Integrand size = 17, antiderivative size = 93 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2}{3 (b c-a d) (c+d x)^{3/2}}+\frac {2 b}{(b c-a d)^2 \sqrt {c+d x}}-\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \] Output:
2/3/(-a*d+b*c)/(d*x+c)^(3/2)+2*b/(-a*d+b*c)^2/(d*x+c)^(1/2)-2*b^(3/2)*arct anh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2))/(-a*d+b*c)^(5/2)
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2 (4 b c-a d+3 b d x)}{3 (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}} \] Input:
Integrate[1/((a + b*x)*(c + d*x)^(5/2)),x]
Output:
(2*(4*b*c - a*d + 3*b*d*x))/(3*(b*c - a*d)^2*(c + d*x)^(3/2)) + (2*b^(3/2) *ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(5/2)
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {61, 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.
\(\displaystyle \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {b \int \frac {1}{(a+b x) (c+d x)^{3/2}}dx}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {c+d x}}dx}{b c-a d}+\frac {2}{\sqrt {c+d x} (b c-a d)}\right )}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (c+d x)}{d}-\frac {b c}{d}}d\sqrt {c+d x}}{d (b c-a d)}+\frac {2}{\sqrt {c+d x} (b c-a d)}\right )}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {b \left (\frac {2}{\sqrt {c+d x} (b c-a d)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}\right )}{b c-a d}+\frac {2}{3 (c+d x)^{3/2} (b c-a d)}\) |
Input:
Int[1/((a + b*x)*(c + d*x)^(5/2)),x]
Output:
2/(3*(b*c - a*d)*(c + d*x)^(3/2)) + (b*(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)))/(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] && 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 = 0.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(-\frac {2}{3 \left (a d -b c \right ) \left (x d +c \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a d -b c \right )^{2} \sqrt {x d +c}}+\frac {2 b^{2} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(90\) |
default | \(-\frac {2}{3 \left (a d -b c \right ) \left (x d +c \right )^{\frac {3}{2}}}+\frac {2 b}{\left (a d -b c \right )^{2} \sqrt {x d +c}}+\frac {2 b^{2} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(90\) |
pseudoelliptic | \(-\frac {2 \left (-3 b^{2} \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right ) \left (x d +c \right )^{\frac {3}{2}}+\sqrt {\left (a d -b c \right ) b}\, \left (\left (-3 x d -4 c \right ) b +a d \right )\right )}{3 \sqrt {\left (a d -b c \right ) b}\, \left (x d +c \right )^{\frac {3}{2}} \left (a d -b c \right )^{2}}\) | \(94\) |
Input:
int(1/(b*x+a)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3/(a*d-b*c)/(d*x+c)^(3/2)+2/(a*d-b*c)^2*b/(d*x+c)^(1/2)+2*b^2/(a*d-b*c) ^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (77) = 154\).
Time = 0.10 (sec) , antiderivative size = 378, normalized size of antiderivative = 4.06 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + 2 \, {\left (3 \, b d x + 4 \, b c - a d\right )} \sqrt {d x + c}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}\right ) + {\left (3 \, b d x + 4 \, b c - a d\right )} \sqrt {d x + c}\right )}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}}\right ] \] Input:
integrate(1/(b*x+a)/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
[1/3*(3*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*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 + 4*b*c - a*d)*sqrt(d*x + c))/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2 *d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^ 2*d^2 + a^2*c*d^3)*x), 2/3*(3*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(-b/(b*c - a*d))*arctan(sqrt(d*x + c)*sqrt(-b/(b*c - a*d))) + (3*b*d*x + 4*b*c - a *d)*sqrt(d*x + c))/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2 *a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x)]
Time = 2.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b d}{\sqrt {c + d x} \left (a d - b c\right )^{2}} + \frac {b d \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{\sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )^{2}} - \frac {d}{3 \left (c + d x\right )^{\frac {3}{2}} \left (a d - b c\right )}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(b*x+a)/(d*x+c)**(5/2),x)
Output:
Piecewise((2*(b*d/(sqrt(c + d*x)*(a*d - b*c)**2) + b*d*atan(sqrt(c + d*x)/ sqrt((a*d - b*c)/b))/(sqrt((a*d - b*c)/b)*(a*d - b*c)**2) - d/(3*(c + d*x) **(3/2)*(a*d - b*c)))/d, Ne(d, 0)), (Piecewise((x/a, Eq(b, 0)), (log(a + b *x)/b, True))/c**(5/2), True))
Exception generated. \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(b*x+a)/(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
Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2 \, b^{2} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )} b + b c - a d\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}}} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(5/2),x, algorithm="giac")
Output:
2*b^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)) + 2/3*(3*(d*x + c)*b + b*c - a*d)/((b^2*c^ 2 - 2*a*b*c*d + a^2*d^2)*(d*x + c)^(3/2))
Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^{5/2}}\right )}{{\left (a\,d-b\,c\right )}^{5/2}}-\frac {\frac {2}{3\,\left (a\,d-b\,c\right )}-\frac {2\,b\,\left (c+d\,x\right )}{{\left (a\,d-b\,c\right )}^2}}{{\left (c+d\,x\right )}^{3/2}} \] Input:
int(1/((a + b*x)*(c + d*x)^(5/2)),x)
Output:
(2*b^(3/2)*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) - (2/(3*(a*d - b*c)) - (2*b*(c + d*x ))/(a*d - b*c)^2)/(c + d*x)^(3/2)
Time = 0.17 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.40 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b c +2 \sqrt {b}\, \sqrt {d x +c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b d x -\frac {2 a^{2} d^{2}}{3}+\frac {10 a b c d}{3}+2 a b \,d^{2} x -\frac {8 b^{2} c^{2}}{3}-2 b^{2} c d x}{\sqrt {d x +c}\, \left (a^{3} d^{4} x -3 a^{2} b c \,d^{3} x +3 a \,b^{2} c^{2} d^{2} x -b^{3} c^{3} d x +a^{3} c \,d^{3}-3 a^{2} b \,c^{2} d^{2}+3 a \,b^{2} c^{3} d -b^{3} c^{4}\right )} \] Input:
int(1/(b*x+a)/(d*x+c)^(5/2),x)
Output:
(2*(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)))*b*c + 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 - a**2*d**2 + 5*a*b*c*d + 3*a*b*d**2*x - 4*b**2*c**2 - 3*b**2*c*d*x))/(3*sqrt(c + d*x)*(a**3*c*d**3 + a**3*d**4*x - 3*a**2*b*c**2*d**2 - 3*a**2*b*c*d**3*x + 3*a*b**2*c**3*d + 3*a*b**2*c**2*d**2*x - b**3*c**4 - b**3*c**3*d*x))