Integrand size = 20, antiderivative size = 98 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {2 c \sqrt {a+b x}}{d^2 \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{d^2}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \] Output:
2*c*(b*x+a)^(1/2)/d^2/(d*x+c)^(1/2)+(b*x+a)^(1/2)*(d*x+c)^(1/2)/d^2-(-a*d+ 3*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/d^(5/2 )
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {a+b x} (3 c+d x)}{d^2 \sqrt {c+d x}}-\frac {(3 b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \] Input:
Integrate[(x*Sqrt[a + b*x])/(c + d*x)^(3/2),x]
Output:
(Sqrt[a + b*x]*(3*c + d*x))/(d^2*Sqrt[c + d*x]) - ((3*b*c - a*d)*ArcTanh[( Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(5/2))
Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.34, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {87, 60, 66, 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.
\(\displaystyle \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(3 b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(3 b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{d (b c-a d)}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {(3 b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{d (b c-a d)}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(3 b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{d (b c-a d)}-\frac {2 c (a+b x)^{3/2}}{d \sqrt {c+d x} (b c-a d)}\) |
Input:
Int[(x*Sqrt[a + b*x])/(c + d*x)^(3/2),x]
Output:
(-2*c*(a + b*x)^(3/2))/(d*(b*c - a*d)*Sqrt[c + d*x]) + ((3*b*c - a*d)*((Sq rt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x] )/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(d*(b*c - a*d))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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. \(263\) vs. \(2(78)=156\).
Time = 0.22 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.69
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,d^{2} x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b c d x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a c d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b \,c^{2}+2 d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+6 c \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\right )}{2 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, d^{2} \sqrt {x d +c}}\) | \(264\) |
Input:
int(x*(b*x+a)^(1/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(b*x+a)^(1/2)*(ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a *d+b*c)/(d*b)^(1/2))*a*d^2*x-3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*( d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b*c*d*x+ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c ))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*c*d-3*ln(1/2*(2*b*d*x+2*((b*x +a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b*c^2+2*d*x*((b*x+a)* (d*x+c))^(1/2)*(d*b)^(1/2)+6*c*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2))/(d*b)^ (1/2)/((b*x+a)*(d*x+c))^(1/2)/d^2/(d*x+c)^(1/2)
Time = 0.13 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.14 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\left [-\frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (b d^{2} x + 3 \, b c d\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b d^{4} x + b c d^{3}\right )}}, \frac {{\left (3 \, b c^{2} - a c d + {\left (3 \, b c d - a d^{2}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (b d^{2} x + 3 \, b c d\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b d^{4} x + b c d^{3}\right )}}\right ] \] Input:
integrate(x*(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="fricas")
Output:
[-1/4*((3*b*c^2 - a*c*d + (3*b*c*d - a*d^2)*x)*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*x + b*c + a*d)*sqrt(b*d)*sqrt( b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(b*d^2*x + 3*b*c*d)* sqrt(b*x + a)*sqrt(d*x + c))/(b*d^4*x + b*c*d^3), 1/2*((3*b*c^2 - a*c*d + (3*b*c*d - a*d^2)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d )*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2) *x)) + 2*(b*d^2*x + 3*b*c*d)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d^4*x + b*c*d ^3)]
\[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\int \frac {x \sqrt {a + b x}}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*(b*x+a)**(1/2)/(d*x+c)**(3/2),x)
Output:
Integral(x*sqrt(a + b*x)/(c + d*x)**(3/2), x)
Exception generated. \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="maxima")
Output:
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
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} {\left | b \right |}}{b d} + \frac {3 \, b^{2} c d {\left | b \right |} - a b d^{2} {\left | b \right |}}{b^{2} d^{3}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b c {\left | b \right |} - a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{2}} \] Input:
integrate(x*(b*x+a)^(1/2)/(d*x+c)^(3/2),x, algorithm="giac")
Output:
sqrt(b*x + a)*((b*x + a)*abs(b)/(b*d) + (3*b^2*c*d*abs(b) - a*b*d^2*abs(b) )/(b^2*d^3))/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) + (3*b*c*abs(b) - a*d*abs (b))*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d )))/(sqrt(b*d)*b*d^2)
Timed out. \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\int \frac {x\,\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((x*(a + b*x)^(1/2))/(c + d*x)^(3/2),x)
Output:
int((x*(a + b*x)^(1/2))/(c + d*x)^(3/2), x)
Time = 0.16 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.65 \[ \int \frac {x \sqrt {a+b x}}{(c+d x)^{3/2}} \, dx=\frac {12 \sqrt {d x +c}\, \sqrt {b x +a}\, b c d +4 \sqrt {d x +c}\, \sqrt {b x +a}\, b \,d^{2} x +4 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a c d +4 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,d^{2} x -12 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b \,c^{2}-12 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b c d x -\sqrt {d}\, \sqrt {b}\, a c d -\sqrt {d}\, \sqrt {b}\, a \,d^{2} x +9 \sqrt {d}\, \sqrt {b}\, b \,c^{2}+9 \sqrt {d}\, \sqrt {b}\, b c d x}{4 b \,d^{3} \left (d x +c \right )} \] Input:
int(x*(b*x+a)^(1/2)/(d*x+c)^(3/2),x)
Output:
(12*sqrt(c + d*x)*sqrt(a + b*x)*b*c*d + 4*sqrt(c + d*x)*sqrt(a + b*x)*b*d* *2*x + 4*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x ))/sqrt(a*d - b*c))*a*c*d + 4*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*d**2*x - 12*sqrt(d)*sqrt(b)*log ((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b*c**2 - 12*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sq rt(a*d - b*c))*b*c*d*x - sqrt(d)*sqrt(b)*a*c*d - sqrt(d)*sqrt(b)*a*d**2*x + 9*sqrt(d)*sqrt(b)*b*c**2 + 9*sqrt(d)*sqrt(b)*b*c*d*x)/(4*b*d**3*(c + d*x ))