Integrand size = 37, antiderivative size = 112 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {3 e \sqrt {d+e x}}{c^2 d^2}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}-\frac {3 e \sqrt {c d^2-a e^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{5/2} d^{5/2}} \] Output:
3*e*(e*x+d)^(1/2)/c^2/d^2-(e*x+d)^(3/2)/c/d/(c*d*x+a*e)-3*e*(-a*e^2+c*d^2) ^(1/2)*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)
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (3 a e^2-c d (d-2 e x)\right )}{c^2 d^2 (a e+c d x)}-\frac {3 e \sqrt {-c d^2+a e^2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{5/2} d^{5/2}} \] Input:
Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
(Sqrt[d + e*x]*(3*a*e^2 - c*d*(d - 2*e*x)))/(c^2*d^2*(a*e + c*d*x)) - (3*e *Sqrt[-(c*d^2) + a*e^2]*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^ 2) + a*e^2]])/(c^(5/2)*d^(5/2))
Time = 0.39 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1121, 51, 60, 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.
\(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \frac {(d+e x)^{3/2}}{(a e+c d x)^2}dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {3 e \int \frac {\sqrt {d+e x}}{a e+c d x}dx}{2 c d}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {3 e \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}}dx}{d}+\frac {2 \sqrt {d+e x}}{c d}\right )}{2 c d}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {3 e \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{-\frac {c d^2}{e}+\frac {c (d+e x) d}{e}+a e}d\sqrt {d+e x}}{d e}+\frac {2 \sqrt {d+e x}}{c d}\right )}{2 c d}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {3 e \left (\frac {2 \sqrt {d+e x}}{c d}-\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} d^{3/2} \sqrt {c d^2-a e^2}}\right )}{2 c d}-\frac {(d+e x)^{3/2}}{c d (a e+c d x)}\) |
Input:
Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
Output:
-((d + e*x)^(3/2)/(c*d*(a*e + c*d*x))) + (3*e*((2*Sqrt[d + e*x])/(c*d) - ( 2*(d^2 - (a*e^2)/c)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a *e^2]])/(Sqrt[c]*d^(3/2)*Sqrt[c*d^2 - a*e^2])))/(2*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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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]
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 = 4.85 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(\frac {3 \sqrt {e x +d}\, \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, \left (-\frac {d \left (-2 e x +d \right ) c}{3}+a \,e^{2}\right )-3 e \left (a \,e^{2}-c \,d^{2}\right ) \left (c d x +a e \right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, c^{2} d^{2} \left (c d x +a e \right )}\) | \(129\) |
derivativedivides | \(2 e \left (\frac {\sqrt {e x +d}}{c^{2} d^{2}}-\frac {\frac {\left (-\frac {a \,e^{2}}{2}+\frac {c \,d^{2}}{2}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}}{c^{2} d^{2}}\right )\) | \(130\) |
default | \(2 e \left (\frac {\sqrt {e x +d}}{c^{2} d^{2}}-\frac {\frac {\left (-\frac {a \,e^{2}}{2}+\frac {c \,d^{2}}{2}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}}+\frac {3 \left (a \,e^{2}-c \,d^{2}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{2 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}}{c^{2} d^{2}}\right )\) | \(130\) |
risch | \(\text {Expression too large to display}\) | \(3814\) |
Input:
int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^2,x,method=_RETURNVERB OSE)
Output:
(3*(e*x+d)^(1/2)*(c*d*(a*e^2-c*d^2))^(1/2)*(-1/3*d*(-2*e*x+d)*c+a*e^2)-3*e *(a*e^2-c*d^2)*(c*d*x+a*e)*arctan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d^2))^(1 /2)))/(c*d*(a*e^2-c*d^2))^(1/2)/c^2/d^2/(c*d*x+a*e)
Time = 0.11 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [\frac {3 \, {\left (c d e x + a e^{2}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, -\frac {3 \, {\left (c d e x + a e^{2}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (2 \, c d e x - c d^{2} + 3 \, a e^{2}\right )} \sqrt {e x + d}}{c^{3} d^{3} x + a c^{2} d^{2} e}\right ] \] Input:
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" fricas")
Output:
[1/2*(3*(c*d*e*x + a*e^2)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d ^2 - a*e^2 - 2*sqrt(e*x + d)*c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e )) + 2*(2*c*d*e*x - c*d^2 + 3*a*e^2)*sqrt(e*x + d))/(c^3*d^3*x + a*c^2*d^2 *e), -(3*(c*d*e*x + a*e^2)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(-sqrt(e*x + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (2*c*d*e*x - c*d^2 + 3*a*e^2)*sqrt(e*x + d))/(c^3*d^3*x + a*c^2*d^2*e)]
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^(7/2)/(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 = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.27 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e}{c^{2} d^{2}} + \frac {3 \, {\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{2} d^{2}} - \frac {\sqrt {e x + d} c d^{2} e - \sqrt {e x + d} a e^{3}}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} c^{2} d^{2}} \] Input:
integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm=" giac")
Output:
2*sqrt(e*x + d)*e/(c^2*d^2) + 3*(c*d^2*e - a*e^3)*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) - (sqrt( e*x + d)*c*d^2*e - sqrt(e*x + d)*a*e^3)/(((e*x + d)*c*d - c*d^2 + a*e^2)*c ^2*d^2)
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\left (a\,e^3-c\,d^2\,e\right )\,\sqrt {d+e\,x}}{c^3\,d^3\,\left (d+e\,x\right )-c^3\,d^4+a\,c^2\,d^2\,e^2}+\frac {2\,e\,\sqrt {d+e\,x}}{c^2\,d^2}-\frac {3\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,\sqrt {a\,e^2-c\,d^2}\,\sqrt {d+e\,x}}{a\,e^3-c\,d^2\,e}\right )\,\sqrt {a\,e^2-c\,d^2}}{c^{5/2}\,d^{5/2}} \] Input:
int((d + e*x)^(7/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)
Output:
((a*e^3 - c*d^2*e)*(d + e*x)^(1/2))/(c^3*d^3*(d + e*x) - c^3*d^4 + a*c^2*d ^2*e^2) + (2*e*(d + e*x)^(1/2))/(c^2*d^2) - (3*e*atan((c^(1/2)*d^(1/2)*e*( a*e^2 - c*d^2)^(1/2)*(d + e*x)^(1/2))/(a*e^3 - c*d^2*e))*(a*e^2 - c*d^2)^( 1/2))/(c^(5/2)*d^(5/2))
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-3 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{2}-3 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) c d e x +3 \sqrt {e x +d}\, a c d \,e^{2}-\sqrt {e x +d}\, c^{2} d^{3}+2 \sqrt {e x +d}\, c^{2} d^{2} e x}{c^{3} d^{3} \left (c d x +a e \right )} \] Input:
int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,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*e**2 - 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*d*e*x + 3*sqrt(d + e*x)*a*c*d*e**2 - sqrt(d + e*x)*c**2*d**3 + 2*sqrt (d + e*x)*c**2*d**2*e*x)/(c**3*d**3*(a*e + c*d*x))