\(\int \frac {x^4 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx\) [189]

Optimal result

Integrand size = 27, antiderivative size = 160 \[ \int \frac {x^4 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {6 d (d-e x)^2}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {(20 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {19 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5} \]

[Out]

-1/5*d^3*(-e*x+d)^4/e^5/(-e^2*x^2+d^2)^(5/2)+19/15*d^2*(-e*x+d)^3/e^5/(-e^2*x^2+d^2)^(3/2)-19/2*d^2*arctan(e*x
/(-e^2*x^2+d^2)^(1/2))/e^5-6*d*(-e*x+d)^2/e^5/(-e^2*x^2+d^2)^(1/2)-1/2*(-e*x+20*d)*(-e^2*x^2+d^2)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {866, 1649, 794, 223, 209} \[ \int \frac {x^4 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {19 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {6 d (d-e x)^2}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {(20 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]

[In]

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

[Out]

-1/5*(d^3*(d - e*x)^4)/(e^5*(d^2 - e^2*x^2)^(5/2)) + (19*d^2*(d - e*x)^3)/(15*e^5*(d^2 - e^2*x^2)^(3/2)) - (6*
d*(d - e*x)^2)/(e^5*Sqrt[d^2 - e^2*x^2]) - ((20*d - e*x)*Sqrt[d^2 - e^2*x^2])/(2*e^5) - (19*d^2*ArcTan[(e*x)/S
qrt[d^2 - e^2*x^2]])/(2*e^5)

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = -\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d-e x)^3 \left (\frac {4 d^4}{e^4}-\frac {5 d^3 x}{e^3}+\frac {5 d^2 x^2}{e^2}-\frac {5 d x^3}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d} \\ & = -\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d-e x)^2 \left (\frac {45 d^4}{e^4}-\frac {30 d^3 x}{e^3}+\frac {15 d^2 x^2}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2} \\ & = -\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {6 d (d-e x)^2}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\left (\frac {135 d^4}{e^4}-\frac {15 d^3 x}{e^3}\right ) (d-e x)}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3} \\ & = -\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {6 d (d-e x)^2}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {(20 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {\left (19 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^4} \\ & = -\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {6 d (d-e x)^2}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {(20 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {\left (19 d^2\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^4} \\ & = -\frac {d^3 (d-e x)^4}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {19 d^2 (d-e x)^3}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {6 d (d-e x)^2}{e^5 \sqrt {d^2-e^2 x^2}}-\frac {(20 d-e x) \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {19 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.70 \[ \int \frac {x^4 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-448 d^4-1059 d^3 e x-713 d^2 e^2 x^2-75 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^3}+\frac {19 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e^5} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-448*d^4 - 1059*d^3*e*x - 713*d^2*e^2*x^2 - 75*d*e^3*x^3 + 15*e^4*x^4))/(30*e^5*(d + e*x
)^3) + (19*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e^5

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.24

method	result	size
risch	\(-\frac {\left (-e x +8 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{5}}-\frac {19 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{4} \sqrt {e^{2}}}-\frac {2 d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {41 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{7} \left (x +\frac {d}{e}\right )^{2}}-\frac {199 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{6} \left (x +\frac {d}{e}\right )}\)	\(199\)
default	\(\frac {\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}}{e^{4}}+\frac {d^{4} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{8}}+\frac {6 d^{2} \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{e^{6}}-\frac {4 d \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{5}}+\frac {4 d^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{8} \left (x +\frac {d}{e}\right )^{3}}\)	\(408\)

[In]

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

[Out]

-1/2*(-e*x+8*d)/e^5*(-e^2*x^2+d^2)^(1/2)-19/2/e^4*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-2
/5/e^8*d^4/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+41/15/e^7*d^3/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e
))^(1/2)-199/15/e^6*d^2/(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 = 190, normalized size of antiderivative = 1.19 \[ \int \frac {x^4 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=-\frac {448 \, d^{2} e^{3} x^{3} + 1344 \, d^{3} e^{2} x^{2} + 1344 \, d^{4} e x + 448 \, d^{5} - 570 \, {\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x + d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, e^{4} x^{4} - 75 \, d e^{3} x^{3} - 713 \, d^{2} e^{2} x^{2} - 1059 \, d^{3} e x - 448 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

[In]

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

[Out]

-1/30*(448*d^2*e^3*x^3 + 1344*d^3*e^2*x^2 + 1344*d^4*e*x + 448*d^5 - 570*(d^2*e^3*x^3 + 3*d^3*e^2*x^2 + 3*d^4*
e*x + d^5)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (15*e^4*x^4 - 75*d*e^3*x^3 - 713*d^2*e^2*x^2 - 1059*d^3
*e*x - 448*d^4)*sqrt(-e^2*x^2 + d^2))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

integrate(x^4*(-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 = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.46 \[ \int \frac {x^4 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx=\frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {x}{e^{4}} - \frac {8 \, d}{e^{5}}\right )} - \frac {19 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, e^{4} {\left | e \right |}} + \frac {2 \, {\left (164 \, d^{2} + \frac {685 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {1025 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}} + \frac {615 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2}}{e^{6} x^{3}} + \frac {135 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2}}{e^{8} x^{4}}\right )}}{15 \, e^{4} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]

[In]

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

[Out]

1/2*sqrt(-e^2*x^2 + d^2)*(x/e^4 - 8*d/e^5) - 19/2*d^2*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^4*abs(e)) + 2/15*(164*d^2
+ 685*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2/(e^2*x) + 1025*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2/(e^4*x
^2) + 615*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2/(e^6*x^3) + 135*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^2/
(e^8*x^4))/(e^4*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^5*abs(e))

Mupad [F(-1)]

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

[In]

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

[Out]

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