Integrand size = 27, antiderivative size = 113 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3} \]
1/12*d*(-3*e*x+4*d)*(-e^2*x^2+d^2)^(3/2)/e^3-1/5*(-e^2*x^2+d^2)^(5/2)/e^3+ 1/8*d^5*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^3+1/8*d^3*x*(-e^2*x^2+d^2)^(1/2 )/e^2
Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (16 d^4-15 d^3 e x+8 d^2 e^2 x^2+30 d e^3 x^3-24 e^4 x^4\right )-30 d^5 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{120 e^3} \]
(Sqrt[d^2 - e^2*x^2]*(16*d^4 - 15*d^3*e*x + 8*d^2*e^2*x^2 + 30*d*e^3*x^3 - 24*e^4*x^4) - 30*d^5*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(12 0*e^3)
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.30, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {562, 533, 25, 27, 533, 25, 27, 455, 211, 224, 216}
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.
\(\displaystyle \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx\) |
\(\Big \downarrow \) 562 |
\(\displaystyle \int x^2 (d-e x) \sqrt {d^2-e^2 x^2}dx\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {\int -d e x (2 d-5 e x) \sqrt {d^2-e^2 x^2}dx}{5 e^2}+\frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {\int d e x (2 d-5 e x) \sqrt {d^2-e^2 x^2}dx}{5 e^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \int x (2 d-5 e x) \sqrt {d^2-e^2 x^2}dx}{5 e}\) |
\(\Big \downarrow \) 533 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {\int -d e (5 d-8 e x) \sqrt {d^2-e^2 x^2}dx}{4 e^2}+\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )}{5 e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {\int d e (5 d-8 e x) \sqrt {d^2-e^2 x^2}dx}{4 e^2}\right )}{5 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \int (5 d-8 e x) \sqrt {d^2-e^2 x^2}dx}{4 e}\right )}{5 e}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (5 d \int \sqrt {d^2-e^2 x^2}dx+\frac {8 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (5 d \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {8 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (5 d \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {8 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {5 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (5 d \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {8 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\) |
(x^2*(d^2 - e^2*x^2)^(3/2))/(5*e) - (d*((5*x*(d^2 - e^2*x^2)^(3/2))/(4*e) - (d*((8*(d^2 - e^2*x^2)^(3/2))/(3*e) + 5*d*((x*Sqrt[d^2 - e^2*x^2])/2 + ( d^2*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e))))/(4*e)))/(5*e)
3.2.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[c^(2*n)/a^n Int[x^m*((a + b*x^2)^(n + p)/(c - d*x)^n), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && IGtQ[n + p + 1/2, 0]
Time = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {\left (-24 e^{4} x^{4}+30 d \,e^{3} x^{3}+8 d^{2} e^{2} x^{2}-15 d^{3} e x +16 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{120 e^{3}}+\frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{2} \sqrt {e^{2}}}\) | \(97\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 e^{3}}-\frac {d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )\right )}{e^{3}}\) | \(238\) |
1/120*(-24*e^4*x^4+30*d*e^3*x^3+8*d^2*e^2*x^2-15*d^3*e*x+16*d^4)/e^3*(-e^2 *x^2+d^2)^(1/2)+1/8*d^5/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2 )^(1/2))
Time = 0.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=-\frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (24 \, e^{4} x^{4} - 30 \, d e^{3} x^{3} - 8 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x - 16 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, e^{3}} \]
-1/120*(30*d^5*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (24*e^4*x^4 - 3 0*d*e^3*x^3 - 8*d^2*e^2*x^2 + 15*d^3*e*x - 16*d^4)*sqrt(-e^2*x^2 + d^2))/e ^3
Time = 1.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.48 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=d \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) - e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]
d*Piecewise((d**4*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e** 2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(8* e**2) + sqrt(d**2 - e**2*x**2)*(-d**2*x/(8*e**2) + x**3/4), Ne(e**2, 0)), (x**3*sqrt(d**2)/3, True)) - e*Piecewise((sqrt(d**2 - e**2*x**2)*(-2*d**4/ (15*e**4) - d**2*x**2/(15*e**2) + x**4/5), Ne(e**2, 0)), (x**4*sqrt(d**2)/ 4, True))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.54 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=-\frac {i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{3}} - \frac {3 \, d^{5} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{3}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{2 \, e^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} x}{8 \, e^{2}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{3 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, e^{3}} \]
-1/2*I*d^5*arcsin(e*x/d + 2)/e^3 - 3/8*d^5*arcsin(e*x/d)/e^3 + 1/2*sqrt(e^ 2*x^2 + 4*d*e*x + 3*d^2)*d^3*x/e^2 - 3/8*sqrt(-e^2*x^2 + d^2)*d^3*x/e^2 + sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4/e^3 - 1/4*(-e^2*x^2 + d^2)^(3/2)*d*x/e ^2 + 1/3*(-e^2*x^2 + d^2)^(3/2)*d^2/e^3 - 1/5*(-e^2*x^2 + d^2)^(5/2)/e^3
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {d^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, e^{2} {\left | e \right |}} - \frac {1}{120} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, {\left (3 \, {\left (4 \, e x - 5 \, d\right )} x - \frac {4 \, d^{2}}{e}\right )} x + \frac {15 \, d^{3}}{e^{2}}\right )} x - \frac {16 \, d^{4}}{e^{3}}\right )} \]
1/8*d^5*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^2*abs(e)) - 1/120*sqrt(-e^2*x^2 + d ^2)*((2*(3*(4*e*x - 5*d)*x - 4*d^2/e)*x + 15*d^3/e^2)*x - 16*d^4/e^3)
Timed out. \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{d+e\,x} \,d x \]