Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\frac {3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 x}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} \sqrt {c}} \]
-3/4*(-a*d+b*c)^2*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^( 5/2)/c^(1/2)-1/2*(d*x+c)^(3/2)*(b*x+a)^(1/2)/a/x^2+3/4*(-a*d+b*c)*(b*x+a)^ (1/2)*(d*x+c)^(1/2)/a^2/x
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} (2 a c-3 b c x+5 a d x)}{4 a^2 x^2}-\frac {3 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} \sqrt {c}} \]
-1/4*(Sqrt[a + b*x]*Sqrt[c + d*x]*(2*a*c - 3*b*c*x + 5*a*d*x))/(a^2*x^2) - (3*(b*c - a*d)^2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] )/(4*a^(5/2)*Sqrt[c])
Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {3 (b c-a d) \left (-\frac {(b c-a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 a x^2}\) |
-1/2*(Sqrt[a + b*x]*(c + d*x)^(3/2))/(a*x^2) - (3*(b*c - a*d)*(-((Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x]) /(Sqrt[a]*Sqrt[c + d*x])])/(a^(3/2)*Sqrt[c])))/(4*a)
3.8.11.3.1 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. \(254\) vs. \(2(93)=186\).
Time = 1.65 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.14
| method | result | size |
| default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}+10 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right )}{8 a^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{2} \sqrt {a c}}\) | \(255\) |
-1/8*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^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*d^2*x^2-6*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+10*(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+4*((b*x+a )*(d*x+c))^(1/2)*a*c*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^2/(a*c)^(1/2)
Time = 0.29 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.81 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - {\left (3 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{3} c x^{2}}, \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - {\left (3 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{3} c x^{2}}\right ] \]
[1/16*(3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(a*c)*x^2*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)*sq rt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(2*a^2*c^2 - (3*a*b*c^2 - 5*a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c*x^2), 1/8* (3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-a*c)*x^2*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(2*a^2*c^2 - (3*a*b*c^2 - 5*a^2*c*d)*x)*sqrt( b*x + a)*sqrt(d*x + c))/(a^3*c*x^2)]
\[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{3} \sqrt {a + b x}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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. 1078 vs. \(2 (93) = 186\).
Time = 0.69 (sec) , antiderivative size = 1078, normalized size of antiderivative = 9.06 \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=-\frac {\frac {3 \, {\left (\sqrt {b d} b^{3} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b^{2} c d {\left | b \right |} + \sqrt {b d} a^{2} b d^{2} {\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^{9} c^{5} {\left | b \right |} - 17 \, \sqrt {b d} a b^{8} c^{4} d {\left | b \right |} + 38 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{2} {\left | b \right |} - 42 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{3} {\left | b \right |} + 23 \, \sqrt {b d} a^{4} b^{5} c d^{4} {\left | b \right |} - 5 \, \sqrt {b d} a^{5} b^{4} d^{5} {\left | b \right |} - 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} b^{7} c^{4} {\left | b \right |} + 20 \, \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^{6} c^{3} 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 )}^{2} a^{2} b^{5} c^{2} d^{2} {\left | b \right |} - 28 \, \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^{4} c d^{3} {\left | b \right |} + 15 \, \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^{3} d^{4} {\left | b \right |} + 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} b^{5} c^{3} {\left | b \right |} - 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 b^{4} c^{2} d {\left | b \right |} - \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^{3} c d^{2} {\left | b \right |} - 15 \, \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^{2} 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 )}^{6} b^{3} c^{2} {\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 )}^{6} a b^{2} c d {\left | b \right |} + 5 \, \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 d^{2} {\left | b \right |}\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} a^{2}}}{4 \, b} \]
-1/4*(3*(sqrt(b*d)*b^3*c^2*abs(b) - 2*sqrt(b*d)*a*b^2*c*d*abs(b) + sqrt(b* d)*a^2*b*d^2*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^9*c^5*abs(b) - 17*sqrt(b*d)*a*b^8*c^4*d*abs (b) + 38*sqrt(b*d)*a^2*b^7*c^3*d^2*abs(b) - 42*sqrt(b*d)*a^3*b^6*c^2*d^3*a bs(b) + 23*sqrt(b*d)*a^4*b^5*c*d^4*abs(b) - 5*sqrt(b*d)*a^5*b^4*d^5*abs(b) - 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b *d))^2*b^7*c^4*abs(b) + 20*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^6*c^3*d*abs(b) + 2*sqrt(b*d)*(sqrt(b*d)*s qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^5*c^2*d^2*abs( b) - 28*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^4*c*d^3*abs(b) + 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - s qrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^3*d^4*abs(b) + 9*sqrt(b*d)*(sq rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^5*c^3*abs (b) - 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*b^4*c^2*d*abs(b) - 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^3*c*d^2*abs(b) - 15*sqrt(b*d)*(sqrt (b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^2*d^3*a bs(b) - 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^3*c^2*abs(b) + 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr...
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^3 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^3\,\sqrt {a+b\,x}} \,d x \]