Integrand size = 22, antiderivative size = 108 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=-\frac {3 (b c-a d) \sqrt {c+d x}}{a^2 \sqrt {a+b x}}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}}+\frac {3 \sqrt {c} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}} \]
3*(-a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))*c^(1/2)/ a^(5/2)-(d*x+c)^(3/2)/a/x/(b*x+a)^(1/2)-3*(-a*d+b*c)*(d*x+c)^(1/2)/a^2/(b* x+a)^(1/2)
Time = 10.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} (-a c-3 b c x+2 a d x)}{a^2 x \sqrt {a+b x}}+\frac {3 \sqrt {c} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{5/2}} \]
(Sqrt[c + d*x]*(-(a*c) - 3*b*c*x + 2*a*d*x))/(a^2*x*Sqrt[a + b*x]) + (3*Sq rt[c]*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] )/a^(5/2)
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {105, 105, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}}dx}{2 a}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 (b c-a d) \left (\frac {c \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}+\frac {2 \sqrt {c+d x}}{a \sqrt {a+b x}}\right )}{2 a}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {3 (b c-a d) \left (\frac {2 c \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}+\frac {2 \sqrt {c+d x}}{a \sqrt {a+b x}}\right )}{2 a}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {3 (b c-a d) \left (\frac {2 \sqrt {c+d x}}{a \sqrt {a+b x}}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}\right )}{2 a}-\frac {(c+d x)^{3/2}}{a x \sqrt {a+b x}}\) |
-((c + d*x)^(3/2)/(a*x*Sqrt[a + b*x])) - (3*(b*c - a*d)*((2*Sqrt[c + d*x]) /(a*Sqrt[a + b*x]) - (2*Sqrt[c]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*S qrt[c + d*x])])/a^(3/2)))/(2*a)
3.8.64.3.1 Defintions of rubi rules used
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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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. \(297\) vs. \(2(88)=176\).
Time = 0.59 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.76
| method | result | size |
| default | \(-\frac {\sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} c d x -3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b \,c^{2} x -4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x +6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{2 a^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {a c}\, \sqrt {b x +a}}\) | \(298\) |
-1/2*(d*x+c)^(1/2)*(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*b*c*d*x^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)*b^2*c^2*x^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^2*c*d*x-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*b*c^2*x-4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d* x+6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c*x+2*((b*x+a)*(d*x+c))^(1/2)*a* c*(a*c)^(1/2))/a^2/((b*x+a)*(d*x+c))^(1/2)/x/(a*c)^(1/2)/(b*x+a)^(1/2)
Time = 0.30 (sec) , antiderivative size = 341, normalized size of antiderivative = 3.16 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{2} c - a b d\right )} x^{2} + {\left (a b c - a^{2} d\right )} x\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (a c + {\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}, -\frac {3 \, {\left ({\left (b^{2} c - a b d\right )} x^{2} + {\left (a b c - a^{2} d\right )} x\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \, {\left (a c + {\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}\right ] \]
[-1/4*(3*((b^2*c - a*b*d)*x^2 + (a*b*c - a^2*d)*x)*sqrt(c/a)*log((8*a^2*c^ 2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)* sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4* (a*c + (3*b*c - 2*a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^2*b*x^2 + a^3*x) , -1/2*(3*((b^2*c - a*b*d)*x^2 + (a*b*c - a^2*d)*x)*sqrt(-c/a)*arctan(1/2* (2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) + 2*(a*c + (3*b*c - 2*a*d)*x)*sqrt(b*x + a)* sqrt(d*x + c))/(a^2*b*x^2 + a^3*x)]
\[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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. 904 vs. \(2 (88) = 176\).
Time = 0.87 (sec) , antiderivative size = 904, normalized size of antiderivative = 8.37 \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {b d} b c^{2} {\left | b \right |} - \sqrt {b d} a 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} a^{2} b} - \frac {2 \, {\left (3 \, \sqrt {b d} b^{6} c^{4} {\left | b \right |} - 11 \, \sqrt {b d} a b^{5} c^{3} d {\left | b \right |} + 15 \, \sqrt {b d} a^{2} b^{4} c^{2} d^{2} {\left | b \right |} - 9 \, \sqrt {b d} a^{3} b^{3} c d^{3} {\left | b \right |} + 2 \, \sqrt {b d} a^{4} b^{2} d^{4} {\left | b \right |} - 6 \, \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} b^{4} c^{3} {\left | b \right |} + 6 \, \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^{3} c^{2} d {\left | b \right |} + 4 \, \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 d^{2} {\left | b \right |} - 4 \, \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 d^{3} {\left | b \right |} + 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 )}^{4} b^{2} c^{2} {\left | b \right |} - 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 )}^{4} a b c d {\left | b \right |} + 2 \, \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} d^{2} {\left | b \right |}\right )}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 3 \, {\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^{4} c^{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} a b^{3} c 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} d^{2} + 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} c + {\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 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}\right )} a^{2} b} \]
3*(sqrt(b*d)*b*c^2*abs(b) - sqrt(b*d)*a*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)*a^2*b) - 2*(3*sqrt(b*d)*b^6*c^4*abs(b) - 11*sqrt(b*d)*a*b^5*c^3*d*abs(b) + 15*sqrt(b*d)*a^2*b^4*c^2*d^2*abs(b) - 9*sqrt(b*d)*a^3*b^3*c*d^3*abs(b) + 2*sqrt(b*d)*a^4*b^2*d^4*abs(b) - 6*sqr t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b ^4*c^3*abs(b) + 6*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^3*c^2*d*abs(b) + 4*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*d^2*abs(b) - 4*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*d^3*abs(b) + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c^2*abs(b) - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b*c*d*abs(b) + 2*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*d^2*abs (b))/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3 - 3*(sqrt(b *d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^4*c^2 + 2*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^3*c*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 *d^2 + 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4 *b^2*c + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)...
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^2\,{\left (a+b\,x\right )}^{3/2}} \,d x \]