\(\int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx\) [34]

Optimal result

Integrand size = 27, antiderivative size = 144 \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\frac {3 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^4} \]

[Out]

3/4*d^5*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4-3/5*d^2*x^2*(-e^2*x^2+d^2)^(1/2)/e^2-1/2*d*x^3*(-e^2*x^2+d^2)^(1/
2)/e-1/5*x^4*(-e^2*x^2+d^2)^(1/2)-3/20*d^3*(5*e*x+8*d)*(-e^2*x^2+d^2)^(1/2)/e^4

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 144, 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 = {1823, 847, 794, 223, 209} \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {3 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^4}-\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4} \]

[In]

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

[Out]

(-3*d^2*x^2*Sqrt[d^2 - e^2*x^2])/(5*e^2) - (d*x^3*Sqrt[d^2 - e^2*x^2])/(2*e) - (x^4*Sqrt[d^2 - e^2*x^2])/5 - (
3*d^3*(8*d + 5*e*x)*Sqrt[d^2 - e^2*x^2])/(20*e^4) + (3*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4*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 1823

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x^3 \left (-9 d^2 e^2-10 d e^3 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{5 e^2} \\ & = -\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {x^2 \left (30 d^3 e^3+36 d^2 e^4 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{20 e^4} \\ & = -\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {x \left (-72 d^4 e^4-90 d^3 e^5 x\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{60 e^6} \\ & = -\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\frac {\left (3 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^3} \\ & = -\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\frac {\left (3 d^5\right ) \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^3} \\ & = -\frac {3 d^2 x^2 \sqrt {d^2-e^2 x^2}}{5 e^2}-\frac {d x^3 \sqrt {d^2-e^2 x^2}}{2 e}-\frac {1}{5} x^4 \sqrt {d^2-e^2 x^2}-\frac {3 d^3 (8 d+5 e x) \sqrt {d^2-e^2 x^2}}{20 e^4}+\frac {3 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.72 \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (24 d^4+15 d^3 e x+12 d^2 e^2 x^2+10 d e^3 x^3+4 e^4 x^4\right )+30 d^5 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{20 e^4} \]

[In]

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

[Out]

-1/20*(Sqrt[d^2 - e^2*x^2]*(24*d^4 + 15*d^3*e*x + 12*d^2*e^2*x^2 + 10*d*e^3*x^3 + 4*e^4*x^4) + 30*d^5*ArcTan[(
e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e^4

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (4 e^{4} x^{4}+10 d \,e^{3} x^{3}+12 d^{2} e^{2} x^{2}+15 d^{3} e x +24 d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{20 e^{4}}+\frac {3 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 e^{3} \sqrt {e^{2}}}\)	\(97\)
default	\(e^{2} \left (-\frac {x^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{5 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}}\right )}{5 e^{2}}\right )+d^{2} \left (-\frac {x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{4}}\right )+2 d e \left (-\frac {x^{3} \sqrt {-e^{2} x^{2}+d^{2}}}{4 e^{2}}+\frac {3 d^{2} \left (-\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )}{4 e^{2}}\right )\)	\(222\)

[In]

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

[Out]

-1/20*(4*e^4*x^4+10*d*e^3*x^3+12*d^2*e^2*x^2+15*d^3*e*x+24*d^4)/e^4*(-e^2*x^2+d^2)^(1/2)+3/4*d^5/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 = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {30 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (4 \, e^{4} x^{4} + 10 \, d e^{3} x^{3} + 12 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x + 24 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{20 \, e^{4}} \]

[In]

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

[Out]

-1/20*(30*d^5*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (4*e^4*x^4 + 10*d*e^3*x^3 + 12*d^2*e^2*x^2 + 15*d^3*
e*x + 24*d^4)*sqrt(-e^2*x^2 + d^2))/e^4

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\begin {cases} \frac {3 d^{5} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{4 e^{3}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {6 d^{4}}{5 e^{4}} - \frac {3 d^{3} x}{4 e^{3}} - \frac {3 d^{2} x^{2}}{5 e^{2}} - \frac {d x^{3}}{2 e} - \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {\frac {d^{2} x^{4}}{4} + \frac {2 d e x^{5}}{5} + \frac {e^{2} x^{6}}{6}}{\sqrt {d^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*d**5*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))/(4*e**3) + sqrt(d**2 - e**2*x**2)*(-6*d**4/(5*e**4) - 3*d**3*x/(4*e**3) - 3
*d**2*x**2/(5*e**2) - d*x**3/(2*e) - x**4/5), Ne(e**2, 0)), ((d**2*x**4/4 + 2*d*e*x**5/5 + e**2*x**6/6)/sqrt(d
**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {1}{5} \, \sqrt {-e^{2} x^{2} + d^{2}} x^{4} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} d x^{3}}{2 \, e} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} x^{2}}{5 \, e^{2}} + \frac {3 \, d^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{4 \, \sqrt {e^{2}} e^{3}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} x}{4 \, e^{3}} - \frac {6 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e^{4}} \]

[In]

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

[Out]

-1/5*sqrt(-e^2*x^2 + d^2)*x^4 - 1/2*sqrt(-e^2*x^2 + d^2)*d*x^3/e - 3/5*sqrt(-e^2*x^2 + d^2)*d^2*x^2/e^2 + 3/4*
d^5*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^3) - 3/4*sqrt(-e^2*x^2 + d^2)*d^3*x/e^3 - 6/5*sqrt(-e^2*x^2 + d^2
)*d^4/e^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.58 \[ \int \frac {x^3 (d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {3 \, d^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{4 \, e^{3} {\left | e \right |}} - \frac {1}{20} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (2 \, {\left ({\left (2 \, x + \frac {5 \, d}{e}\right )} x + \frac {6 \, d^{2}}{e^{2}}\right )} x + \frac {15 \, d^{3}}{e^{3}}\right )} x + \frac {24 \, d^{4}}{e^{4}}\right )} \]

[In]

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

[Out]

3/4*d^5*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) - 1/20*sqrt(-e^2*x^2 + d^2)*((2*((2*x + 5*d/e)*x + 6*d^2/e^2)
*x + 15*d^3/e^3)*x + 24*d^4/e^4)

Mupad [F(-1)]

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

[In]

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

[Out]

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