3.33 \(\int \frac{x^4 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{d^4 (256 d+165 e x) \sqrt{d^2-e^2 x^2}}{240 e^5}-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{11 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^5} \]

[Out]

(-8*d^3*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) - (11*d^2*x^3*Sqrt[d^2 - e^2*x^2])/(24*e^2) - (2*d*x^4*Sqrt[d^2 - e^
2*x^2])/(5*e) - (x^5*Sqrt[d^2 - e^2*x^2])/6 - (d^4*(256*d + 165*e*x)*Sqrt[d^2 - e^2*x^2])/(240*e^5) + (11*d^6*
ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^5)

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Rubi [A]  time = 0.226878, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1809, 833, 780, 217, 203} \[ -\frac{d^4 (256 d+165 e x) \sqrt{d^2-e^2 x^2}}{240 e^5}-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{11 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]

[Out]

(-8*d^3*x^2*Sqrt[d^2 - e^2*x^2])/(15*e^3) - (11*d^2*x^3*Sqrt[d^2 - e^2*x^2])/(24*e^2) - (2*d*x^4*Sqrt[d^2 - e^
2*x^2])/(5*e) - (x^5*Sqrt[d^2 - e^2*x^2])/6 - (d^4*(256*d + 165*e*x)*Sqrt[d^2 - e^2*x^2])/(240*e^5) + (11*d^6*
ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e^5)

Rule 1809

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])

Rule 833

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 780

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 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 \frac{x^4 (d+e x)^2}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{x^4 \left (-11 d^2 e^2-12 d e^3 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{6 e^2}\\ &=-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{x^3 \left (48 d^3 e^3+55 d^2 e^4 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{30 e^4}\\ &=-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{x^2 \left (-165 d^4 e^4-192 d^3 e^5 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{120 e^6}\\ &=-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{x \left (384 d^5 e^5+495 d^4 e^6 x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{360 e^8}\\ &=-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{d^4 (256 d+165 e x) \sqrt{d^2-e^2 x^2}}{240 e^5}+\frac{\left (11 d^6\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^4}\\ &=-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{d^4 (256 d+165 e x) \sqrt{d^2-e^2 x^2}}{240 e^5}+\frac{\left (11 d^6\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^4}\\ &=-\frac{8 d^3 x^2 \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{11 d^2 x^3 \sqrt{d^2-e^2 x^2}}{24 e^2}-\frac{2 d x^4 \sqrt{d^2-e^2 x^2}}{5 e}-\frac{1}{6} x^5 \sqrt{d^2-e^2 x^2}-\frac{d^4 (256 d+165 e x) \sqrt{d^2-e^2 x^2}}{240 e^5}+\frac{11 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^5}\\ \end{align*}

Mathematica [A]  time = 0.111076, size = 103, normalized size = 0.6 \[ \frac{165 d^6 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\sqrt{d^2-e^2 x^2} \left (128 d^3 e^2 x^2+110 d^2 e^3 x^3+165 d^4 e x+256 d^5+96 d e^4 x^4+40 e^5 x^5\right )}{240 e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x)^2)/Sqrt[d^2 - e^2*x^2],x]

[Out]

(-(Sqrt[d^2 - e^2*x^2]*(256*d^5 + 165*d^4*e*x + 128*d^3*e^2*x^2 + 110*d^2*e^3*x^3 + 96*d*e^4*x^4 + 40*e^5*x^5)
) + 165*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(240*e^5)

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Maple [A]  time = 0.07, size = 174, normalized size = 1. \begin{align*} -{\frac{{x}^{5}}{6}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{11\,{d}^{2}{x}^{3}}{24\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{11\,{d}^{4}x}{16\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{11\,{d}^{6}}{16\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{2\,d{x}^{4}}{5\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{8\,{d}^{3}{x}^{2}}{15\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{16\,{d}^{5}}{15\,{e}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)^2/(-e^2*x^2+d^2)^(1/2),x)

[Out]

