\(\int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx\) [93]

Optimal result

Integrand size = 27, antiderivative size = 118 \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {3 d^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \]

[Out]

-3/8*d^4*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4-1/3*d*x^2*(-e^2*x^2+d^2)^(1/2)/e^2+1/4*x^3*(-e^2*x^2+d^2)^(1/2)/
e-1/24*d^2*(-9*e*x+16*d)*(-e^2*x^2+d^2)^(1/2)/e^4

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {864, 847, 794, 223, 209} \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {3 d^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4}-\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4} \]

[In]

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

[Out]

-1/3*(d*x^2*Sqrt[d^2 - e^2*x^2])/e^2 + (x^3*Sqrt[d^2 - e^2*x^2])/(4*e) - (d^2*(16*d - 9*e*x)*Sqrt[d^2 - e^2*x^
2])/(24*e^4) - (3*d^4*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e^4)

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 847

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

Rule 864

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (d-e x)}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {\int \frac {x^2 \left (3 d^2 e-4 d e^2 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^2} \\ & = -\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}+\frac {\int \frac {x \left (8 d^3 e^2-9 d^2 e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{12 e^4} \\ & = -\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {\left (3 d^4\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^3} \\ & = -\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {\left (3 d^4\right ) \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^3} \\ & = -\frac {d x^2 \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {x^3 \sqrt {d^2-e^2 x^2}}{4 e}-\frac {d^2 (16 d-9 e x) \sqrt {d^2-e^2 x^2}}{24 e^4}-\frac {3 d^4 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-16 d^3+9 d^2 e x-8 d e^2 x^2+6 e^3 x^3\right )+18 d^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{24 e^4} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-16*d^3 + 9*d^2*e*x - 8*d*e^2*x^2 + 6*e^3*x^3) + 18*d^4*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d
^2 - e^2*x^2])])/(24*e^4)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {\left (-6 e^{3} x^{3}+8 d \,e^{2} x^{2}-9 d^{2} e x +16 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{24 e^{4}}-\frac {3 d^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{3} \sqrt {e^{2}}}\)	\(86\)
default	\(\frac {-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 e^{2}}+\frac {d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4 e^{2}}}{e}+\frac {d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{e^{3}}+\frac {d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{4}}-\frac {d^{3} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{e^{4}}\)	\(243\)

[In]

int(x^3*(-e^2*x^2+d^2)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

-1/24*(-6*e^3*x^3+8*d*e^2*x^2-9*d^2*e*x+16*d^3)/e^4*(-e^2*x^2+d^2)^(1/2)-3/8*d^4/e^3/(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 = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\frac {18 \, d^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (6 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 9 \, d^{2} e x - 16 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, e^{4}} \]

[In]

integrate(x^3*(-e^2*x^2+d^2)^(1/2)/(e*x+d),x, algorithm="fricas")

[Out]

1/24*(18*d^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (6*e^3*x^3 - 8*d*e^2*x^2 + 9*d^2*e*x - 16*d^3)*sqrt(-
e^2*x^2 + d^2))/e^4

Sympy [F]

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

[In]

integrate(x**3*(-e**2*x**2+d**2)**(1/2)/(e*x+d),x)

[Out]

Integral(x**3*sqrt(-(-d + e*x)*(d + e*x))/(d + e*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {3 \, d^{4} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{4}} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} x}{8 \, e^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} x}{4 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{3 \, e^{4}} \]

[In]

integrate(x^3*(-e^2*x^2+d^2)^(1/2)/(e*x+d),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.64 \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=-\frac {3 \, d^{4} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, e^{3} {\left | e \right |}} + \frac {1}{24} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, x {\left (\frac {3 \, x}{e} - \frac {4 \, d}{e^{2}}\right )} + \frac {9 \, d^{2}}{e^{3}}\right )} x - \frac {16 \, d^{3}}{e^{4}}\right )} \]

[In]

integrate(x^3*(-e^2*x^2+d^2)^(1/2)/(e*x+d),x, algorithm="giac")

[Out]

-3/8*d^4*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) + 1/24*sqrt(-e^2*x^2 + d^2)*((2*x*(3*x/e - 4*d/e^2) + 9*d^2/
e^3)*x - 16*d^3/e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {d^2-e^2 x^2}}{d+e x} \, dx=\int \frac {x^3\,\sqrt {d^2-e^2\,x^2}}{d+e\,x} \,d x \]

[In]

int((x^3*(d^2 - e^2*x^2)^(1/2))/(d + e*x),x)

[Out]

int((x^3*(d^2 - e^2*x^2)^(1/2))/(d + e*x), x)