Integrand size = 22, antiderivative size = 177 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {a (7 b c-3 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c^2 x}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 c^{5/2}}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}} \]
-1/4*(3*a^2*d^2-10*a*b*c*d+15*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/ 2)/(d*x+c)^(1/2))*a^(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^(1/2)-1/2*a*(b*x+a)^(3/2)*(d*x+c)^(1/2)/c/x^2-1/4 *a*(-3*a*d+7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2/x
Time = 0.53 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\frac {1}{4} \left (\frac {a \sqrt {a+b x} \sqrt {c+d x} (-2 a c-9 b c x+3 a d x)}{c^2 x^2}-\frac {\sqrt {a} \left (15 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{5/2}}+\frac {8 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {d}}\right ) \]
((a*Sqrt[a + b*x]*Sqrt[c + d*x]*(-2*a*c - 9*b*c*x + 3*a*d*x))/(c^2*x^2) - (Sqrt[a]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d *x])/(Sqrt[c]*Sqrt[a + b*x])])/c^(5/2) + (8*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[ c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/Sqrt[d])/4
Time = 0.31 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {109, 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.
| \(\displaystyle \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int -\frac {\sqrt {a+b x} \left (4 c x b^2+a (7 b c-3 a d)\right )}{2 x^2 \sqrt {c+d x}}dx}{2 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b x} \left (4 c x b^2+a (7 b c-3 a d)\right )}{x^2 \sqrt {c+d x}}dx}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {\frac {\int \frac {8 c^2 x b^3+a \left (15 b^2 c^2-10 a b d c+3 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{c x}}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {8 c^2 x b^3+a \left (15 b^2 c^2-10 a b d c+3 a^2 d^2\right )}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{c x}}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\frac {a \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+8 b^3 c^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{c x}}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {a \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+16 b^3 c^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{c x}}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\frac {2 a \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+16 b^3 c^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{c x}}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {16 b^{5/2} c^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {2 \sqrt {a} \left (3 a^2 d^2-10 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}}{2 c}-\frac {a \sqrt {a+b x} \sqrt {c+d x} (7 b c-3 a d)}{c x}}{4 c}-\frac {a (a+b x)^{3/2} \sqrt {c+d x}}{2 c x^2}\) |
-1/2*(a*(a + b*x)^(3/2)*Sqrt[c + d*x])/(c*x^2) + (-((a*(7*b*c - 3*a*d)*Sqr t[a + b*x]*Sqrt[c + d*x])/(c*x)) + ((-2*Sqrt[a]*(15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt [c] + (16*b^(5/2)*c^2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d* x])])/Sqrt[d])/(2*c))/(4*c)
3.7.79.3.1 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] && 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. \(353\) vs. \(2(139)=278\).
Time = 0.56 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.00
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x^{2} \sqrt {b d}-10 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c d \,x^{2} \sqrt {b d}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} x^{2} \sqrt {b d}-8 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} x^{2} \sqrt {a c}-6 a^{2} d x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+18 a b c x \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+4 a^{2} c \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{8 c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {b d}\, \sqrt {a c}}\) | \(354\) |
-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2*(3*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^2*x^2*(b*d)^(1/2)-10*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*d*x^2*(b*d)^(1/2)+1 5*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^ 2*x^2*(b*d)^(1/2)-8*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*x^2*(a*c)^(1/2)-6*a^2*d*x*(b*d)^(1/2)*(a*c)^ (1/2)*((b*x+a)*(d*x+c))^(1/2)+18*a*b*c*x*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)* (d*x+c))^(1/2)+4*a^2*c*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(( b*x+a)*(d*x+c))^(1/2)/x^2/(b*d)^(1/2)/(a*c)^(1/2)
Time = 1.12 (sec) , antiderivative size = 1035, normalized size of antiderivative = 5.85 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\left [\frac {8 \, b^{2} c^{2} x^{2} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, c^{2} x^{2}}, -\frac {16 \, b^{2} c^{2} x^{2} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, c^{2} x^{2}}, \frac {4 \, b^{2} c^{2} x^{2} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, c^{2} x^{2}}, -\frac {8 \, b^{2} c^{2} x^{2} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) - {\left (15 \, b^{2} c^{2} - 10 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \, {\left (2 \, a^{2} c + 3 \, {\left (3 \, a b c - a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, c^{2} x^{2}}\right ] \]
[1/16*(8*b^2*c^2*x^2*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) + (15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^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*(2*a^2*c + 3*(3*a*b*c - a^2*d)*x)*sqrt(b*x + a)*sqr t(d*x + c))/(c^2*x^2), -1/16*(16*b^2*c^2*x^2*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)) - (15*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^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*(2*a^2*c + 3*(3*a*b*c - a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c ))/(c^2*x^2), 1/8*(4*b^2*c^2*x^2*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) + (15*b^2*c^2 - 10*a*b*c*d + 3*a ^2*d^2)*x^2*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sq rt(d*x + c)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 2*(2*a^2 *c + 3*(3*a*b*c - a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(c^2*x^2), -1/8*( 8*b^2*c^2*x^2*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sq rt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) - (15*b...
