Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
2*d^(3/2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)+(-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^( 3/2)-c*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/x
Time = 0.33 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}+\frac {\sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}} \]
-((c*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) + (Sqrt[c]*(b*c - 3*a*d)*ArcTanh[ (Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/a^(3/2) + (2*d^(3/2)*Arc Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b]
Time = 0.23 (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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {\int \frac {c (b c-3 a d)-2 a d^2 x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {c (b c-3 a d)-2 a d^2 x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {c (b c-3 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 a d^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {c (b c-3 a d) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-4 a d^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {2 c (b c-3 a d) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-4 a d^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {4 a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {2 \sqrt {c} (b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}}{2 a}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{a x}\) |
-((c*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x)) - ((-2*Sqrt[c]*(b*c - 3*a*d)*ArcT anh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] - (4*a*d^(3/ 2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b])/(2*a )
3.8.10.3.1 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. \(222\) vs. \(2(91)=182\).
Time = 0.57 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.87
| method | result | size |
| default | \(\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,d^{2} x \sqrt {a c}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a c d x \sqrt {b d}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b \,c^{2} x \sqrt {b d}-2 c \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 a \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\) | \(223\) |
1/2*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a*(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))*a*d^2*x*(a*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*c*d*x*(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)*b*c^2*x*(b*d) ^(1/2)-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/(b*d)^(1/2)/(a*c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (91) = 182\).
Time = 0.47 (sec) , antiderivative size = 858, normalized size of antiderivative = 7.21 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\left [\frac {2 \, a d x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - {\left (b c - 3 \, a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} c}{4 \, a x}, -\frac {4 \, a d x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + {\left (b c - 3 \, a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} c}{4 \, a x}, \frac {a d x \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - {\left (b c - 3 \, a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} c}{2 \, a x}, -\frac {2 \, a d x \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + {\left (b c - 3 \, a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} c}{2 \, a x}\right ] \]
[1/4*(2*a*d*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8* (b^2*c*d + a*b*d^2)*x) - (b*c - 3*a*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*sqrt(b*x + a)*sqrt(d*x + c)*c)/(a*x), -1/4*(4*a*d*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b *c*d + a*d^2)*x)) + (b*c - 3*a*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)*s qrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*s qrt(d*x + c)*c)/(a*x), 1/2*(a*d*x*sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) - (b*c - 3*a*d)*x*sqrt(-c/a)*ar ctan(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*sqrt(b*x + a)*sqrt(d*x + c)*c)/ (a*x), -1/2*(2*a*d*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) + ( b*c - 3*a*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*sqr t(b*x + a)*sqrt(d*x + c)*c)/(a*x)]
\[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{2} \sqrt {a + b x}}\, dx \]
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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. 484 vs. \(2 (91) = 182\).
Time = 0.42 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.07 \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=-\frac {\frac {\sqrt {b d} d {\left | b \right |} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b} - \frac {{\left (\sqrt {b d} b^{2} c^{2} {\left | b \right |} - 3 \, \sqrt {b d} a b c d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b} + \frac {2 \, {\left (\sqrt {b d} b^{4} c^{3} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c^{2} d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} c d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c d {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a}}{b} \]
-(sqrt(b*d)*d*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a) *b*d - a*b*d))^2)/b - (sqrt(b*d)*b^2*c^2*abs(b) - 3*sqrt(b*d)*a*b*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*b) + 2*(sq rt(b*d)*b^4*c^3*abs(b) - 2*sqrt(b*d)*a*b^3*c^2*d*abs(b) + sqrt(b*d)*a^2*b^ 2*c*d^2*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c^2*abs(b) - 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*b*c*d*abs(b))/((b^4*c^2 - 2*a*b^3* c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)* b*d - a*b*d))^4)*a))/b
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x^2 \sqrt {a+b x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x^2\,\sqrt {a+b\,x}} \,d x \]