\(\int \frac {x (d^2-e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\) [161]

Optimal result

Integrand size = 25, antiderivative size = 136 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^2} \]

[Out]

-1/6*d*x*(-e^2*x^2+d^2)^(3/2)/e-2/15*(-e^2*x^2+d^2)^(5/2)/e^2-1/3*(-e^2*x^2+d^2)^(7/2)/e^2/(e*x+d)^2-1/4*d^5*a
rctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^2-1/4*d^3*x*(-e^2*x^2+d^2)^(1/2)/e

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {807, 679, 201, 223, 209} \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^2}-\frac {d x \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {\left (d^2-e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2}-\frac {2 \left (d^2-e^2 x^2\right )^{5/2}}{15 e^2}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{4 e} \]

[In]

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

[Out]

-1/4*(d^3*x*Sqrt[d^2 - e^2*x^2])/e - (d*x*(d^2 - e^2*x^2)^(3/2))/(6*e) - (2*(d^2 - e^2*x^2)^(5/2))/(15*e^2) -
(d^2 - e^2*x^2)^(7/2)/(3*e^2*(d + e*x)^2) - (d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*e^2)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 679

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 2*p + 1))), x] - Dist[2*c*d*(p/(e^2*(m + 2*p + 1))), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[
m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 807

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

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {e \sqrt {d^2-e^2 x^2} \left (-28 d^4+15 d^3 e x+16 d^2 e^2 x^2-30 d e^3 x^3+12 e^4 x^4\right )-15 d^5 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{60 e^3} \]

[In]

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

[Out]

(e*Sqrt[d^2 - e^2*x^2]*(-28*d^4 + 15*d^3*e*x + 16*d^2*e^2*x^2 - 30*d*e^3*x^3 + 12*e^4*x^4) - 15*d^5*Sqrt[-e^2]
*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(60*e^3)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\left (-12 e^{4} x^{4}+30 d \,e^{3} x^{3}-16 d^{2} e^{2} x^{2}-15 d^{3} e x +28 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{60 e^{2}}-\frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 e \sqrt {e^{2}}}\)	\(97\)
default	\(\frac {\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{e^{2}}-\frac {d \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{e^{3}}\)	\(438\)

[In]

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

[Out]

-1/60*(-12*e^4*x^4+30*d*e^3*x^3-16*d^2*e^2*x^2-15*d^3*e*x+28*d^4)/e^2*(-e^2*x^2+d^2)^(1/2)-1/4*d^5/e/(e^2)^(1/
2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.69 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (12 \, e^{4} x^{4} - 30 \, d e^{3} x^{3} + 16 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x - 28 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{60 \, e^{2}} \]

[In]

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

[Out]

1/60*(30*d^5*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (12*e^4*x^4 - 30*d*e^3*x^3 + 16*d^2*e^2*x^2 + 15*d^3*
e*x - 28*d^4)*sqrt(-e^2*x^2 + d^2))/e^2

Sympy [A] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=d^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**2*Piecewise((sqrt(d**2 - e**2*x**2)*(-d**2/(3*e**2) + x**2/3), Ne(e**2, 0)), (x**2*sqrt(d**2)/2, True)) - 2
*d*e*Piecewise((d**4*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))/(8*e**2) + sqrt(d**2 - e**2*x**2)*(-d**2*x/(8*e**2) + x**3/4), Ne(e**2,
0)), (x**3*sqrt(d**2)/3, True)) + e**2*Piecewise((sqrt(d**2 - e**2*x**2)*(-2*d**4/(15*e**4) - d**2*x**2/(15*e*
*2) + x**4/5), Ne(e**2, 0)), (x**4*sqrt(d**2)/4, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.23 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{4 \, e^{2}} - \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{4 \, e} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{4 \, {\left (e^{3} x + d e^{2}\right )}} - \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{2 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{12 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, e^{2}} \]

[In]

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

[Out]

1/4*I*d^5*arcsin(e*x/d + 2)/e^2 - 1/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^3*x/e - 1/4*(-e^2*x^2 + d^2)^(5/2)*d/(
e^3*x + d*e^2) - 1/2*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4/e^2 + 1/4*(-e^2*x^2 + d^2)^(3/2)*d*x/e - 5/12*(-e^2*x
^2 + d^2)^(3/2)*d^2/e^2 + 1/5*(-e^2*x^2 + d^2)^(5/2)/e^2

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (480 \, d^{6} e^{6} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (15 \, d^{6} e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 250 \, d^{6} e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 128 \, d^{6} e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 70 \, d^{6} e^{6} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 15 \, d^{6} e^{6} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{5}}{d^{5}}\right )} {\left | e \right |}}{960 \, d e^{9}} \]

[In]

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

[Out]

1/960*(480*d^6*e^6*arctan(sqrt(2*d/(e*x + d) - 1))*sgn(1/(e*x + d))*sgn(e) + (15*d^6*e^6*(2*d/(e*x + d) - 1)^(
9/2)*sgn(1/(e*x + d))*sgn(e) - 250*d^6*e^6*(2*d/(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 128*d^6*e^6*(2*
d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))*sgn(e) - 70*d^6*e^6*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))*sgn(e)
- 15*d^6*e^6*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn(e))*(e*x + d)^5/d^5)*abs(e)/(d*e^9)

Mupad [F(-1)]

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

[In]

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

[Out]

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