + 3*(b^9*c^6*d - 2*a*b^8*c^5*d^2 + 2*a^3*b^6*c^3*d^4 - a^4*b^5*c^2*d^5)/(b^3*c^5*d^3*abs(b) - a*b^2*c^4*d^ 4*abs(b)))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2)
Timed out. \[ \int \frac {(a+b x)^{5/2}}{x (c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x\,{\left (c+d\,x\right )}^{5/2}} \,d x \]