Integrand size = 37, antiderivative size = 120 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {2 c d}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {2 c^{3/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]
2/3/(-a*e^2+c*d^2)/(e*x+d)^(3/2)-2*c^(3/2)*d^(3/2)*arctanh(c^(1/2)*d^(1/2) *(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/(-a*e^2+c*d^2)^(5/2)+2*c*d/(-a*e^2+c* d^2)^2/(e*x+d)^(1/2)
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2 \left (-a e^2+c d (4 d+3 e x)\right )}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 c^{3/2} d^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{5/2}} \]
(2*(-(a*e^2) + c*d*(4*d + 3*e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (2*c^(3/2)*d^(3/2)*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(-(c*d^2) + a*e^2)^(5/2)
Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1121, 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.
\(\displaystyle \int \frac {1}{(d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \frac {1}{(d+e x)^{5/2} (a e+c d x)}dx\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {c d \int \frac {1}{(a e+c d x) (d+e x)^{3/2}}dx}{c d^2-a e^2}+\frac {2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {c d \left (\frac {c d \int \frac {1}{(a e+c d x) \sqrt {d+e x}}dx}{c d^2-a e^2}+\frac {2}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{c d^2-a e^2}+\frac {2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {c d \left (\frac {2 c d \int \frac {1}{-\frac {c d^2}{e}+\frac {c (d+e x) d}{e}+a e}d\sqrt {d+e x}}{e \left (c d^2-a e^2\right )}+\frac {2}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{c d^2-a e^2}+\frac {2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {c d \left (\frac {2}{\sqrt {d+e x} \left (c d^2-a e^2\right )}-\frac {2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}\right )}{c d^2-a e^2}+\frac {2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )}\) |
2/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (c*d*(2/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - (2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c* d^2 - a*e^2]])/(c*d^2 - a*e^2)^(3/2)))/(c*d^2 - a*e^2)
3.21.6.3.1 Defintions of rubi rules used
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]
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 = 3.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {2 c^{2} d^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c d}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}\) | \(117\) |
default | \(\frac {2 c^{2} d^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c d}{\left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}\) | \(117\) |
pseudoelliptic | \(-\frac {2 \left (-3 c^{2} d^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) \left (e x +d \right )^{\frac {3}{2}}+\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \left (\left (-3 d e x -4 d^{2}\right ) c +e^{2} a \right )\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \left (e^{2} a -c \,d^{2}\right )^{2}}\) | \(122\) |
2*c^2*d^2/(a*e^2-c*d^2)^2/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/ 2)/((a*e^2-c*d^2)*c*d)^(1/2))-2/3/(a*e^2-c*d^2)/(e*x+d)^(3/2)+2/(a*e^2-c*d ^2)^2*c*d/(e*x+d)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (100) = 200\).
Time = 0.53 (sec) , antiderivative size = 461, normalized size of antiderivative = 3.84 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\left [\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 2 \, {\left (3 \, c d e x + 4 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - {\left (3 \, c d e x + 4 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}\right )}}{3 \, {\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}}\right ] \]
[1/3*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(c*d/(c*d^2 - a*e^2))*log( (c*d*e*x + 2*c*d^2 - a*e^2 - 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d/(c*d ^2 - a*e^2)))/(c*d*x + a*e)) + 2*(3*c*d*e*x + 4*c*d^2 - a*e^2)*sqrt(e*x + d))/(c^2*d^6 - 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x), -2/3*(3*(c* d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*e*x + c*d^2)) - (3 *c*d*e*x + 4*c*d^2 - a*e^2)*sqrt(e*x + d))/(c^2*d^6 - 2*a*c*d^4*e^2 + a^2* d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e - 2*a *c*d^3*e^3 + a^2*d*e^5)*x)]
Time = 2.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {c d e}{\sqrt {d + e x} \left (a e^{2} - c d^{2}\right )^{2}} + \frac {c d e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac {e}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e^{2} - c d^{2}\right )}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\log {\left (x \right )}}{c d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(c*d*e/(sqrt(d + e*x)*(a*e**2 - c*d**2)**2) + c*d*e*atan(sqrt (d + e*x)/sqrt((a*e**2 - c*d**2)/(c*d)))/(sqrt((a*e**2 - c*d**2)/(c*d))*(a *e**2 - c*d**2)**2) - e/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)))/e, Ne(e, 0 )), (log(x)/(c*d**(7/2)), True))
Exception generated. \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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 = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )}}{3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]
2*c^2*d^2*arctan(sqrt(e*x + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(-c^2*d^3 + a*c*d*e^2)) + 2/3*(3*(e*x + d)*c *d + c*d^2 - a*e^2)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(3/2))
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2\,c^{3/2}\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^{5/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{5/2}}-\frac {\frac {2}{3\,\left (a\,e^2-c\,d^2\right )}-\frac {2\,c\,d\,\left (d+e\,x\right )}{{\left (a\,e^2-c\,d^2\right )}^2}}{{\left (d+e\,x\right )}^{3/2}} \]