3.21 \(\int \frac{x^5 (d+e x)}{(d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=122 \[ \frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

[Out]

(x^4*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^2*(4*d + 5*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (8*d + 15
*e*x)/(15*e^6*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e^6

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Rubi [A]  time = 0.080827, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {819, 778, 217, 203} \[ \frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(x^4*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^2*(4*d + 5*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (8*d + 15
*e*x)/(15*e^6*Sqrt[d^2 - e^2*x^2]) - ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]]/e^6

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[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^5 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x^3 \left (4 d^3+5 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{x \left (8 d^5+15 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac{x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d+15 e x}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.0869623, size = 130, normalized size = 1.07 \[ \frac{-27 d^2 e^2 x^2-15 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+7 d^3 e x+8 d^4-8 d e^3 x^3+23 e^4 x^4}{15 e^6 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(d + e*x))/(d^2 - e^2*x^2)^(7/2),x]

[Out]

(8*d^4 + 7*d^3*e*x - 27*d^2*e^2*x^2 - 8*d*e^3*x^3 + 23*e^4*x^4 - 15*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*
ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(15*e^6*(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2])

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Maple [A]  time = 0.076, size = 166, normalized size = 1.4 \begin{align*}{\frac{{x}^{5}}{5\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{{x}^{3}}{3\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{x}{{e}^{5}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{1}{{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{d{x}^{4}}{{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{4\,{d}^{3}{x}^{2}}{3\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{8\,{d}^{5}}{15\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

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

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Maxima [B]  time = 1.51158, size = 355, normalized size = 2.91 \begin{align*} \frac{1}{15} \, e x{\left (\frac{15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}}\right )} - \frac{x{\left (\frac{3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{2}} - \frac{2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{4}}\right )}}{3 \, e} + \frac{d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{4 \, d^{3} x^{2}}{3 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} + \frac{8 \, d^{5}}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{4 \, d^{2} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{5}} - \frac{7 \, x}{15 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{5}} - \frac{\arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}} e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*e*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d
^2)^(5/2)*e^6)) - 1/3*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e + d*x^4/((
-e^2*x^2 + d^2)^(5/2)*e^2) - 4/3*d^3*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8/15*d^5/((-e^2*x^2 + d^2)^(5/2)*e^6)
+ 4/15*d^2*x/((-e^2*x^2 + d^2)^(3/2)*e^5) - 7/15*x/(sqrt(-e^2*x^2 + d^2)*e^5) - arcsin(e^2*x/sqrt(d^2*e^2))/(s
qrt(e^2)*e^5)

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Fricas [B]  time = 2.09142, size = 497, normalized size = 4.07 \begin{align*} \frac{8 \, e^{5} x^{5} - 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 8 \, d^{5} + 30 \,{\left (e^{5} x^{5} - d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} + 2 \, d^{3} e^{2} x^{2} + d^{4} e x - d^{5}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (23 \, e^{4} x^{4} - 8 \, d e^{3} x^{3} - 27 \, d^{2} e^{2} x^{2} + 7 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (e^{11} x^{5} - d e^{10} x^{4} - 2 \, d^{2} e^{9} x^{3} + 2 \, d^{3} e^{8} x^{2} + d^{4} e^{7} x - d^{5} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(8*e^5*x^5 - 8*d*e^4*x^4 - 16*d^2*e^3*x^3 + 16*d^3*e^2*x^2 + 8*d^4*e*x - 8*d^5 + 30*(e^5*x^5 - d*e^4*x^4
- 2*d^2*e^3*x^3 + 2*d^3*e^2*x^2 + d^4*e*x - d^5)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (23*e^4*x^4 - 8*d
*e^3*x^3 - 27*d^2*e^2*x^2 + 7*d^3*e*x + 8*d^4)*sqrt(-e^2*x^2 + d^2))/(e^11*x^5 - d*e^10*x^4 - 2*d^2*e^9*x^3 +
2*d^3*e^8*x^2 + d^4*e^7*x - d^5*e^6)

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Sympy [B]  time = 43.4936, size = 1741, normalized size = 14.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(e*x+d)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((8*d**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10
*x**4*sqrt(d**2 - e**2*x**2)) - 20*d**2*e**2*x**2/(15*d**4*e**6*sqrt(d**2 - e**2*x**2) - 30*d**2*e**8*x**2*sqr
t(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)) + 15*e**4*x**4/(15*d**4*e**6*sqrt(d**2 - e**2*x**2
) - 30*d**2*e**8*x**2*sqrt(d**2 - e**2*x**2) + 15*e**10*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**6/(6*(d**
2)**(7/2)), True)) + e*Piecewise((30*I*d**5*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**
2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 15*p
i*d**5*sqrt(-1 + e**2*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x*
*2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 30*I*d**4*e*x/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2)
- 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 60*I*d**3*e**2*x*
*2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1
+ e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*pi*d**3*e**2*x**2*sqrt(-1 + e**2*x**2/d**2
)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt
(-1 + e**2*x**2/d**2)) + 70*I*d**2*e**3*x**3/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(
-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*I*d*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)
*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**
11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 15*pi*d*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*
x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 46*I*e
**5*x**5/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x*
*4*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (-15*d**5*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(
15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 -
e**2*x**2/d**2)) + 15*d**4*e*x/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d
**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) + 30*d**3*e**2*x**2*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(15*
d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**
2*x**2/d**2)) - 35*d**2*e**3*x**3/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**
2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)) - 15*d*e**4*x**4*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(15*
d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**2) + 15*d*e**11*x**4*sqrt(1 - e**
2*x**2/d**2)) + 23*e**5*x**5/(15*d**5*e**7*sqrt(1 - e**2*x**2/d**2) - 30*d**3*e**9*x**2*sqrt(1 - e**2*x**2/d**
2) + 15*d*e**11*x**4*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [A]  time = 1.18063, size = 131, normalized size = 1.07 \begin{align*} -\arcsin \left (\frac{x e}{d}\right ) e^{\left (-6\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (8 \, d^{5} e^{\left (-6\right )} +{\left (15 \, d^{4} e^{\left (-5\right )} -{\left (20 \, d^{3} e^{\left (-4\right )} +{\left (35 \, d^{2} e^{\left (-3\right )} -{\left (23 \, x e^{\left (-1\right )} + 15 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{15 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-arcsin(x*e/d)*e^(-6)*sgn(d) - 1/15*(8*d^5*e^(-6) + (15*d^4*e^(-5) - (20*d^3*e^(-4) + (35*d^2*e^(-3) - (23*x*e
^(-1) + 15*d*e^(-2))*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^2*e^2 - d^2)^3