Integrand size = 17, antiderivative size = 114 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=-\frac {\sqrt {c+d x}}{2 (b c-a d) (a+b x)^2}+\frac {3 d \sqrt {c+d x}}{4 (b c-a d)^2 (a+b x)}-\frac {3 d^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{5/2}} \] Output:
-1/2*(d*x+c)^(1/2)/(-a*d+b*c)/(b*x+a)^2+3/4*d*(d*x+c)^(1/2)/(-a*d+b*c)^2/( b*x+a)-3/4*d^2*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(1/2)/(-a *d+b*c)^(5/2)
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {1}{4} \left (\frac {\sqrt {c+d x} (-2 b c+5 a d+3 b d x)}{(b c-a d)^2 (a+b x)^2}+\frac {3 d^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{5/2}}\right ) \] Input:
Integrate[1/((a + b*x)^3*Sqrt[c + d*x]),x]
Output:
((Sqrt[c + d*x]*(-2*b*c + 5*a*d + 3*b*d*x))/((b*c - a*d)^2*(a + b*x)^2) + (3*d^2*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(Sqrt[b]*(-(b*c ) + a*d)^(5/2)))/4
Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {52, 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.
\(\displaystyle \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {3 d \int \frac {1}{(a+b x)^2 \sqrt {c+d x}}dx}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {3 d \left (-\frac {d \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{2 (b c-a d)}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {3 d \left (-\frac {\int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{b c-a d}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {3 d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{\sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x}}{(a+b x) (b c-a d)}\right )}{4 (b c-a d)}-\frac {\sqrt {c+d x}}{2 (a+b x)^2 (b c-a d)}\) |
Input:
Int[1/((a + b*x)^3*Sqrt[c + d*x]),x]
Output:
-1/2*Sqrt[c + d*x]/((b*c - a*d)*(a + b*x)^2) - (3*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*c - a*d))
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 = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\frac {3 d^{2} \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{4}+\frac {5 \left (\frac {\left (3 x d -2 c \right ) b}{5}+a d \right ) \sqrt {x d +c}\, \sqrt {\left (a d -b c \right ) b}}{4}}{\left (b x +a \right )^{2} \left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(102\) |
derivativedivides | \(2 d^{2} \left (\frac {\sqrt {x d +c}}{4 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {\frac {3 \sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}}{a d -b c}\right )\) | \(138\) |
default | \(2 d^{2} \left (\frac {\sqrt {x d +c}}{4 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )^{2}}+\frac {\frac {3 \sqrt {x d +c}}{8 \left (a d -b c \right ) \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}}{a d -b c}\right )\) | \(138\) |
Input:
int(1/(b*x+a)^3/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
5/4*(3/5*d^2*(b*x+a)^2*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))+(1/5*(3 *d*x-2*c)*b+a*d)*(d*x+c)^(1/2)*((a*d-b*c)*b)^(1/2))/((a*d-b*c)*b)^(1/2)/(b *x+a)^2/(a*d-b*c)^2
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (94) = 188\).
