Integrand size = 24, antiderivative size = 92 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}-\frac {5 d (4 d+e x) \sqrt {d^2-e^2 x^2}}{6 e}+\frac {5 d^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \] Output:
-1/3*(e*x+d)^2*(-e^2*x^2+d^2)^(1/2)/e-5/6*d*(e*x+4*d)*(-e^2*x^2+d^2)^(1/2) /e+5/2*d^3*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {\left (-22 d^2-9 d e x-2 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{6 e}-\frac {5 d^3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e} \] Input:
Integrate[(d + e*x)^3/Sqrt[d^2 - e^2*x^2],x]
Output:
((-22*d^2 - 9*d*e*x - 2*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(6*e) - (5*d^3*ArcTa n[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e
Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {469, 469, 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.
\(\displaystyle \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle \frac {5}{3} d \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}}dx-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 469 |
\(\displaystyle \frac {5}{3} d \left (\frac {3}{2} d \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}}dx-\frac {(d+e x) \sqrt {d^2-e^2 x^2}}{2 e}\right )-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {5}{3} d \left (\frac {3}{2} d \left (d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )-\frac {(d+e x) \sqrt {d^2-e^2 x^2}}{2 e}\right )-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {5}{3} d \left (\frac {3}{2} d \left (d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )-\frac {(d+e x) \sqrt {d^2-e^2 x^2}}{2 e}\right )-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {5}{3} d \left (\frac {3}{2} d \left (\frac {d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )-\frac {(d+e x) \sqrt {d^2-e^2 x^2}}{2 e}\right )-\frac {(d+e x)^2 \sqrt {d^2-e^2 x^2}}{3 e}\) |
Input:
Int[(d + e*x)^3/Sqrt[d^2 - e^2*x^2],x]
Output:
-1/3*((d + e*x)^2*Sqrt[d^2 - e^2*x^2])/e + (5*d*(-1/2*((d + e*x)*Sqrt[d^2 - e^2*x^2])/e + (3*d*(-(Sqrt[d^2 - e^2*x^2]/e) + (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/2))/3
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] :> 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 = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\left (2 e^{2} x^{2}+9 d e x +22 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6 e}+\frac {5 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\) | \(72\) |
default | \(\frac {d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+e^{3} \left (-\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}}\right )+3 d \,e^{2} \left (-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )-\frac {3 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{e}\) | \(166\) |
Input:
int((e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(2*e^2*x^2+9*d*e*x+22*d^2)/e*(-e^2*x^2+d^2)^(1/2)+5/2*d^3/(e^2)^(1/2) *arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {30 \, d^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (2 \, e^{2} x^{2} + 9 \, d e x + 22 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, e} \] Input:
integrate((e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")
Output:
-1/6*(30*d^3*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (2*e^2*x^2 + 9*d* e*x + 22*d^2)*sqrt(-e^2*x^2 + d^2))/e
Time = 0.52 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.37 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=\begin {cases} \frac {5 d^{3} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {11 d^{2}}{3 e} - \frac {3 d x}{2} - \frac {e x^{2}}{3}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {\begin {cases} d^{3} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{4}}{4 e} & \text {otherwise} \end {cases}}{\sqrt {d^{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**3/(-e**2*x**2+d**2)**(1/2),x)
Output:
Piecewise((5*d**3*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 + sqrt(d**2 - e**2*x**2)*(-11*d**2/(3*e) - 3*d*x/2 - e*x**2/3), Ne(e**2, 0) ), (Piecewise((d**3*x, Eq(e, 0)), ((d + e*x)**4/(4*e), True))/sqrt(d**2), True))
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {1}{3} \, \sqrt {-e^{2} x^{2} + d^{2}} e x^{2} + \frac {5 \, d^{3} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d x - \frac {11 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e} \] Input:
integrate((e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")
Output:
-1/3*sqrt(-e^2*x^2 + d^2)*e*x^2 + 5/2*d^3*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt (e^2) - 3/2*sqrt(-e^2*x^2 + d^2)*d*x - 11/3*sqrt(-e^2*x^2 + d^2)*d^2/e
Time = 0.14 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.61 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {5 \, d^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} - \frac {1}{6} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, e x + 9 \, d\right )} x + \frac {22 \, d^{2}}{e}\right )} \] Input:
integrate((e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")
Output:
5/2*d^3*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/6*sqrt(-e^2*x^2 + d^2)*((2* e*x + 9*d)*x + 22*d^2/e)
Timed out. \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{\sqrt {d^2-e^2\,x^2}} \,d x \] Input:
int((d + e*x)^3/(d^2 - e^2*x^2)^(1/2),x)
Output:
int((d + e*x)^3/(d^2 - e^2*x^2)^(1/2), x)
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {15 \mathit {asin} \left (\frac {e x}{d}\right ) d^{3}-22 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}-9 \sqrt {-e^{2} x^{2}+d^{2}}\, d e x -2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}+22 d^{3}}{6 e} \] Input:
int((e*x+d)^3/(-e^2*x^2+d^2)^(1/2),x)
Output:
(15*asin((e*x)/d)*d**3 - 22*sqrt(d**2 - e**2*x**2)*d**2 - 9*sqrt(d**2 - e* *2*x**2)*d*e*x - 2*sqrt(d**2 - e**2*x**2)*e**2*x**2 + 22*d**3)/(6*e)