Integrand size = 24, antiderivative size = 74 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {4 d (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e}-\frac {3 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \] Output:
4*d*(e*x+d)/e/(-e^2*x^2+d^2)^(1/2)+(-e^2*x^2+d^2)^(1/2)/e-3*d*arctan(e*x/( -e^2*x^2+d^2)^(1/2))/e
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {(-5 d+e x) \sqrt {d^2-e^2 x^2}}{e (d-e x)}+\frac {3 d \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \] Input:
Integrate[(d + e*x)^3/(d^2 - e^2*x^2)^(3/2),x]
Output:
-(((-5*d + e*x)*Sqrt[d^2 - e^2*x^2])/(e*(d - e*x))) + (3*d*Log[-(Sqrt[-e^2 ]*x) + Sqrt[d^2 - e^2*x^2]])/Sqrt[-e^2]
Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 455, 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.
\(\displaystyle \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 462 |
\(\displaystyle \frac {4 d (d+e x)}{e \sqrt {d^2-e^2 x^2}}-\int \frac {3 d+e x}{\sqrt {d^2-e^2 x^2}}dx\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -3 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {4 d (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -3 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {4 d (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {3 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {4 d (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e}\) |
Input:
Int[(d + e*x)^3/(d^2 - e^2*x^2)^(3/2),x]
Output:
(4*d*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) + Sqrt[d^2 - e^2*x^2]/e - (3*d*Arc Tan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e
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)^(3/2), x_Symbol] :> Simp [(-2^(n - 1))*d*c^(n - 2)*((c + d*x)/(b*Sqrt[a + b*x^2])), x] + Simp[d^2/b Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(n - 1)*c^(n - 1) - (c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[n, 2]
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e}-\frac {3 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {4 d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{e^{2} \left (x -\frac {d}{e}\right )}\) | \(96\) |
default | \(\frac {d x}{\sqrt {-e^{2} x^{2}+d^{2}}}+e^{3} \left (-\frac {x^{2}}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 d^{2}}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )+3 d \,e^{2} \left (\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}\right )+\frac {3 d^{2}}{e \sqrt {-e^{2} x^{2}+d^{2}}}\) | \(148\) |
Input:
int((e*x+d)^3/(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-e^2*x^2+d^2)^(1/2)/e-3*d/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2) ^(1/2))-4*d/e^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {5 \, d e x - 5 \, d^{2} + 6 \, {\left (d e x - d^{2}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt {-e^{2} x^{2} + d^{2}} {\left (e x - 5 \, d\right )}}{e^{2} x - d e} \] Input:
integrate((e*x+d)^3/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")
Output:
(5*d*e*x - 5*d^2 + 6*(d*e*x - d^2)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x )) + sqrt(-e^2*x^2 + d^2)*(e*x - 5*d))/(e^2*x - d*e)
\[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**3/(-e**2*x**2+d**2)**(3/2),x)
Output:
Integral((d + e*x)**3/(-(-d + e*x)*(d + e*x))**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {e x^{2}}{\sqrt {-e^{2} x^{2} + d^{2}}} + \frac {4 \, d x}{\sqrt {-e^{2} x^{2} + d^{2}}} - \frac {3 \, d \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {5 \, d^{2}}{\sqrt {-e^{2} x^{2} + d^{2}} e} \] Input:
integrate((e*x+d)^3/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")
Output:
-e*x^2/sqrt(-e^2*x^2 + d^2) + 4*d*x/sqrt(-e^2*x^2 + d^2) - 3*d*arcsin(e^2* x/(d*sqrt(e^2)))/sqrt(e^2) + 5*d^2/(sqrt(-e^2*x^2 + d^2)*e)
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {3 \, d \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{e} + \frac {8 \, d}{{\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )} {\left | e \right |}} \] Input:
integrate((e*x+d)^3/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
Output:
-3*d*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + sqrt(-e^2*x^2 + d^2)/e + 8*d/((( d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)*abs(e))
Timed out. \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((d + e*x)^3/(d^2 - e^2*x^2)^(3/2),x)
Output:
int((d + e*x)^3/(d^2 - e^2*x^2)^(3/2), x)
Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^3}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {-3 \sqrt {-e^{2} x^{2}+d^{2}}\, \mathit {asin} \left (\frac {e x}{d}\right ) d +3 \mathit {asin} \left (\frac {e x}{d}\right ) d^{2}-3 \mathit {asin} \left (\frac {e x}{d}\right ) d e x -8 \sqrt {-e^{2} x^{2}+d^{2}}\, d +\sqrt {-e^{2} x^{2}+d^{2}}\, e x +8 d^{2}+d e x -e^{2} x^{2}}{e \left (\sqrt {-e^{2} x^{2}+d^{2}}-d +e x \right )} \] Input:
int((e*x+d)^3/(-e^2*x^2+d^2)^(3/2),x)
Output:
( - 3*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d + 3*asin((e*x)/d)*d**2 - 3*as in((e*x)/d)*d*e*x - 8*sqrt(d**2 - e**2*x**2)*d + sqrt(d**2 - e**2*x**2)*e* x + 8*d**2 + d*e*x - e**2*x**2)/(e*(sqrt(d**2 - e**2*x**2) - d + e*x))