Integrand size = 22, antiderivative size = 119 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \]
-2*a^(3/2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(3/2)+2* b^(3/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(3/2)-2*(-a *d+b*c)*(b*x+a)^(1/2)/c/d/(d*x+c)^(1/2)
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.61 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {2 i a^{3/2} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b c^{5/2}}-\frac {2 a^{3/2} \left (\sqrt {a} \sqrt {d}+i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b c^{5/2}}-\frac {4 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{d^{3/2}} \]
(-2*(b*c - a*d)*Sqrt[a + b*x])/(c*d*Sqrt[c + d*x]) + ((2*I)*a^(3/2)*(I*Sqr t[a]*Sqrt[d] + Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*S qrt[b*c - a*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] ))])/(b*c^(5/2)) - (2*a^(3/2)*(Sqrt[a]*Sqrt[d] + I*Sqrt[b*c - a*d])*Sqrt[b *c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*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]))])/(b*c^(5/2)) - (4*b^(3/2)*ArcTanh[ (Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/d^ (3/2)
Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {109, 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.
| \(\displaystyle \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \int \frac {d a^2+b^2 c x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c d}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {d a^2+b^2 c x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c d}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {a^2 d \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+b^2 c \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{c d}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {a^2 d \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+2 b^2 c \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c d}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {2 a^2 d \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 b^2 c \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c d}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 b^{3/2} c \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {2 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}}{c d}-\frac {2 \sqrt {a+b x} (b c-a d)}{c d \sqrt {c+d x}}\) |
(-2*(b*c - a*d)*Sqrt[a + b*x])/(c*d*Sqrt[c + d*x]) + ((-2*a^(3/2)*d*ArcTan h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] + (2*b^(3/2)*c *ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[d])/(c*d)
3.7.37.3.1 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[(((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. \(305\) vs. \(2(91)=182\).
Time = 0.58 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.57
| method | result | size |
| default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x \sqrt {b d}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c d x \sqrt {a c}-\sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{2} c^{2}+2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d -2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c \right ) \sqrt {b x +a}}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\, \sqrt {d x +c}\, c d}\) | \(306\) |
(-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*d^2* x*(b*d)^(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^2*c*d*x*(a*c)^(1/2)-(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*c*d+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^2*c^2+2*(b *d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d-2*(b*d)^(1/2)*(a*c)^(1/2 )*((b*x+a)*(d*x+c))^(1/2)*b*c)*(b*x+a)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/(b*d) ^(1/2)/(a*c)^(1/2)/(d*x+c)^(1/2)/c/d
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (91) = 182\).
Time = 0.51 (sec) , antiderivative size = 956, normalized size of antiderivative = 8.03 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\left [\frac {{\left (b c d x + b c^{2}\right )} \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 (a d^{2} x + a c d\right )} \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 (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, -\frac {2 \, {\left (b c d x + b c^{2}\right )} \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 (a d^{2} x + a c d\right )} \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 (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, \frac {2 \, {\left (a d^{2} x + a c d\right )} \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 ) + {\left (b c d x + b c^{2}\right )} \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 ) - 4 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (c d^{2} x + c^{2} d\right )}}, \frac {{\left (a d^{2} x + a c d\right )} \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 ) - {\left (b c d x + b c^{2}\right )} \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 ) - 2 \, {\left (b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{c d^{2} x + c^{2} d}\right ] \]
[1/2*((b*c*d*x + b*c^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) + (a*d^2*x + a*c*d)*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*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d^2*x + c^2*d), -1/2*(2*(b*c *d*x + b*c^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)) - (a*d^2*x + a*c*d)*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*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/ (c*d^2*x + c^2*d), 1/2*(2*(a*d^2*x + a*c*d)*sqrt(-a/c)*arctan(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)) + (b*c*d*x + b*c^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) - 4*(b*c - a*d)*sqrt(b* x + a)*sqrt(d*x + c))/(c*d^2*x + c^2*d), ((a*d^2*x + a*c*d)*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)) - (b*c*d*x + b*c^2)*sqrt(-b/d)*arcta n(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2...
\[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}}}{x \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 205 vs. \(2 (91) = 182\).
Time = 0.35 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b d} a^{2} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c {\left | b \right |}} - \frac {\sqrt {b d} b^{2} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2} {\left | b \right |}} - \frac {2 \, {\left (b^{3} c {\left | b \right |} - a b^{2} d {\left | b \right |}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} b^{2} c d} \]
-2*sqrt(b*d)*a^2*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*abs(b)) - 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^2*abs(b)) - 2*(b^3*c*abs(b) - a*b^2*d*abs(b)) *sqrt(b*x + a)/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*b^2*c*d)
Timed out. \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}}{x\,{\left (c+d\,x\right )}^{3/2}} \,d x \]