Integrand size = 37, antiderivative size = 130 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {3 e \sqrt {d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{3/2}}{2 c d (a e+c d x)^2}-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt {c d^2-a e^2}} \] Output:
-3/4*e*(e*x+d)^(1/2)/c^2/d^2/(c*d*x+a*e)-1/2*(e*x+d)^(3/2)/c/d/(c*d*x+a*e) ^2-3/4*e^2*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/c^( 5/2)/d^(5/2)/(-a*e^2+c*d^2)^(1/2)
Time = 0.44 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (3 a e^2+c d (2 d+5 e x)\right )}{4 c^2 d^2 (a e+c d x)^2}+\frac {3 e^2 \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt {-c d^2+a e^2}} \] Input:
Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
-1/4*(Sqrt[d + e*x]*(3*a*e^2 + c*d*(2*d + 5*e*x)))/(c^2*d^2*(a*e + c*d*x)^ 2) + (3*e^2*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]] )/(4*c^(5/2)*d^(5/2)*Sqrt[-(c*d^2) + a*e^2])
Time = 0.38 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1121, 51, 51, 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.
\(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \frac {(d+e x)^{3/2}}{(a e+c d x)^3}dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {3 e \int \frac {\sqrt {d+e x}}{(a e+c d x)^2}dx}{4 c d}-\frac {(d+e x)^{3/2}}{2 c d (a e+c d x)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {3 e \left (\frac {e \int \frac {1}{(a e+c d x) \sqrt {d+e x}}dx}{2 c d}-\frac {\sqrt {d+e x}}{c d (a e+c d x)}\right )}{4 c d}-\frac {(d+e x)^{3/2}}{2 c d (a e+c d x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 e \left (\frac {\int \frac {1}{-\frac {c d^2}{e}+\frac {c (d+e x) d}{e}+a e}d\sqrt {d+e x}}{c d}-\frac {\sqrt {d+e x}}{c d (a e+c d x)}\right )}{4 c d}-\frac {(d+e x)^{3/2}}{2 c d (a e+c d x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 e \left (-\frac {e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt {c d^2-a e^2}}-\frac {\sqrt {d+e x}}{c d (a e+c d x)}\right )}{4 c d}-\frac {(d+e x)^{3/2}}{2 c d (a e+c d x)^2}\) |
Input:
Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
-1/2*(d + e*x)^(3/2)/(c*d*(a*e + c*d*x)^2) + (3*e*(-(Sqrt[d + e*x]/(c*d*(a *e + c*d*x))) - (e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a* e^2]])/(c^(3/2)*d^(3/2)*Sqrt[c*d^2 - a*e^2])))/(4*c*d)
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] :> 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 = 15.54 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {e^{2} \left (-\frac {\sqrt {e x +d}\, \left (5 c d x e +3 a \,e^{2}+2 c \,d^{2}\right )}{e^{2} \left (c d x +a e \right )^{2}}+\frac {3 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{4 c^{2} d^{2}}\) | \(101\) |
derivativedivides | \(2 e^{2} \left (\frac {-\frac {5 \left (e x +d \right )^{\frac {3}{2}}}{8 c d}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}}{8 c^{2} d^{2}}}{\left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {3 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{8 c^{2} d^{2} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )\) | \(126\) |
default | \(2 e^{2} \left (\frac {-\frac {5 \left (e x +d \right )^{\frac {3}{2}}}{8 c d}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) \sqrt {e x +d}}{8 c^{2} d^{2}}}{\left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {3 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{8 c^{2} d^{2} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )\) | \(126\) |
Input:
int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERB OSE)
Output:
1/4*e^2/c^2/d^2*(-(e*x+d)^(1/2)*(5*c*d*e*x+3*a*e^2+2*c*d^2)/e^2/(c*d*x+a*e )^2+3/(c*d*(a*e^2-c*d^2))^(1/2)*arctan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d^2 ))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (106) = 212\).
