3.4 \(\int x^2 (d+e x) (d^2-e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=159 \[ \frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3} \]

[Out]

(d^5*x*Sqrt[d^2 - e^2*x^2])/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) - (d^2*(d^2 - e^2*x^2)^(5/2))/(5
*e^3) - (d*x*(d^2 - e^2*x^2)^(5/2))/(6*e^2) + (d^2 - e^2*x^2)^(7/2)/(7*e^3) + (d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2
*x^2]])/(16*e^3)

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Rubi [A]  time = 0.10801, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {797, 641, 195, 217, 203} \[ \frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^5*x*Sqrt[d^2 - e^2*x^2])/(16*e^2) + (d^3*x*(d^2 - e^2*x^2)^(3/2))/(24*e^2) - (d^2*(d^2 - e^2*x^2)^(5/2))/(5
*e^3) - (d*x*(d^2 - e^2*x^2)^(5/2))/(6*e^2) + (d^2 - e^2*x^2)^(7/2)/(7*e^3) + (d^7*ArcTan[(e*x)/Sqrt[d^2 - e^2
*x^2]])/(16*e^3)

Rule 797

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/c, Int[(f + g*x)*(a + c*x^2)^(p
 + 1), x], x] - Dist[a/c, Int[(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*
c, 0]

Rule 641

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

Rule 195

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 217

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
 FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac{\int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac{d^2 \int (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{d \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2}+\frac{d^3 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{e^2}\\ &=\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{\left (5 d^3\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{6 e^2}+\frac{\left (3 d^5\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac{3 d^5 x \sqrt{d^2-e^2 x^2}}{8 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{\left (5 d^5\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{8 e^2}+\frac{\left (3 d^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}-\frac{\left (5 d^7\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^2}+\frac{\left (3 d^7\right ) \operatorname{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^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{3 d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}-\frac{\left (5 d^7\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^2}\\ &=\frac{d^5 x \sqrt{d^2-e^2 x^2}}{16 e^2}+\frac{d^3 x \left (d^2-e^2 x^2\right )^{3/2}}{24 e^2}-\frac{d^2 \left (d^2-e^2 x^2\right )^{5/2}}{5 e^3}-\frac{d x \left (d^2-e^2 x^2\right )^{5/2}}{6 e^2}+\frac{\left (d^2-e^2 x^2\right )^{7/2}}{7 e^3}+\frac{d^7 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3}\\ \end{align*}

Mathematica [A]  time = 0.191435, size = 135, normalized size = 0.85 \[ \frac{\sqrt{d^2-e^2 x^2} \left (105 d^6 \sin ^{-1}\left (\frac{e x}{d}\right )-\sqrt{1-\frac{e^2 x^2}{d^2}} \left (48 d^4 e^2 x^2-490 d^3 e^3 x^3-384 d^2 e^4 x^4+105 d^5 e x+96 d^6+280 d e^5 x^5+240 e^6 x^6\right )\right )}{1680 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-(Sqrt[1 - (e^2*x^2)/d^2]*(96*d^6 + 105*d^5*e*x + 48*d^4*e^2*x^2 - 490*d^3*e^3*x^3 - 384
*d^2*e^4*x^4 + 280*d*e^5*x^5 + 240*e^6*x^6)) + 105*d^6*ArcSin[(e*x)/d]))/(1680*e^3*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.059, size = 148, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{7\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{d}^{2}}{35\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{dx}{6\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}x}{24\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{5}x}{16\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{7}}{16\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*x^2*(-e^2*x^2+d^2)^(5/2)/e-2/35*d^2*(-e^2*x^2+d^2)^(5/2)/e^3-1/6*d*x*(-e^2*x^2+d^2)^(5/2)/e^2+1/24*d^3*x*
(-e^2*x^2+d^2)^(3/2)/e^2+1/16*d^5*x*(-e^2*x^2+d^2)^(1/2)/e^2+1/16*d^7/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e
^2*x^2+d^2)^(1/2))

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Maxima [A]  time = 1.57686, size = 189, normalized size = 1.19 \begin{align*} \frac{d^{7} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{2}} + \frac{\sqrt{-e^{2} x^{2} + d^{2}} d^{5} x}{16 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{3} x}{24 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} x^{2}}{7 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x}{6 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{2}}{35 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*d^7*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) + 1/16*sqrt(-e^2*x^2 + d^2)*d^5*x/e^2 + 1/24*(-e^2*x^2 +
d^2)^(3/2)*d^3*x/e^2 - 1/7*(-e^2*x^2 + d^2)^(5/2)*x^2/e - 1/6*(-e^2*x^2 + d^2)^(5/2)*d*x/e^2 - 2/35*(-e^2*x^2
+ d^2)^(5/2)*d^2/e^3

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Fricas [A]  time = 1.91371, size = 262, normalized size = 1.65 \begin{align*} -\frac{210 \, d^{7} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (240 \, e^{6} x^{6} + 280 \, d e^{5} x^{5} - 384 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} + 48 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x + 96 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{1680 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(210*d^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (240*e^6*x^6 + 280*d*e^5*x^5 - 384*d^2*e^4*x^4 -
490*d^3*e^3*x^3 + 48*d^4*e^2*x^2 + 105*d^5*e*x + 96*d^6)*sqrt(-e^2*x^2 + d^2))/e^3

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Sympy [C]  time = 14.9523, size = 656, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**3*Piecewise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 3*I*d*x**3/(8*sq
rt(-1 + e**2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**4*a
sin(e*x/d)/(8*e**3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2*
x**5/(4*d*sqrt(1 - e**2*x**2/d**2)), True)) + d**2*e*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**
2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True))
 - d*e**2*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x*
*3/(48*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1
+ e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2
*x**2/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x*
*7/(6*d*sqrt(1 - e**2*x**2/d**2)), True)) - e**3*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4
*x**2*sqrt(d**2 - e**2*x**2)/(105*e**4) - d**2*x**4*sqrt(d**2 - e**2*x**2)/(35*e**2) + x**6*sqrt(d**2 - e**2*x
**2)/7, Ne(e, 0)), (x**6*sqrt(d**2)/6, True))

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Giac [A]  time = 1.30189, size = 130, normalized size = 0.82 \begin{align*} \frac{1}{16} \, d^{7} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{1680} \,{\left (96 \, d^{6} e^{\left (-3\right )} +{\left (105 \, d^{5} e^{\left (-2\right )} + 2 \,{\left (24 \, d^{4} e^{\left (-1\right )} -{\left (245 \, d^{3} + 4 \,{\left (48 \, d^{2} e - 5 \,{\left (6 \, x e^{3} + 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*d^7*arcsin(x*e/d)*e^(-3)*sgn(d) - 1/1680*(96*d^6*e^(-3) + (105*d^5*e^(-2) + 2*(24*d^4*e^(-1) - (245*d^3 +
 4*(48*d^2*e - 5*(6*x*e^3 + 7*d*e^2)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)