Integrand size = 24, antiderivative size = 103 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {4 d \sqrt {d^2-e^2 x^2}}{e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}-\frac {15 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \] Output:
8*d^2*(e*x+d)/e/(-e^2*x^2+d^2)^(1/2)+4*d*(-e^2*x^2+d^2)^(1/2)/e+1/2*x*(-e^ 2*x^2+d^2)^(1/2)-15/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e
Time = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (-24 d^2+7 d e x+e^2 x^2\right )}{2 e (d-e x)}+\frac {15 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e} \] Input:
Integrate[(d + e*x)^4/(d^2 - e^2*x^2)^(3/2),x]
Output:
-1/2*(Sqrt[d^2 - e^2*x^2]*(-24*d^2 + 7*d*e*x + e^2*x^2))/(e*(d - e*x)) + ( 15*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e
Time = 0.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {462, 2346, 25, 27, 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.
\(\displaystyle \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 462 |
\(\displaystyle \frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}-\int \frac {7 d^2+4 e x d+e^2 x^2}{\sqrt {d^2-e^2 x^2}}dx\) |
\(\Big \downarrow \) 2346 |
\(\displaystyle \frac {\int -\frac {d e^2 (15 d+8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {d e^2 (15 d+8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{2} d \int \frac {15 d+8 e x}{\sqrt {d^2-e^2 x^2}}dx+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {1}{2} d \left (15 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {1}{2} d \left (15 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {1}{2} d \left (\frac {15 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\) |
Input:
Int[(d + e*x)^4/(d^2 - e^2*x^2)^(3/2),x]
Output:
(8*d^2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) + (x*Sqrt[d^2 - e^2*x^2])/2 - (d *((-8*Sqrt[d^2 - e^2*x^2])/e + (15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e) )/2
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)^(-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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05
method | result | size |
risch | \(\frac {\left (e x +8 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e}-\frac {15 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {8 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{e^{2} \left (x -\frac {d}{e}\right )}\) | \(108\) |
default | \(\frac {d^{2} x}{\sqrt {-e^{2} x^{2}+d^{2}}}+e^{4} \left (-\frac {x^{3}}{2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {3 d^{2} \left (\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}\right )}{2 e^{2}}\right )+4 d \,e^{3} \left (-\frac {x^{2}}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 d^{2}}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}\right )+\frac {4 d^{3}}{e \sqrt {-e^{2} x^{2}+d^{2}}}\) | \(241\) |
Input:
int((e*x+d)^4/(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(e*x+8*d)/e*(-e^2*x^2+d^2)^(1/2)-15/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/ 2)*x/(-e^2*x^2+d^2)^(1/2))-8*d^2/e^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e) )^(1/2)
Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {24 \, d^{2} e x - 24 \, d^{3} + 30 \, {\left (d^{2} e x - d^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (e^{2} x^{2} + 7 \, d e x - 24 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, {\left (e^{2} x - d e\right )}} \] Input:
integrate((e*x+d)^4/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")
Output:
1/2*(24*d^2*e*x - 24*d^3 + 30*(d^2*e*x - d^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (e^2*x^2 + 7*d*e*x - 24*d^2)*sqrt(-e^2*x^2 + d^2))/(e^2*x - d*e)
\[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**4/(-e**2*x**2+d**2)**(3/2),x)
Output:
Integral((d + e*x)**4/(-(-d + e*x)*(d + e*x))**(3/2), x)
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {e^{2} x^{3}}{2 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {4 \, d e x^{2}}{\sqrt {-e^{2} x^{2} + d^{2}}} + \frac {17 \, d^{2} x}{2 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {15 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} + \frac {12 \, d^{3}}{\sqrt {-e^{2} x^{2} + d^{2}} e} \] Input:
integrate((e*x+d)^4/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")
Output:
-1/2*e^2*x^3/sqrt(-e^2*x^2 + d^2) - 4*d*e*x^2/sqrt(-e^2*x^2 + d^2) + 17/2* d^2*x/sqrt(-e^2*x^2 + d^2) - 15/2*d^2*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2 ) + 12*d^3/(sqrt(-e^2*x^2 + d^2)*e)
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {15 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x + \frac {8 \, d}{e}\right )} + \frac {16 \, d^{2}}{{\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )} {\left | e \right |}} \] Input:
integrate((e*x+d)^4/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
Output:
-15/2*d^2*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 1/2*sqrt(-e^2*x^2 + d^2)*(x + 8*d/e) + 16*d^2/(((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)*abs( e))
Timed out. \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (d^2-e^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((d + e*x)^4/(d^2 - e^2*x^2)^(3/2),x)
Output:
int((d + e*x)^4/(d^2 - e^2*x^2)^(3/2), x)
Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.59 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {-15 \sqrt {-e^{2} x^{2}+d^{2}}\, \mathit {asin} \left (\frac {e x}{d}\right ) d^{2}+15 \mathit {asin} \left (\frac {e x}{d}\right ) d^{3}-15 \mathit {asin} \left (\frac {e x}{d}\right ) d^{2} e x -34 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}+7 \sqrt {-e^{2} x^{2}+d^{2}}\, d e x +\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}+34 d^{3}+7 d^{2} e x -8 d \,e^{2} x^{2}-e^{3} x^{3}}{2 e \left (\sqrt {-e^{2} x^{2}+d^{2}}-d +e x \right )} \] Input:
int((e*x+d)^4/(-e^2*x^2+d^2)^(3/2),x)
Output:
( - 15*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**2 + 15*asin((e*x)/d)*d**3 - 15*asin((e*x)/d)*d**2*e*x - 34*sqrt(d**2 - e**2*x**2)*d**2 + 7*sqrt(d**2 - e**2*x**2)*d*e*x + sqrt(d**2 - e**2*x**2)*e**2*x**2 + 34*d**3 + 7*d**2*e *x - 8*d*e**2*x**2 - e**3*x**3)/(2*e*(sqrt(d**2 - e**2*x**2) - d + e*x))