\(\int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx\) [188]

Optimal result

Integrand size = 27, antiderivative size = 204 \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {18 d^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]

[Out]

1/5*d^4*(-e*x+d)^4/e^6/(-e^2*x^2+d^2)^(5/2)-8/5*d^3*(-e*x+d)^3/e^6/(-e^2*x^2+d^2)^(3/2)+18*d^3*arctan(e*x/(-e^
2*x^2+d^2)^(1/2))/e^6+10*d^2*(-e*x+d)^2/e^6/(-e^2*x^2+d^2)^(1/2)+59/3*d^2*(-e^2*x^2+d^2)^(1/2)/e^6-2*d*x*(-e^2
*x^2+d^2)^(1/2)/e^5+1/3*x^2*(-e^2*x^2+d^2)^(1/2)/e^4

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {866, 1649, 1829, 655, 223, 209} \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {18 d^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

Int[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

(d^4*(d - e*x)^4)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - (8*d^3*(d - e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(3/2)) + (10*d^2*
(d - e*x)^2)/(e^6*Sqrt[d^2 - e^2*x^2]) + (59*d^2*Sqrt[d^2 - e^2*x^2])/(3*e^6) - (2*d*x*Sqrt[d^2 - e^2*x^2])/e^
5 + (x^2*Sqrt[d^2 - e^2*x^2])/(3*e^4) + (18*d^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^6

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

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]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 866

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[(f + g*x)^n*((a + c*x^2)^(m + p)/(d - e*x)^m), x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rule 1829

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Si
mp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*(q + 2*p + 1))), x] + Dist[1/(b*(q + 2*p + 1)), Int[(a + b*x^2)^p*Expan
dToSum[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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^5}{e^5}+\frac {5 d^4 x}{e^4}-\frac {5 d^3 x^2}{e^3}+\frac {5 d^2 x^3}{e^2}-\frac {5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d-e x)^2 \left (-\frac {60 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {30 d^3 x^2}{e^3}+\frac {15 d^2 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x) \left (-\frac {240 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {15 d^3 x^2}{e^3}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\int \frac {\frac {720 d^6}{e^3}-\frac {885 d^5 x}{e^2}+\frac {180 d^4 x^2}{e}}{\sqrt {d^2-e^2 x^2}} \, dx}{45 d^3 e^2} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}-\frac {\int \frac {-\frac {1620 d^6}{e}+1770 d^5 x}{\sqrt {d^2-e^2 x^2}} \, dx}{90 d^3 e^4} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^5} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5} \\ & = \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.64 \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right )}{15 e^6 (d+e x)^3}+\frac {18 d^3 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^7} \]

[In]

Integrate[(x^5*Sqrt[d^2 - e^2*x^2])/(d + e*x)^4,x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(424*d^5 + 1002*d^4*e*x + 674*d^3*e^2*x^2 + 70*d^2*e^3*x^3 - 15*d*e^4*x^4 + 5*e^5*x^5))/(
15*e^6*(d + e*x)^3) + (18*d^3*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/e^7

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {\left (e^{2} x^{2}-6 d e x +29 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{6}}+\frac {18 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{5} \sqrt {e^{2}}}+\frac {2 d^{5} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{9} \left (x +\frac {d}{e}\right )^{3}}-\frac {17 d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{8} \left (x +\frac {d}{e}\right )^{2}}+\frac {108 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{7} \left (x +\frac {d}{e}\right )}\)	\(209\)
default	\(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{6}}-\frac {4 d \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{e^{5}}-\frac {d^{5} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{9}}-\frac {5 d^{3} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{9} \left (x +\frac {d}{e}\right )^{3}}-\frac {10 d^{3} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{e^{7}}+\frac {10 d^{2} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{6}}\)	\(432\)

[In]

int(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/3*(e^2*x^2-6*d*e*x+29*d^2)/e^6*(-e^2*x^2+d^2)^(1/2)+18/e^5*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^
2)^(1/2))+2/5/e^9*d^5/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-17/5/e^8*d^4/(x+d/e)^2*(-(x+d/e)^2*e^2+2*
d*e*(x+d/e))^(1/2)+108/5/e^7*d^3/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.98 \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {424 \, d^{3} e^{3} x^{3} + 1272 \, d^{4} e^{2} x^{2} + 1272 \, d^{5} e x + 424 \, d^{6} - 540 \, {\left (d^{3} e^{3} x^{3} + 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x + d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (5 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} + 70 \, d^{2} e^{3} x^{3} + 674 \, d^{3} e^{2} x^{2} + 1002 \, d^{4} e x + 424 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/15*(424*d^3*e^3*x^3 + 1272*d^4*e^2*x^2 + 1272*d^5*e*x + 424*d^6 - 540*(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3*d^5*e
*x + d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (5*e^5*x^5 - 15*d*e^4*x^4 + 70*d^2*e^3*x^3 + 674*d^3*e^2
*x^2 + 1002*d^4*e*x + 424*d^5)*sqrt(-e^2*x^2 + d^2))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

Sympy [F]

\[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\int \frac {x^{5} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate(x**5*(-e**2*x**2+d**2)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(x**5*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x)**4, x)

Maxima [F(-1)]

Timed out. \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\text {Timed out} \]

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Timed out

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.20 \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {1}{3} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x {\left (\frac {x}{e^{4}} - \frac {6 \, d}{e^{5}}\right )} + \frac {29 \, d^{2}}{e^{6}}\right )} + \frac {18 \, d^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{e^{5} {\left | e \right |}} - \frac {2 \, {\left (93 \, d^{3} + \frac {385 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3}}{e^{2} x} + \frac {575 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3}}{e^{4} x^{2}} + \frac {355 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3}}{e^{6} x^{3}} + \frac {80 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3}}{e^{8} x^{4}}\right )}}{5 \, e^{5} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]

[In]

integrate(x^5*(-e^2*x^2+d^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/3*sqrt(-e^2*x^2 + d^2)*(x*(x/e^4 - 6*d/e^5) + 29*d^2/e^6) + 18*d^3*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^5*abs(e))
- 2/5*(93*d^3 + 385*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^3/(e^2*x) + 575*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^
2*d^3/(e^4*x^2) + 355*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^3/(e^6*x^3) + 80*(d*e + sqrt(-e^2*x^2 + d^2)*abs
(e))^4*d^3/(e^8*x^4))/(e^5*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^5*abs(e))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\int \frac {x^5\,\sqrt {d^2-e^2\,x^2}}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

int((x^5*(d^2 - e^2*x^2)^(1/2))/(d + e*x)^4,x)

[Out]

int((x^5*(d^2 - e^2*x^2)^(1/2))/(d + e*x)^4, x)