Integrand size = 17, antiderivative size = 99 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=-\frac {3 d}{(b c-a d)^2 \sqrt {c+d x}}-\frac {1}{(b c-a d) (a+b x) \sqrt {c+d x}}+\frac {3 \sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \] Output:
-3*d/(-a*d+b*c)^2/(d*x+c)^(1/2)-1/(-a*d+b*c)/(b*x+a)/(d*x+c)^(1/2)+3*b^(1/ 2)*d*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/(-a*d+b*c)^(5/2)
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=-\frac {2 a d+b (c+3 d x)}{(b c-a d)^2 (a+b x) \sqrt {c+d x}}-\frac {3 \sqrt {b} d \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}} \] Input:
Integrate[1/((a + b*x)^2*(c + d*x)^(3/2)),x]
Output:
-((2*a*d + b*(c + 3*d*x))/((b*c - a*d)^2*(a + b*x)*Sqrt[c + d*x])) - (3*Sq rt[b]*d*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d) ^(5/2)
Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {52, 61, 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.
\(\displaystyle \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {3 d \int \frac {1}{(a+b x) (c+d x)^{3/2}}dx}{2 (b c-a d)}-\frac {1}{(a+b x) \sqrt {c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {3 d \left (\frac {b \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{b c-a d}+\frac {2}{\sqrt {c+d x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) \sqrt {c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {3 d \left (\frac {2 b \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d (b c-a d)}+\frac {2}{\sqrt {c+d x} (b c-a d)}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) \sqrt {c+d x} (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {3 d \left (\frac {2}{\sqrt {c+d x} (b c-a d)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}\right )}{2 (b c-a d)}-\frac {1}{(a+b x) \sqrt {c+d x} (b c-a d)}\) |
Input:
Int[1/((a + b*x)^2*(c + d*x)^(3/2)),x]
Output:
-(1/((b*c - a*d)*(a + b*x)*Sqrt[c + d*x])) - (3*d*(2/((b*c - a*d)*Sqrt[c + d*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2)))/(2*(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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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 = 78, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {-\frac {b \sqrt {x d +c}}{b x +a}-\frac {3 d b \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}-\frac {2 d}{\sqrt {x d +c}}}{\left (a d -b c \right )^{2}}\) | \(78\) |
derivativedivides | \(2 d \left (-\frac {b \left (\frac {\sqrt {x d +c}}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2}}-\frac {1}{\left (a d -b c \right )^{2} \sqrt {x d +c}}\right )\) | \(100\) |
default | \(2 d \left (-\frac {b \left (\frac {\sqrt {x d +c}}{2 \left (x d +c \right ) b +2 a d -2 b c}+\frac {3 \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2}}-\frac {1}{\left (a d -b c \right )^{2} \sqrt {x d +c}}\right )\) | \(100\) |
Input:
int(1/(b*x+a)^2/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/(a*d-b*c)^2*(-b*(d*x+c)^(1/2)/(b*x+a)-3*d*b/((a*d-b*c)*b)^(1/2)*arctan(b *(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))-2*d/(d*x+c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (85) = 170\).
Time = 0.16 (sec) , antiderivative size = 404, normalized size of antiderivative = 4.08 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) - 2 \, {\left (3 \, b d x + b c + 2 \, a d\right )} \sqrt {d x + c}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )}}, -\frac {3 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}\right ) + {\left (3 \, b d x + b c + 2 \, a d\right )} \sqrt {d x + c}}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x}\right ] \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^(3/2),x, algorithm="fricas")
Output:
[1/2*(3*(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)*sqrt(b/(b*c - a*d))*log((b *d*x + 2*b*c - a*d + 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) - 2*(3*b*d*x + b*c + 2*a*d)*sqrt(d*x + c))/(a*b^2*c^3 - 2*a^2*b*c^2 *d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x), -(3*(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)*sqrt(-b/(b*c - a*d))*arctan(sqrt(d*x + c)*sqrt(-b/(b*c - a*d))) + (3*b*d*x + b*c + 2*a*d)*sqrt(d*x + c))/(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3 *c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + (b^3*c^3 - a*b^2*c^ 2*d - a^2*b*c*d^2 + a^3*d^3)*x)]
\[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(b*x+a)**2/(d*x+c)**(3/2),x)
Output:
Integral(1/((a + b*x)**2*(c + d*x)**(3/2)), x)
Exception generated. \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a)^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 = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=-\frac {3 \, b d \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {3 \, {\left (d x + c\right )} b d - 2 \, b c d + 2 \, a d^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x + c\right )}^{\frac {3}{2}} b - \sqrt {d x + c} b c + \sqrt {d x + c} a d\right )}} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^(3/2),x, algorithm="giac")
Output:
-3*b*d*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)) - (3*(d*x + c)*b*d - 2*b*c*d + 2*a*d^2)/( (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*((d*x + c)^(3/2)*b - sqrt(d*x + c)*b*c + s qrt(d*x + c)*a*d))
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=-\frac {\frac {2\,d}{a\,d-b\,c}+\frac {3\,b\,d\,\left (c+d\,x\right )}{{\left (a\,d-b\,c\right )}^2}}{b\,{\left (c+d\,x\right )}^{3/2}+\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}-\frac {3\,\sqrt {b}\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^{5/2}}\right )}{{\left (a\,d-b\,c\right )}^{5/2}} \] Input:
int(1/((a + b*x)^2*(c + d*x)^(3/2)),x)
Output:
- ((2*d)/(a*d - b*c) + (3*b*d*(c + d*x))/(a*d - b*c)^2)/(b*(c + d*x)^(3/2) + (a*d - b*c)*(c + d*x)^(1/2)) - (3*b^(1/2)*d*atan((b^(1/2)*(c + d*x)^(1/ 2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^(5/2)))/(a*d - b*c)^(5/2)
Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx=\frac {-3 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a d -3 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b d x -2 a^{2} d^{2}+a b c d -3 a b \,d^{2} x +b^{2} c^{2}+3 b^{2} c d x}{\sqrt {d x +c}\, \left (a^{3} b \,d^{3} x -3 a^{2} b^{2} c \,d^{2} x +3 a \,b^{3} c^{2} d x -b^{4} c^{3} x +a^{4} d^{3}-3 a^{3} b c \,d^{2}+3 a^{2} b^{2} c^{2} d -a \,b^{3} c^{3}\right )} \] Input:
int(1/(b*x+a)^2/(d*x+c)^(3/2),x)
Output:
( - 3*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b )*sqrt(a*d - b*c)))*a*d - 3*sqrt(b)*sqrt(c + d*x)*sqrt(a*d - b*c)*atan((sq rt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b*d*x - 2*a**2*d**2 + a*b*c*d - 3*a*b*d**2*x + b**2*c**2 + 3*b**2*c*d*x)/(sqrt(c + d*x)*(a**4*d**3 - 3*a** 3*b*c*d**2 + a**3*b*d**3*x + 3*a**2*b**2*c**2*d - 3*a**2*b**2*c*d**2*x - a *b**3*c**3 + 3*a*b**3*c**2*d*x - b**4*c**3*x))