Integrand size = 22, antiderivative size = 112 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x}}{3 c (c+d x)^{3/2}}+\frac {2 (2 b c-3 a d) \sqrt {a+b x}}{3 c^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \] Output:
2/3*(b*x+a)^(1/2)/c/(d*x+c)^(3/2)+2/3*(-3*a*d+2*b*c)*(b*x+a)^(1/2)/c^2/(-a *d+b*c)/(d*x+c)^(1/2)-2*a^(1/2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x +c)^(1/2))/c^(5/2)
Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x} \left (3 b c-3 a d-\frac {c d (a+b x)}{c+d x}\right )}{3 c^2 (b c-a d) \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{5/2}} \] Input:
Integrate[Sqrt[a + b*x]/(x*(c + d*x)^(5/2)),x]
Output:
(2*Sqrt[a + b*x]*(3*b*c - 3*a*d - (c*d*(a + b*x))/(c + d*x)))/(3*c^2*(b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a ]*Sqrt[c + d*x])])/c^(5/2)
Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {107, 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.
\(\displaystyle \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {\int \frac {\sqrt {a+b x}}{x (c+d x)^{3/2}}dx}{c}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\frac {a \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}}{c}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {\frac {2 a \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}}{c}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 \sqrt {a+b x}}{c \sqrt {c+d x}}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}}{c}-\frac {2 d (a+b x)^{3/2}}{3 c (c+d x)^{3/2} (b c-a d)}\) |
Input:
Int[Sqrt[a + b*x]/(x*(c + d*x)^(5/2)),x]
Output:
(-2*d*(a + b*x)^(3/2))/(3*c*(b*c - a*d)*(c + d*x)^(3/2)) + ((2*Sqrt[a + b* x])/(c*Sqrt[c + d*x]) - (2*Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a ]*Sqrt[c + d*x])])/c^(3/2))/c
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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. \(429\) vs. \(2(88)=176\).
Time = 0.22 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.84
method | result | size |
default | \(-\frac {\left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} d^{3} x^{2}-3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b c \,d^{2} x^{2}+6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} c \,d^{2} x -6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b \,c^{2} d x +3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} c^{2} d -3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b \,c^{3}-6 a \,d^{2} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b c d x -8 a c d \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b \,c^{2}\right ) \sqrt {b x +a}}{3 \left (a d -b c \right ) \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \left (x d +c \right )^{\frac {3}{2}} c^{2}}\) | \(430\) |
Input:
int((b*x+a)^(1/2)/x/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(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^3*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*b*c*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^2*c*d^2*x-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^2*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^2*c^2*d-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^3-6*a*d^2*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+4* (a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c*d*x-8*a*c*d*(a*c)^(1/2)*((b*x+a)*( d*x+c))^(1/2)+6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c^2)*(b*x+a)^(1/2)/( a*d-b*c)/(a*c)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(3/2)/c^2
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 479, normalized size of antiderivative = 4.28 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, b c^{2} - 4 \, a c d + {\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b c^{5} - a c^{4} d + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}, \frac {3 \, {\left (b c^{3} - a c^{2} d + {\left (b c d^{2} - a d^{3}\right )} x^{2} + 2 \, {\left (b c^{2} d - a c d^{2}\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} - 4 \, a c d + {\left (2 \, b c d - 3 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b c^{5} - a c^{4} d + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{2} + 2 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} x\right )}}\right ] \] Input:
integrate((b*x+a)^(1/2)/x/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
[1/6*(3*(b*c^3 - a*c^2*d + (b*c*d^2 - a*d^3)*x^2 + 2*(b*c^2*d - a*c*d^2)*x )*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*(3*b*c^2 - 4*a*c*d + (2*b*c*d - 3*a*d^2)*x)*sqrt( b*x + a)*sqrt(d*x + c))/(b*c^5 - a*c^4*d + (b*c^3*d^2 - a*c^2*d^3)*x^2 + 2 *(b*c^4*d - a*c^3*d^2)*x), 1/3*(3*(b*c^3 - a*c^2*d + (b*c*d^2 - a*d^3)*x^2 + 2*(b*c^2*d - a*c*d^2)*x)*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)) + 2*(3*b*c^2 - 4*a*c*d + (2*b*c*d - 3*a*d^2)*x)*sqrt(b*x + a)*sqrt(d *x + c))/(b*c^5 - a*c^4*d + (b*c^3*d^2 - a*c^2*d^3)*x^2 + 2*(b*c^4*d - a*c ^3*d^2)*x)]
\[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a + b x}}{x \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((b*x+a)**(1/2)/x/(d*x+c)**(5/2),x)
Output:
Integral(sqrt(a + b*x)/(x*(c + d*x)**(5/2)), x)
Exception generated. \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^(1/2)/x/(d*x+c)^(5/2),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
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (88) = 176\).