Time = 0.09 (sec) , antiderivative size = 549, normalized size of antiderivative = 4.82 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{8 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (2 \, b^{3} c^{2} - 7 \, a b^{2} c d + 5 \, a^{2} b d^{2} - 3 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{4 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x\right )}}\right ] \] Input:
integrate(1/(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
[1/8*(3*(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 - 7*a*b^2*c*d + 5*a^2*b*d^2 - 3*(b^3*c*d - a*b^2*d^2)*x)*sqrt(d*x + c))/(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3 + (b^6* c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^2 + 2*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*x), 1/4*(3*(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)*s qrt(d*x + c)/(b*d*x + b*c)) - (2*b^3*c^2 - 7*a*b^2*c*d + 5*a^2*b*d^2 - 3*( b^3*c*d - a*b^2*d^2)*x)*sqrt(d*x + c))/(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3* a^4*b^2*c*d^2 - a^5*b*d^3 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a ^3*b^3*d^3)*x^2 + 2*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b ^2*d^3)*x)]
\[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{3} \sqrt {c + d x}}\, dx \] Input:
integrate(1/(b*x+a)**3/(d*x+c)**(1/2),x)
Output:
Integral(1/((a + b*x)**3*sqrt(c + d*x)), x)
Exception generated. \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a)^3/(d*x+c)^(1/2),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 = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {3 \, d^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {3 \, {\left (d x + c\right )}^{\frac {3}{2}} b d^{2} - 5 \, \sqrt {d x + c} b c d^{2} + 5 \, \sqrt {d x + c} a d^{3}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \] Input:
integrate(1/(b*x+a)^3/(d*x+c)^(1/2),x, algorithm="giac")
Output:
3/4*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-b^2*c + a*b*d)) + 1/4*(3*(d*x + c)^(3/2)*b*d^2 - 5*sqrt( d*x + c)*b*c*d^2 + 5*sqrt(d*x + c)*a*d^3)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2) *((d*x + c)*b - b*c + a*d)^2)
Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {\frac {5\,d^2\,\sqrt {c+d\,x}}{4\,\left (a\,d-b\,c\right )}+\frac {3\,b\,d^2\,{\left (c+d\,x\right )}^{3/2}}{4\,{\left (a\,d-b\,c\right )}^2}}{b^2\,{\left (c+d\,x\right )}^2-\left (2\,b^2\,c-2\,a\,b\,d\right )\,\left (c+d\,x\right )+a^2\,d^2+b^2\,c^2-2\,a\,b\,c\,d}+\frac {3\,d^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{4\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}} \] Input:
int(1/((a + b*x)^3*(c + d*x)^(1/2)),x)
Output:
((5*d^2*(c + d*x)^(1/2))/(4*(a*d - b*c)) + (3*b*d^2*(c + d*x)^(3/2))/(4*(a *d - b*c)^2))/(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) + (3*d^2*atan((b^(1/2)*(c + d*x)^(1/2))/(a*d - b*c)^ (1/2)))/(4*b^(1/2)*(a*d - b*c)^(5/2))
Time = 0.16 (sec) , antiderivative size = 352, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx=\frac {3 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} d^{2}+6 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a b \,d^{2} x +3 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{2} d^{2} x^{2}+5 \sqrt {d x +c}\, a^{2} b \,d^{2}-7 \sqrt {d x +c}\, a \,b^{2} c d +3 \sqrt {d x +c}\, a \,b^{2} d^{2} x +2 \sqrt {d x +c}\, b^{3} c^{2}-3 \sqrt {d x +c}\, b^{3} c d x}{4 b \left (a^{3} b^{2} d^{3} x^{2}-3 a^{2} b^{3} c \,d^{2} x^{2}+3 a \,b^{4} c^{2} d \,x^{2}-b^{5} c^{3} x^{2}+2 a^{4} b \,d^{3} x -6 a^{3} b^{2} c \,d^{2} x +6 a^{2} b^{3} c^{2} d x -2 a \,b^{4} c^{3} x +a^{5} d^{3}-3 a^{4} b c \,d^{2}+3 a^{3} b^{2} c^{2} d -a^{2} b^{3} c^{3}\right )} \] Input:
int(1/(b*x+a)^3/(d*x+c)^(1/2),x)
Output:
(3*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c) ))*a**2*d**2 + 6*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*s qrt(a*d - b*c)))*a*b*d**2*x + 3*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 + 5*sqrt(c + d*x)*a**2*b*d* *2 - 7*sqrt(c + d*x)*a*b**2*c*d + 3*sqrt(c + d*x)*a*b**2*d**2*x + 2*sqrt(c + d*x)*b**3*c**2 - 3*sqrt(c + d*x)*b**3*c*d*x)/(4*b*(a**5*d**3 - 3*a**4*b *c*d**2 + 2*a**4*b*d**3*x + 3*a**3*b**2*c**2*d - 6*a**3*b**2*c*d**2*x + a* *3*b**2*d**3*x**2 - a**2*b**3*c**3 + 6*a**2*b**3*c**2*d*x - 3*a**2*b**3*c* d**2*x**2 - 2*a*b**4*c**3*x + 3*a*b**4*c**2*d*x**2 - b**5*c**3*x**2))