\(\int \frac {x^2 (d^2-e^2 x^2)^{3/2}}{d+e x} \, dx\) [102]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3} \]

[Out]

1/12*d*(-3*e*x+4*d)*(-e^2*x^2+d^2)^(3/2)/e^3-1/5*(-e^2*x^2+d^2)^(5/2)/e^3+1/8*d^5*arctan(e*x/(-e^2*x^2+d^2)^(1
/2))/e^3+1/8*d^3*x*(-e^2*x^2+d^2)^(1/2)/e^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1653, 12, 799, 794, 201, 223, 209} \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2} \]

[In]

Int[(x^2*(d^2 - e^2*x^2)^(3/2))/(d + e*x),x]

[Out]

(d^3*x*Sqrt[d^2 - e^2*x^2])/(8*e^2) + (d*(4*d - 3*e*x)*(d^2 - e^2*x^2)^(3/2))/(12*e^3) - (d^2 - e^2*x^2)^(5/2)
/(5*e^3) + (d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 799

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

Rule 1653

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\int \frac {5 d e^3 x \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{5 e^4} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {d \int \frac {x \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e} \\ & = -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac {\int x \left (d^2 e-d e^2 x\right ) \sqrt {d^2-e^2 x^2} \, dx}{e^2} \\ & = \frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^3 \int \sqrt {d^2-e^2 x^2} \, dx}{4 e^2} \\ & = \frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^2} \\ & = \frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\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 )}{8 e^2} \\ & = \frac {d^3 x \sqrt {d^2-e^2 x^2}}{8 e^2}+\frac {d (4 d-3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 e^3}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac {d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (16 d^4-15 d^3 e x+8 d^2 e^2 x^2+30 d e^3 x^3-24 e^4 x^4\right )-30 d^5 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{120 e^3} \]

[In]

Integrate[(x^2*(d^2 - e^2*x^2)^(3/2))/(d + e*x),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(16*d^4 - 15*d^3*e*x + 8*d^2*e^2*x^2 + 30*d*e^3*x^3 - 24*e^4*x^4) - 30*d^5*ArcTan[(e*x)/(
Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(120*e^3)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {\left (-24 e^{4} x^{4}+30 d \,e^{3} x^{3}+8 d^{2} e^{2} x^{2}-15 d^{3} e x +16 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{120 e^{3}}+\frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{2} \sqrt {e^{2}}}\)	\(97\)
default	\(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 e^{3}}-\frac {d \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{e^{2}}+\frac {d^{2} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )\right )}{e^{3}}\)	\(238\)

[In]

int(x^2*(-e^2*x^2+d^2)^(3/2)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/120*(-24*e^4*x^4+30*d*e^3*x^3+8*d^2*e^2*x^2-15*d^3*e*x+16*d^4)/e^3*(-e^2*x^2+d^2)^(1/2)+1/8*d^5/e^2/(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 = 0.35 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=-\frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (24 \, e^{4} x^{4} - 30 \, d e^{3} x^{3} - 8 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x - 16 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{120 \, e^{3}} \]

[In]

integrate(x^2*(-e^2*x^2+d^2)^(3/2)/(e*x+d),x, algorithm="fricas")

[Out]

-1/120*(30*d^5*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (24*e^4*x^4 - 30*d*e^3*x^3 - 8*d^2*e^2*x^2 + 15*d^3
*e*x - 16*d^4)*sqrt(-e^2*x^2 + d^2))/e^3

Sympy [A] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.48 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=d \left (\begin {cases} \frac {d^{4} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) - e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate(x**2*(-e**2*x**2+d**2)**(3/2)/(e*x+d),x)

[Out]

d*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*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 = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.54 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=-\frac {i \, d^{5} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{3}} - \frac {3 \, d^{5} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{3}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{3} x}{2 \, e^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} x}{8 \, e^{2}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4}}{e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d x}{4 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{3 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{5 \, e^{3}} \]

[In]

integrate(x^2*(-e^2*x^2+d^2)^(3/2)/(e*x+d),x, algorithm="maxima")

[Out]

-1/2*I*d^5*arcsin(e*x/d + 2)/e^3 - 3/8*d^5*arcsin(e*x/d)/e^3 + 1/2*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^3*x/e^2 -
3/8*sqrt(-e^2*x^2 + d^2)*d^3*x/e^2 + sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4/e^3 - 1/4*(-e^2*x^2 + d^2)^(3/2)*d*x
/e^2 + 1/3*(-e^2*x^2 + d^2)^(3/2)*d^2/e^3 - 1/5*(-e^2*x^2 + d^2)^(5/2)/e^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\frac {d^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, e^{2} {\left | e \right |}} - \frac {1}{120} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, {\left (3 \, {\left (4 \, e x - 5 \, d\right )} x - \frac {4 \, d^{2}}{e}\right )} x + \frac {15 \, d^{3}}{e^{2}}\right )} x - \frac {16 \, d^{4}}{e^{3}}\right )} \]

[In]

integrate(x^2*(-e^2*x^2+d^2)^(3/2)/(e*x+d),x, algorithm="giac")

[Out]

1/8*d^5*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^2*abs(e)) - 1/120*sqrt(-e^2*x^2 + d^2)*((2*(3*(4*e*x - 5*d)*x - 4*d^2/e
)*x + 15*d^3/e^2)*x - 16*d^4/e^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {x^2\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{d+e\,x} \,d x \]

[In]

int((x^2*(d^2 - e^2*x^2)^(3/2))/(d + e*x),x)

[Out]

int((x^2*(d^2 - e^2*x^2)^(3/2))/(d + e*x), x)