Integrand size = 37, antiderivative size = 101 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}+\frac {e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}} \] Output:
-(e*x+d)^(1/2)/(-a*e^2+c*d^2)/(c*d*x+a*e)+e*arctanh(c^(1/2)*d^(1/2)*(e*x+d )^(1/2)/(-a*e^2+c*d^2)^(1/2))/c^(1/2)/d^(1/2)/(-a*e^2+c*d^2)^(3/2)
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d+e x}}{\left (-c d^2+a e^2\right ) (a e+c d x)}+\frac {e \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (-c d^2+a e^2\right )^{3/2}} \] Input:
Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
Sqrt[d + e*x]/((-(c*d^2) + a*e^2)*(a*e + c*d*x)) + (e*ArcTan[(Sqrt[c]*Sqrt [d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(Sqrt[c]*Sqrt[d]*(-(c*d^2) + a *e^2)^(3/2))
Time = 0.36 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1121, 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.
\(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \frac {1}{\sqrt {d+e x} (a e+c d x)^2}dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {e \int \frac {1}{(a e+c d x) \sqrt {d+e x}}dx}{2 \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\int \frac {1}{-\frac {c d^2}{e}+\frac {c (d+e x) d}{e}+a e}d\sqrt {d+e x}}{c d^2-a e^2}-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}}-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}\) |
Input:
Int[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
-(Sqrt[d + e*x]/((c*d^2 - a*e^2)*(a*e + c*d*x))) + (e*ArcTanh[(Sqrt[c]*Sqr t[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2) ^(3/2))
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]
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 1.86 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {e x +d}}{c d x +a e}+\frac {e \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}}{a \,e^{2}-c \,d^{2}}\) | \(82\) |
derivativedivides | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (a \,e^{2}-c \,d^{2}\right ) \left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )}+\frac {\arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )\) | \(111\) |
default | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (a \,e^{2}-c \,d^{2}\right ) \left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )}+\frac {\arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )\) | \(111\) |
Input:
int((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERB OSE)
Output:
1/(a*e^2-c*d^2)*((e*x+d)^(1/2)/(c*d*x+a*e)+e/(c*d*(a*e^2-c*d^2))^(1/2)*arc tan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d^2))^(1/2)))
Time = 0.11 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.50 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [-\frac {\sqrt {c^{2} d^{3} - a c d e^{2}} {\left (c d e x + a e^{2}\right )} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {e x + d}}{c d x + a e}\right ) + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x\right )}}, -\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d e x + a e^{2}\right )} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {e x + d}}{c d e x + c d^{2}}\right ) + {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x}\right ] \] Input:
integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" fricas")
Output:
[-1/2*(sqrt(c^2*d^3 - a*c*d*e^2)*(c*d*e*x + a*e^2)*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(c*d*x + a*e)) + 2*(c ^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(a*c^3*d^5*e - 2*a^2*c^2*d^3*e^3 + a^3* c*d*e^5 + (c^4*d^6 - 2*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*x), -(sqrt(-c^2*d^ 3 + a*c*d*e^2)*(c*d*e*x + a*e^2)*arctan(sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e* x + d)/(c*d*e*x + c*d^2)) + (c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(a*c^3*d^ 5*e - 2*a^2*c^2*d^3*e^3 + a^3*c*d*e^5 + (c^4*d^6 - 2*a*c^3*d^4*e^2 + a^2*c ^2*d^2*e^4)*x)]
\[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\int \frac {1}{\sqrt {d + e x} \left (a e + c d x\right )^{2}}\, dx \] Input:
integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
Integral(1/(sqrt(d + e*x)*(a*e + c*d*x)**2), x)
Exception generated. \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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*e^2-c*d^2>0)', see `assume?` f or more de
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {e \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d^{2} - a e^{2}\right )}} - \frac {\sqrt {e x + d} e}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} {\left (c d^{2} - a e^{2}\right )}} \] Input:
integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" giac")
Output:
-e*arctan(sqrt(e*x + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/(sqrt(-c^2*d^3 + a *c*d*e^2)*(c*d^2 - a*e^2)) - sqrt(e*x + d)*e/(((e*x + d)*c*d - c*d^2 + a*e ^2)*(c*d^2 - a*e^2))
Time = 5.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e\,\mathrm {atan}\left (\frac {c\,d\,\sqrt {d+e\,x}}{\sqrt {c\,d}\,\sqrt {a\,e^2-c\,d^2}}\right )}{\sqrt {c\,d}\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}+\frac {e\,\sqrt {d+e\,x}}{\left (a\,e^2-c\,d^2\right )\,\left (a\,e^2-c\,d^2+c\,d\,\left (d+e\,x\right )\right )} \] Input:
int((d + e*x)^(3/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
(e*atan((c*d*(d + e*x)^(1/2))/((c*d)^(1/2)*(a*e^2 - c*d^2)^(1/2))))/((c*d) ^(1/2)*(a*e^2 - c*d^2)^(3/2)) + (e*(d + e*x)^(1/2))/((a*e^2 - c*d^2)*(a*e^ 2 - c*d^2 + c*d*(d + e*x)))
Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.05 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{2}+\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) c d e x +\sqrt {e x +d}\, a c d \,e^{2}-\sqrt {e x +d}\, c^{2} d^{3}}{c d \left (a^{2} c d \,e^{4} x -2 a \,c^{2} d^{3} e^{2} x +c^{3} d^{5} x +a^{3} e^{5}-2 a^{2} c \,d^{2} e^{3}+a \,c^{2} d^{4} e \right )} \] Input:
int((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
Output:
(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*s qrt(c)*sqrt(a*e**2 - c*d**2)))*a*e**2 + sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d* *2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*c*d* e*x + sqrt(d + e*x)*a*c*d*e**2 - sqrt(d + e*x)*c**2*d**3)/(c*d*(a**3*e**5 - 2*a**2*c*d**2*e**3 + a**2*c*d*e**4*x + a*c**2*d**4*e - 2*a*c**2*d**3*e** 2*x + c**3*d**5*x))