3.2 \(\int x^4 (d+e x) (d^2-e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=201 \[ \frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5} \]

[Out]

(3*d^7*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4) - (4*d^2*x^2*(d^2 - e^2*x^2)^
(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8*e^2) - (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) - (d^3*(128*d + 31
5*e*x)*(d^2 - e^2*x^2)^(5/2))/(5040*e^5) + (3*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

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Rubi [A]  time = 0.149388, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {833, 780, 195, 217, 203} \[ \frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x)*(d^2 - e^2*x^2)^(3/2),x]

[Out]

(3*d^7*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4) - (4*d^2*x^2*(d^2 - e^2*x^2)^
(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8*e^2) - (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) - (d^3*(128*d + 31
5*e*x)*(d^2 - e^2*x^2)^(5/2))/(5040*e^5) + (3*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

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 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^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx &=-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{\int x^3 \left (-4 d^2 e-9 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{9 e^2}\\ &=-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac{\int x^2 \left (27 d^3 e^2+32 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{72 e^4}\\ &=-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{\int x \left (-64 d^4 e^3-189 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{504 e^6}\\ &=-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{d^5 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^4}\\ &=\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{\left (3 d^7\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{\left (3 d^9\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{\left (3 d^9\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^4}\\ &=\frac{3 d^7 x \sqrt{d^2-e^2 x^2}}{128 e^4}+\frac{d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac{4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac{d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac{x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac{d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac{3 d^9 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{128 e^5}\\ \end{align*}

Mathematica [A]  time = 0.239684, size = 157, normalized size = 0.78 \[ \frac{\sqrt{d^2-e^2 x^2} \left (945 d^8 \sin ^{-1}\left (\frac{e x}{d}\right )-\sqrt{1-\frac{e^2 x^2}{d^2}} \left (512 d^6 e^2 x^2+630 d^5 e^3 x^3+384 d^4 e^4 x^4-7560 d^3 e^5 x^5-6400 d^2 e^6 x^6+945 d^7 e x+1024 d^8+5040 d e^7 x^7+4480 e^8 x^8\right )\right )}{40320 e^5 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(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]*(1024*d^8 + 945*d^7*e*x + 512*d^6*e^2*x^2 + 630*d^5*e^3*x^3 +
384*d^4*e^4*x^4 - 7560*d^3*e^5*x^5 - 6400*d^2*e^6*x^6 + 5040*d*e^7*x^7 + 4480*e^8*x^8)) + 945*d^8*ArcSin[(e*x)
/d]))/(40320*e^5*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.067, size = 198, normalized size = 1. \begin{align*} -{\frac{{x}^{4}}{9\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{4\,{d}^{2}{x}^{2}}{63\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{8\,{d}^{4}}{315\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{d{x}^{3}}{8\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{d}^{3}x}{16\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{5}x}{64\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{7}x}{128\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,{d}^{9}}{128\,{e}^{4}}\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^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x)

[Out]

-1/9*x^4*(-e^2*x^2+d^2)^(5/2)/e-4/63*d^2*x^2*(-e^2*x^2+d^2)^(5/2)/e^3-8/315*d^4/e^5*(-e^2*x^2+d^2)^(5/2)-1/8*d
*x^3*(-e^2*x^2+d^2)^(5/2)/e^2-1/16*d^3/e^4*x*(-e^2*x^2+d^2)^(5/2)+1/64*d^5*x*(-e^2*x^2+d^2)^(3/2)/e^4+3/128*d^
7*x*(-e^2*x^2+d^2)^(1/2)/e^4+3/128*d^9/e^4/(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 = 1.75911, size = 257, normalized size = 1.28 \begin{align*} -\frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} x^{4}}{9 \, e} + \frac{3 \, d^{9} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{128 \, \sqrt{e^{2}} e^{4}} + \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d x^{3}}{8 \, e^{2}} + \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{5} x}{64 \, e^{4}} - \frac{4 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{2} x^{2}}{63 \, e^{3}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{3} x}{16 \, e^{4}} - \frac{8 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{4}}{315 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")

