Integrand size = 24, antiderivative size = 142 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {7}{8} d^3 x \sqrt {d^2-e^2 x^2}+\frac {7 d^2 \left (d^2-e^2 x^2\right )^{3/2}}{12 e}+\frac {7 d (d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{20 e}+\frac {(d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac {7 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e} \]
7/12*d^2*(-e^2*x^2+d^2)^(3/2)/e+7/20*d*(-e*x+d)*(-e^2*x^2+d^2)^(3/2)/e+1/5 *(-e*x+d)^2*(-e^2*x^2+d^2)^(3/2)/e+7/8*d^5*arctan(e*x/(-e^2*x^2+d^2)^(1/2) )/e+7/8*d^3*x*(-e^2*x^2+d^2)^(1/2)
Time = 0.45 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (136 d^4+15 d^3 e x-112 d^2 e^2 x^2+90 d e^3 x^3-24 e^4 x^4\right )}{120 e}-\frac {7 d^5 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{8 \sqrt {-e^2}} \]
(Sqrt[d^2 - e^2*x^2]*(136*d^4 + 15*d^3*e*x - 112*d^2*e^2*x^2 + 90*d*e^3*x^ 3 - 24*e^4*x^4))/(120*e) - (7*d^5*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2 ]])/(8*Sqrt[-e^2])
Time = 0.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {464, 469, 469, 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.
| \(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 464 |
| \(\displaystyle \int (d-e x)^3 \sqrt {d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle \frac {7}{5} d \int (d-e x)^2 \sqrt {d^2-e^2 x^2}dx+\frac {\left (d^2-e^2 x^2\right )^{3/2} (d-e x)^2}{5 e}\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle \frac {7}{5} d \left (\frac {5}{4} d \int (d-e x) \sqrt {d^2-e^2 x^2}dx+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2} (d-e x)^2}{5 e}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {7}{5} d \left (\frac {5}{4} d \left (d \int \sqrt {d^2-e^2 x^2}dx+\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2} (d-e x)^2}{5 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {7}{5} d \left (\frac {5}{4} d \left (d \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2} (d-e x)^2}{5 e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {7}{5} d \left (\frac {5}{4} d \left (d \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2} (d-e x)^2}{5 e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {7}{5} d \left (\frac {5}{4} d \left (d \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )+\frac {\left (d^2-e^2 x^2\right )^{3/2} (d-e x)^2}{5 e}\) |
((d - e*x)^2*(d^2 - e^2*x^2)^(3/2))/(5*e) + (7*d*(((d - e*x)*(d^2 - e^2*x^ 2)^(3/2))/(4*e) + (5*d*((d^2 - e^2*x^2)^(3/2)/(3*e) + 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))/5
3.9.5.3.1 Defintions of rubi rules used
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[( a + b*x^2)^(n + p)/(a/c + b*(x/d))^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c^2 + a*d^2, 0] && IntegerQ[n] && RationalQ[p] && (LtQ[0, -n, p] || LtQ[p, -n, 0]) && NeQ[n, 2] && NeQ[n, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* ((n + p)/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* p + 1, 0] && IntegerQ[2*p]
Time = 2.44 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {\left (-24 e^{4} x^{4}+90 d \,e^{3} x^{3}-112 d^{2} e^{2} x^{2}+15 d^{3} e x +136 d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{120 e}+\frac {7 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}\) | \(94\) |
| default | \(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}}{e^{3}}\) | \(352\) |
1/120*(-24*e^4*x^4+90*d*e^3*x^3-112*d^2*e^2*x^2+15*d^3*e*x+136*d^4)/e*(-e^ 2*x^2+d^2)^(1/2)+7/8*d^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^( 1/2))
Time = 0.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=-\frac {210 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (24 \, e^{4} x^{4} - 90 \, d e^{3} x^{3} + 112 \, d^{2} e^{2} x^{2} - 15 \, d^{3} e x - 136 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, e} \]
-1/120*(210*d^5*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (24*e^4*x^4 - 90*d*e^3*x^3 + 112*d^2*e^2*x^2 - 15*d^3*e*x - 136*d^4)*sqrt(-e^2*x^2 + d^2 ))/e
Time = 3.33 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.23 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=d^{3} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) - 3 d^{2} e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) + 3 d e^{2} \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]
d**3*Piecewise((d**2*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))/ 2 + x*sqrt(d**2 - e**2*x**2)/2, Ne(e**2, 0)), (x*sqrt(d**2), True)) - 3*d* *2*e*Piecewise((sqrt(d**2 - e**2*x**2)*(-d**2/(3*e**2) + x**2/3), Ne(e**2, 0)), (x**2*sqrt(d**2)/2, True)) + 3*d*e**2*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** 3*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 = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=-\frac {7 i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e} + \frac {7}{8} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x + \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{4 \, e} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{5 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{20 \, {\left (e^{2} x + d e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{12 \, e} \]
-7/8*I*d^5*arcsin(e*x/d + 2)/e + 7/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^3*x + 7/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4/e + 1/5*(-e^2*x^2 + d^2)^(7/2)/ (e^3*x^2 + 2*d*e^2*x + d^2*e) + 7/20*(-e^2*x^2 + d^2)^(5/2)*d/(e^2*x + d*e ) + 7/12*(-e^2*x^2 + d^2)^(3/2)*d^2/e
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {7 \, d^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} + \frac {1}{120} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {136 \, d^{4}}{e} + {\left (15 \, d^{3} - 2 \, {\left (56 \, d^{2} e + 3 \, {\left (4 \, e^{3} x - 15 \, d e^{2}\right )} x\right )} x\right )} x\right )} \]
7/8*d^5*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 1/120*sqrt(-e^2*x^2 + d^2)*(1 36*d^4/e + (15*d^3 - 2*(56*d^2*e + 3*(4*e^3*x - 15*d*e^2)*x)*x)*x)
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^3} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^3} \,d x \]