-1/6*x^5*(-e^2*x^2+d^2)^(1/2)-11/24*d^2*x^3*(-e^2*x^2+d^2)^(1/2)/e^2-11/16*d^4*x*(-e^2*x^2+d^2)^(1/2)/e^4+11/1
6/e^4*d^6/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-2/5*d*x^4*(-e^2*x^2+d^2)^(1/2)/e-8/15*d^3*x^2
*(-e^2*x^2+d^2)^(1/2)/e^3-16/15*d^5*(-e^2*x^2+d^2)^(1/2)/e^5

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Maxima [A]  time = 1.46733, size = 224, normalized size = 1.29 \begin{align*} -\frac{1}{6} \, \sqrt{-e^{2} x^{2} + d^{2}} x^{5} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} d x^{4}}{5 \, e} - \frac{11 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x^{3}}{24 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x^{2}}{15 \, e^{3}} + \frac{11 \, d^{6} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{4}} - \frac{11 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{4} x}{16 \, e^{4}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{5}}{15 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^2/(-e^2*x^2+d^2)^(1/2),x, algorithm="maxima")

[Out]

-1/6*sqrt(-e^2*x^2 + d^2)*x^5 - 2/5*sqrt(-e^2*x^2 + d^2)*d*x^4/e - 11/24*sqrt(-e^2*x^2 + d^2)*d^2*x^3/e^2 - 8/
15*sqrt(-e^2*x^2 + d^2)*d^3*x^2/e^3 + 11/16*d^6*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^4) - 11/16*sqrt(-e^2*
x^2 + d^2)*d^4*x/e^4 - 16/15*sqrt(-e^2*x^2 + d^2)*d^5/e^5

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Fricas [A]  time = 1.82931, size = 236, normalized size = 1.36 \begin{align*} -\frac{330 \, d^{6} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (40 \, e^{5} x^{5} + 96 \, d e^{4} x^{4} + 110 \, d^{2} e^{3} x^{3} + 128 \, d^{3} e^{2} x^{2} + 165 \, d^{4} e x + 256 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^2/(-e^2*x^2+d^2)^(1/2),x, algorithm="fricas")

[Out]

-1/240*(330*d^6*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (40*e^5*x^5 + 96*d*e^4*x^4 + 110*d^2*e^3*x^3 + 128
*d^3*e^2*x^2 + 165*d^4*e*x + 256*d^5)*sqrt(-e^2*x^2 + d^2))/e^5

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Sympy [C]  time = 13.5183, size = 561, normalized size = 3.24 \begin{align*} d^{2} \left (\begin{cases} - \frac{3 i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{5}} + \frac{3 i d^{3} x}{8 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d x^{3}}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{3 d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{5}} - \frac{3 d^{3} x}{8 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d x^{3}}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{5}}{4 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{8 d^{4} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{6}} - \frac{4 d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac{x^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{6}}{6 \sqrt{d^{2}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{5 i d^{6} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{16 e^{7}} + \frac{5 i d^{5} x}{16 e^{6} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{5 i d^{3} x^{3}}{48 e^{4} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i d x^{5}}{24 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{i x^{7}}{6 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{5 d^{6} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{16 e^{7}} - \frac{5 d^{5} x}{16 e^{6} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{5 d^{3} x^{3}}{48 e^{4} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{d x^{5}}{24 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{7}}{6 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)**2/(-e**2*x**2+d**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.18962, size = 113, normalized size = 0.65 \begin{align*} \frac{11}{16} \, d^{6} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-5\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{240} \,{\left (256 \, d^{5} e^{\left (-5\right )} +{\left (165 \, d^{4} e^{\left (-4\right )} + 2 \,{\left (64 \, d^{3} e^{\left (-3\right )} +{\left (55 \, d^{2} e^{\left (-2\right )} + 4 \,{\left (12 \, d e^{\left (-1\right )} + 5 \, 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^4*(e*x+d)^2/(-e^2*x^2+d^2)^(1/2),x, algorithm="giac")

[Out]

11/16*d^6*arcsin(x*e/d)*e^(-5)*sgn(d) - 1/240*(256*d^5*e^(-5) + (165*d^4*e^(-4) + 2*(64*d^3*e^(-3) + (55*d^2*e
^(-2) + 4*(12*d*e^(-1) + 5*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)