Integrand size = 17, antiderivative size = 110 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}-\frac {d \sqrt {c+d x}}{4 b (b c-a d) (a+b x)}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}} \] Output:
-1/2*(d*x+c)^(1/2)/b/(b*x+a)^2-1/4*d*(d*x+c)^(1/2)/b/(-a*d+b*c)/(b*x+a)+1/ 4*d^2*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(-a*d+b*c)^( 3/2)
Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {c+d x} (2 b c-a d+b d x)}{(-b c+a d) (a+b x)^2}+\frac {d^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{4 b^{3/2}} \] Input:
Integrate[Sqrt[c + d*x]/(a + b*x)^3,x]
Output:
((Sqrt[b]*Sqrt[c + d*x]*(2*b*c - a*d + b*d*x))/((-(b*c) + a*d)*(a + b*x)^2 ) + (d^2*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d )^(3/2))/(4*b^(3/2))
Time = 0.20 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 52, 73, 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.
\(\displaystyle \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}}dx}{4 b}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {d \left (-\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{2 (b c-a d)}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 b}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {d \left (-\frac {\int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b c-a d}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 b}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 b}-\frac {\sqrt {c+d x}}{2 b (a+b x)^2}\) |
Input:
Int[Sqrt[c + d*x]/(a + b*x)^3,x]
Output:
-1/2*Sqrt[c + d*x]/(b*(a + b*x)^2) + (d*(-(Sqrt[c + d*x]/((b*c - a*d)*(a + b*x))) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(Sqrt[b]*(b *c - a*d)^(3/2))))/(4*b)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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]
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {d^{2} \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\sqrt {\left (a d -b c \right ) b}\, \left (\left (-x d -2 c \right ) b +a d \right ) \sqrt {x d +c}}{4 \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right ) b \left (b x +a \right )^{2}}\) | \(104\) |
derivativedivides | \(2 d^{2} \left (\frac {\frac {\left (x d +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {x d +c}}{8 b}}{\left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(106\) |
default | \(2 d^{2} \left (\frac {\frac {\left (x d +c \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\sqrt {x d +c}}{8 b}}{\left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {\arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) b \sqrt {\left (a d -b c \right ) b}}\right )\) | \(106\) |
Input:
int((d*x+c)^(1/2)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/4*(d^2*(b*x+a)^2*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-((a*d-b*c)* b)^(1/2)*((-d*x-2*c)*b+a*d)*(d*x+c)^(1/2))/((a*d-b*c)*b)^(1/2)/(a*d-b*c)/b /(b*x+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (90) = 180\).
Time = 0.11 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.15 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\left [-\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}, -\frac {{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} + {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}\right ] \] Input:
integrate((d*x+c)^(1/2)/(b*x+a)^3,x, algorithm="fricas")
Output:
[-1/8*((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqrt(b^2*c - a*b*d)*log((b*d* x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sqrt(d*x + c))/(b*x + a)) + 2*(2*b ^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x)*sqrt(d*x + c)) /(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2 *b^4*d^2)*x^2 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x), -1/4*((b^2 *d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) + (2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d ^2 + (b^3*c*d - a*b^2*d^2)*x)*sqrt(d*x + c))/(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^2 + 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x)]
\[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{3}}\, dx \] Input:
integrate((d*x+c)**(1/2)/(b*x+a)**3,x)
Output:
Integral(sqrt(c + d*x)/(a + b*x)**3, x)
Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(1/2)/(b*x+a)^3,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 = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=-\frac {d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{2} c - a b d\right )} \sqrt {-b^{2} c + a b d}} - \frac {{\left (d x + c\right )}^{\frac {3}{2}} b d^{2} + \sqrt {d x + c} b c d^{2} - \sqrt {d x + c} a d^{3}}{4 \, {\left (b^{2} c - a b d\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \] Input:
integrate((d*x+c)^(1/2)/(b*x+a)^3,x, algorithm="giac")
Output:
-1/4*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c - a*b*d)*sqr t(-b^2*c + a*b*d)) - 1/4*((d*x + c)^(3/2)*b*d^2 + sqrt(d*x + c)*b*c*d^2 - sqrt(d*x + c)*a*d^3)/((b^2*c - a*b*d)*((d*x + c)*b - b*c + a*d)^2)
Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {\frac {d^2\,\sqrt {c+d\,x}}{4\,b}-\frac {d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,\left (a\,d-b\,c\right )}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d} \] Input:
int((c + d*x)^(1/2)/(a + b*x)^3,x)
Output:
(d^2*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^(1/2)))/(4*b^(3/2)*(a*d - b*c)^(3/2)) - ((d^2*(c + d*x)^(1/2))/(4*b) - (d^2*(c + d*x)^(3/2))/(4*(a*d - b*c)))/(b^2*(c + d*x)^2 - (2*b^2*c - 2*a*b*d)*(c + d*x) + a^2*d^2 + b^2 *c^2 - 2*a*b*c*d)
Time = 0.17 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.74 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx=\frac {\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} d^{2}+2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a b \,d^{2} x +\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{2} d^{2} x^{2}-\sqrt {d x +c}\, a^{2} b \,d^{2}+3 \sqrt {d x +c}\, a \,b^{2} c d +\sqrt {d x +c}\, a \,b^{2} d^{2} x -2 \sqrt {d x +c}\, b^{3} c^{2}-\sqrt {d x +c}\, b^{3} c d x}{4 b^{2} \left (a^{2} b^{2} d^{2} x^{2}-2 a \,b^{3} c d \,x^{2}+b^{4} c^{2} x^{2}+2 a^{3} b \,d^{2} x -4 a^{2} b^{2} c d x +2 a \,b^{3} c^{2} x +a^{4} d^{2}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right )} \] Input:
int((d*x+c)^(1/2)/(b*x+a)^3,x)
Output:
(sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c))) *a**2*d**2 + 2*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqr t(a*d - b*c)))*a*b*d**2*x + sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b) /(sqrt(b)*sqrt(a*d - b*c)))*b**2*d**2*x**2 - sqrt(c + d*x)*a**2*b*d**2 + 3 *sqrt(c + d*x)*a*b**2*c*d + sqrt(c + d*x)*a*b**2*d**2*x - 2*sqrt(c + d*x)* b**3*c**2 - sqrt(c + d*x)*b**3*c*d*x)/(4*b**2*(a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x + a**2*b**2*c**2 - 4*a**2*b**2*c*d*x + a**2*b**2*d**2*x**2 + 2*a*b**3*c**2*x - 2*a*b**3*c*d*x**2 + b**4*c**2*x**2))