Time = 0.19 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.17 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (2 \, b^{4} c^{3} d^{2} {\left | b \right |} - 3 \, a b^{3} c^{2} d^{3} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{5} d - a b^{2} c^{4} d^{2}} + \frac {3 \, {\left (b^{5} c^{4} d {\left | b \right |} - 2 \, a b^{4} c^{3} d^{2} {\left | b \right |} + a^{2} b^{3} c^{2} d^{3} {\left | b \right |}\right )}}{b^{3} c^{5} d - a b^{2} c^{4} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, \sqrt {b d} a b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c^{2} {\left | b \right |}} \] Input:
integrate((b*x+a)^(1/2)/x/(d*x+c)^(5/2),x, algorithm="giac")
Output:
2/3*sqrt(b*x + a)*((2*b^4*c^3*d^2*abs(b) - 3*a*b^3*c^2*d^3*abs(b))*(b*x + a)/(b^3*c^5*d - a*b^2*c^4*d^2) + 3*(b^5*c^4*d*abs(b) - 2*a*b^4*c^3*d^2*abs (b) + a^2*b^3*c^2*d^3*abs(b))/(b^3*c^5*d - a*b^2*c^4*d^2))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 2*sqrt(b*d)*a*b*arctan(-1/2*(b^2*c + a*b*d - (sqr t(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^2*abs(b))
Timed out. \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx=\int \frac {\sqrt {a+b\,x}}{x\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*x)^(1/2)/(x*(c + d*x)^(5/2)),x)
Output:
int((a + b*x)^(1/2)/(x*(c + d*x)^(5/2)), x)
Time = 0.29 (sec) , antiderivative size = 1126, normalized size of antiderivative = 10.05 \[ \int \frac {\sqrt {a+b x}}{x (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^(1/2)/x/(d*x+c)^(5/2),x)
Output:
(8*sqrt(c + d*x)*sqrt(a + b*x)*a*c**2*d**2 + 6*sqrt(c + d*x)*sqrt(a + b*x) *a*c*d**3*x - 6*sqrt(c + d*x)*sqrt(a + b*x)*b*c**3*d - 4*sqrt(c + d*x)*sqr t(a + b*x)*b*c**2*d**2*x + 3*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c) *sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d *x))*a*c**2*d**2 + 6*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b) *sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*c *d**3*x + 3*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*d**4*x**2 - 3*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b*c**3*d - 6*sqrt(c) *sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt (d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b*c**2*d**2*x - 3*sqrt(c)*sqrt( a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sq rt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b*c*d**3*x**2 + 3*sqrt(c)*sqrt(a)*log (sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b* x) + sqrt(b)*sqrt(c + d*x))*a*c**2*d**2 + 6*sqrt(c)*sqrt(a)*log(sqrt(2*sqr t(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b )*sqrt(c + d*x))*a*c*d**3*x + 3*sqrt(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c) *sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d *x))*a*d**4*x**2 - 3*sqrt(c)*sqrt(a)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)...