\[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{3} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 1107 vs. \(2 (139) = 278\).
Time = 0.79 (sec) , antiderivative size = 1107, normalized size of antiderivative = 6.25 \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=-\frac {{\left (\frac {4 \, \sqrt {b d} b^{2} \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} + \frac {{\left (15 \, \sqrt {b d} a b^{3} c^{2} - 10 \, \sqrt {b d} a^{2} b^{2} c d + 3 \, \sqrt {b d} a^{3} b d^{2}\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 c^{2}} + \frac {2 \, {\left (9 \, \sqrt {b d} a b^{9} c^{5} - 39 \, \sqrt {b d} a^{2} b^{8} c^{4} d + 66 \, \sqrt {b d} a^{3} b^{7} c^{3} d^{2} - 54 \, \sqrt {b d} a^{4} b^{6} c^{2} d^{3} + 21 \, \sqrt {b d} a^{5} b^{5} c d^{4} - 3 \, \sqrt {b d} a^{6} b^{4} d^{5} - 27 \, \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 b^{7} c^{4} + 40 \, \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^{6} c^{3} d + 10 \, \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^{5} c^{2} d^{2} - 32 \, \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^{4} b^{4} c d^{3} + 9 \, \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^{5} b^{3} d^{4} + 27 \, \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 )}^{4} a b^{5} c^{3} + 9 \, \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 )}^{4} a^{2} b^{4} c^{2} d + 21 \, \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 )}^{4} a^{3} b^{3} c d^{2} - 9 \, \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 )}^{4} a^{4} b^{2} d^{3} - 9 \, \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 )}^{6} a b^{3} c^{2} - 10 \, \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 )}^{6} a^{2} b^{2} c d + 3 \, \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 )}^{6} a^{3} b d^{2}\right )}}{{\left (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}\right )}^{2} c^{2}}\right )} b}{4 \, {\left | b \right |}} \]
-1/4*(4*sqrt(b*d)*b^2*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a )*b*d - a*b*d))^2)/d + (15*sqrt(b*d)*a*b^3*c^2 - 10*sqrt(b*d)*a^2*b^2*c*d + 3*sqrt(b*d)*a^3*b*d^2)*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*c^2) + 2*(9*sqrt(b*d)*a*b^9*c^5 - 39*sqrt(b*d)*a^2*b^8*c^4*d + 66*sqrt(b*d)*a^3*b^7*c^3*d^2 - 54*sqrt(b*d)*a^4*b^6*c^2*d^3 + 21*sqrt(b*d) *a^5*b^5*c*d^4 - 3*sqrt(b*d)*a^6*b^4*d^5 - 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^4 + 40*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 ^3*d + 10*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^2*d^2 - 32*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*d^3 + 9*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*d^4 + 27*sq rt(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^3 + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^4*a^2*b^4*c^2*d + 21*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^3*c*d^2 - 9*sqrt(b*d)*(sqrt(b*d )*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^4*b^2*d^3 - 9*s qrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6 *a*b^3*c^2 - 10*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x ...
Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^3 \sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^3\,\sqrt {c+d\,x}} \,d x \]