Integrand size = 24, antiderivative size = 81 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e} \] Output:
2/3*(e*x+d)^3/e/(-e^2*x^2+d^2)^(3/2)-2*(e*x+d)/e/(-e^2*x^2+d^2)^(1/2)+arct an(e*x/(-e^2*x^2+d^2)^(1/2))/e
Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {2 \left (-\frac {2 (d-2 e x) \sqrt {d^2-e^2 x^2}}{(d-e x)^2}-3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )\right )}{3 e} \] Input:
Integrate[(d + e*x)^4/(d^2 - e^2*x^2)^(5/2),x]
Output:
(2*((-2*(d - 2*e*x)*Sqrt[d^2 - e^2*x^2])/(d - e*x)^2 - 3*ArcTan[(e*x)/(Sqr t[d^2] - Sqrt[d^2 - e^2*x^2])]))/(3*e)
Time = 0.31 (sec) , antiderivative size = 81, 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 = {468, 457, 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.
\(\displaystyle \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 468 |
\(\displaystyle \frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 457 |
\(\displaystyle \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {2 (d+e x)^3}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}\) |
Input:
Int[(d + e*x)^4/(d^2 - e^2*x^2)^(5/2),x]
Output:
(2*(d + e*x)^3)/(3*e*(d^2 - e^2*x^2)^(3/2)) - (2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) + ArcTan[(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_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[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*(p + 1))), x] - Simp[d^2*((n + p)/(b*(p + 1))) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[p, -1] && GtQ[n, 1] && I ntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(73)=146\).
Time = 0.25 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.47
method | result | size |
default | \(d^{4} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )+e^{4} \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )+4 d \,e^{3} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )+6 d^{2} e^{2} \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )+\frac {4 d^{3}}{3 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(281\) |
Input:
int((e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
d^4*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2))+e^4*(1 /3*x^3/e^2/(-e^2*x^2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1/e^2/(e ^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))+4*d*e^3*(x^2/e^2/(- e^2*x^2+d^2)^(3/2)-2/3*d^2/e^4/(-e^2*x^2+d^2)^(3/2))+6*d^2*e^2*(1/2*x/e^2/ (-e^2*x^2+d^2)^(3/2)-1/2*d^2/e^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4 /(-e^2*x^2+d^2)^(1/2)))+4/3*d^3/e/(-e^2*x^2+d^2)^(3/2)
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (2 \, e^{2} x^{2} - 4 \, d e x + 2 \, d^{2} + 3 \, {\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (2 \, e x - d\right )}\right )}}{3 \, {\left (e^{3} x^{2} - 2 \, d e^{2} x + d^{2} e\right )}} \] Input:
integrate((e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")
Output:
-2/3*(2*e^2*x^2 - 4*d*e*x + 2*d^2 + 3*(e^2*x^2 - 2*d*e*x + d^2)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 2*sqrt(-e^2*x^2 + d^2)*(2*e*x - d))/(e^3 *x^2 - 2*d*e^2*x + d^2*e)
\[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**4/(-e**2*x**2+d**2)**(5/2),x)
Output:
Integral((d + e*x)**4/(-(-d + e*x)*(d + e*x))**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (73) = 146\).
Time = 0.12 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {1}{3} \, e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {4 \, d e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {7 \, d^{2} x}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {4 \, d^{3}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {5 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}}} + \frac {\arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} \] Input:
integrate((e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")
Output:
1/3*e^4*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3 /2)*e^4)) + 4*d*e*x^2/(-e^2*x^2 + d^2)^(3/2) + 7/3*d^2*x/(-e^2*x^2 + d^2)^ (3/2) - 4/3*d^3/((-e^2*x^2 + d^2)^(3/2)*e) - 5/3*x/sqrt(-e^2*x^2 + d^2) + arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2)
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {8 \, {\left (\frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - 1\right )}}{3 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{3} {\left | e \right |}} \] Input:
integrate((e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")
Output:
arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 8/3*(3*(d*e + sqrt(-e^2*x^2 + d^2)*ab s(e))/(e^2*x) - 1)/(((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^3*ab s(e))
Timed out. \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (d^2-e^2\,x^2\right )}^{5/2}} \,d x \] Input:
int((d + e*x)^4/(d^2 - e^2*x^2)^(5/2),x)
Output:
int((d + e*x)^4/(d^2 - e^2*x^2)^(5/2), x)
Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.62 \[ \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {3 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{3}-9 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{2}+9 \mathit {asin} \left (\frac {e x}{d}\right ) \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )-3 \mathit {asin} \left (\frac {e x}{d}\right )-24 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )+8}{3 e \left (\tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{3}-3 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )^{2}+3 \tan \left (\frac {\mathit {asin} \left (\frac {e x}{d}\right )}{2}\right )-1\right )} \] Input:
int((e*x+d)^4/(-e^2*x^2+d^2)^(5/2),x)
Output:
(3*asin((e*x)/d)*tan(asin((e*x)/d)/2)**3 - 9*asin((e*x)/d)*tan(asin((e*x)/ d)/2)**2 + 9*asin((e*x)/d)*tan(asin((e*x)/d)/2) - 3*asin((e*x)/d) - 24*tan (asin((e*x)/d)/2) + 8)/(3*e*(tan(asin((e*x)/d)/2)**3 - 3*tan(asin((e*x)/d) /2)**2 + 3*tan(asin((e*x)/d)/2) - 1))