Time = 0.11 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.78 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt {c^{2} d^{3} - a c d e^{2}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {e x + d}}{c d x + a e}\right ) - 2 \, {\left (2 \, c^{3} d^{5} + a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4} + 5 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} c^{4} d^{5} e^{2} - a^{3} c^{3} d^{3} e^{4} + {\left (c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{2} + 2 \, {\left (a c^{5} d^{6} e - a^{2} c^{4} d^{4} e^{3}\right )} x\right )}}, \frac {3 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {e x + d}}{c d e x + c d^{2}}\right ) - {\left (2 \, c^{3} d^{5} + a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4} + 5 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} c^{4} d^{5} e^{2} - a^{3} c^{3} d^{3} e^{4} + {\left (c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{2} + 2 \, {\left (a c^{5} d^{6} e - a^{2} c^{4} d^{4} e^{3}\right )} x\right )}}\right ] \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm=" fricas")
Output:
[1/8*(3*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*sqrt(c^2*d^3 - a*c*d*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*(2*c^3*d^5 + a*c^2*d^3*e^2 - 3*a^2*c*d*e^4 + 5*(c ^3*d^4*e - a*c^2*d^2*e^3)*x)*sqrt(e*x + d))/(a^2*c^4*d^5*e^2 - a^3*c^3*d^3 *e^4 + (c^6*d^7 - a*c^5*d^5*e^2)*x^2 + 2*(a*c^5*d^6*e - a^2*c^4*d^4*e^3)*x ), 1/4*(3*(c^2*d^2*e^2*x^2 + 2*a*c*d*e^3*x + a^2*e^4)*sqrt(-c^2*d^3 + a*c* d*e^2)*arctan(sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d)/(c*d*e*x + c*d^2)) - (2*c^3*d^5 + a*c^2*d^3*e^2 - 3*a^2*c*d*e^4 + 5*(c^3*d^4*e - a*c^2*d^2*e^ 3)*x)*sqrt(e*x + d))/(a^2*c^4*d^5*e^2 - a^3*c^3*d^3*e^4 + (c^6*d^7 - a*c^5 *d^5*e^2)*x^2 + 2*(a*c^5*d^6*e - a^2*c^4*d^4*e^3)*x)]
Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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*e^2-c*d^2>0)', see `assume?` f or more de
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {3 \, e^{2} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} c^{2} d^{2}} - \frac {5 \, {\left (e x + d\right )}^{\frac {3}{2}} c d e^{2} - 3 \, \sqrt {e x + d} c d^{2} e^{2} + 3 \, \sqrt {e x + d} a e^{4}}{4 \, {\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{2} d^{2}} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm=" giac")
Output:
3/4*e^2*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^2*d^2) - 1/4*(5*(e*x + d)^(3/2)*c*d*e^2 - 3*sqrt(e*x + d) *c*d^2*e^2 + 3*sqrt(e*x + d)*a*e^4)/(((e*x + d)*c*d - c*d^2 + a*e^2)^2*c^2 *d^2)
Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {3\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}}{\sqrt {a\,e^2-c\,d^2}}\right )}{4\,c^{5/2}\,d^{5/2}\,\sqrt {a\,e^2-c\,d^2}}-\frac {\frac {5\,e^2\,{\left (d+e\,x\right )}^{3/2}}{4\,c\,d}+\frac {3\,e^2\,\left (a\,e^2-c\,d^2\right )\,\sqrt {d+e\,x}}{4\,c^2\,d^2}}{a^2\,e^4+c^2\,d^4-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\left (d+e\,x\right )+c^2\,d^2\,{\left (d+e\,x\right )}^2-2\,a\,c\,d^2\,e^2} \] Input:
int((d + e*x)^(9/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
Output:
(3*e^2*atan((c^(1/2)*d^(1/2)*(d + e*x)^(1/2))/(a*e^2 - c*d^2)^(1/2)))/(4*c ^(5/2)*d^(5/2)*(a*e^2 - c*d^2)^(1/2)) - ((5*e^2*(d + e*x)^(3/2))/(4*c*d) + (3*e^2*(a*e^2 - c*d^2)*(d + e*x)^(1/2))/(4*c^2*d^2))/(a^2*e^4 + c^2*d^4 - (2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x) + c^2*d^2*(d + e*x)^2 - 2*a*c*d^2*e^2 )
Time = 0.21 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.62 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {3 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}+6 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c d \,e^{3} x +3 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{2} e^{2} x^{2}-3 \sqrt {e x +d}\, a^{2} c d \,e^{4}+\sqrt {e x +d}\, a \,c^{2} d^{3} e^{2}-5 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{3} x +2 \sqrt {e x +d}\, c^{3} d^{5}+5 \sqrt {e x +d}\, c^{3} d^{4} e x}{4 c^{3} d^{3} \left (a \,c^{2} d^{2} e^{2} x^{2}-c^{3} d^{4} x^{2}+2 a^{2} c d \,e^{3} x -2 a \,c^{2} d^{3} e x +a^{3} e^{4}-a^{2} c \,d^{2} e^{2}\right )} \] Input:
int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
Output:
(3*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)))*a**2*e**4 + 6*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) ))*a*c*d*e**3*x + 3*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**2*d**2*e**2*x**2 - 3* sqrt(d + e*x)*a**2*c*d*e**4 + sqrt(d + e*x)*a*c**2*d**3*e**2 - 5*sqrt(d + e*x)*a*c**2*d**2*e**3*x + 2*sqrt(d + e*x)*c**3*d**5 + 5*sqrt(d + e*x)*c**3 *d**4*e*x)/(4*c**3*d**3*(a**3*e**4 - a**2*c*d**2*e**2 + 2*a**2*c*d*e**3*x - 2*a*c**2*d**3*e*x + a*c**2*d**2*e**2*x**2 - c**3*d**4*x**2))