[Out]

-1/9*(-e^2*x^2 + d^2)^(5/2)*x^4/e + 3/128*d^9*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^4) + 3/128*sqrt(-e^2*x^
2 + d^2)*d^7*x/e^4 - 1/8*(-e^2*x^2 + d^2)^(5/2)*d*x^3/e^2 + 1/64*(-e^2*x^2 + d^2)^(3/2)*d^5*x/e^4 - 4/63*(-e^2
*x^2 + d^2)^(5/2)*d^2*x^2/e^3 - 1/16*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^4 - 8/315*(-e^2*x^2 + d^2)^(5/2)*d^4/e^5

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Fricas [A]  time = 1.82994, size = 323, normalized size = 1.61 \begin{align*} -\frac{1890 \, d^{9} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (4480 \, e^{8} x^{8} + 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} - 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} + 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} + 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")

[Out]

-1/40320*(1890*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (4480*e^8*x^8 + 5040*d*e^7*x^7 - 6400*d^2*e^6*x
^6 - 7560*d^3*e^5*x^5 + 384*d^4*e^4*x^4 + 630*d^5*e^3*x^3 + 512*d^6*e^2*x^2 + 945*d^7*e*x + 1024*d^8)*sqrt(-e^
2*x^2 + d^2))/e^5

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Sympy [C]  time = 22.206, size = 833, normalized size = 4.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)*(-e**2*x**2+d**2)**(3/2),x)

[Out]

d**3*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/(4
8*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)) + d**2*e*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)) - d*e**2*Piecewise((-5*I*d**8*acosh(e*x/d)/(128*e**7) + 5*I*d**7*x/
(128*e**6*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d**5*x**3/(384*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**5/(192*e
**2*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d*x**7/(48*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**9/(8*d*sqrt(-1 + e**2*x
**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (5*d**8*asin(e*x/d)/(128*e**7) - 5*d**7*x/(128*e**6*sqrt(1 - e**2*x
**2/d**2)) + 5*d**5*x**3/(384*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**5/(192*e**2*sqrt(1 - e**2*x**2/d**2)) +
7*d*x**7/(48*sqrt(1 - e**2*x**2/d**2)) - e**2*x**9/(8*d*sqrt(1 - e**2*x**2/d**2)), True)) - e**3*Piecewise((-
16*d**8*sqrt(d**2 - e**2*x**2)/(315*e**8) - 8*d**6*x**2*sqrt(d**2 - e**2*x**2)/(315*e**6) - 2*d**4*x**4*sqrt(d
**2 - e**2*x**2)/(105*e**4) - d**2*x**6*sqrt(d**2 - e**2*x**2)/(63*e**2) + x**8*sqrt(d**2 - e**2*x**2)/9, Ne(e
, 0)), (x**8*sqrt(d**2)/8, True))

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Giac [A]  time = 1.29974, size = 158, normalized size = 0.79 \begin{align*} \frac{3}{128} \, d^{9} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-5\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{40320} \,{\left (1024 \, d^{8} e^{\left (-5\right )} +{\left (945 \, d^{7} e^{\left (-4\right )} + 2 \,{\left (256 \, d^{6} e^{\left (-3\right )} +{\left (315 \, d^{5} e^{\left (-2\right )} + 4 \,{\left (48 \, d^{4} e^{\left (-1\right )} - 5 \,{\left (189 \, d^{3} + 2 \,{\left (80 \, d^{2} e - 7 \,{\left (8 \, x e^{3} + 9 \, d e^{2}\right )} x\right )} x\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^4*(e*x+d)*(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")

[Out]

3/128*d^9*arcsin(x*e/d)*e^(-5)*sgn(d) - 1/40320*(1024*d^8*e^(-5) + (945*d^7*e^(-4) + 2*(256*d^6*e^(-3) + (315*
d^5*e^(-2) + 4*(48*d^4*e^(-1) - 5*(189*d^3 + 2*(80*d^2*e - 7*(8*x*e^3 + 9*d*e^2)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^
2*e^2 + d^2)