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

Optimal. Leaf size=161 \[ \frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8} \]

[Out]

(x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*
d + 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d + 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) - (7*d^2*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/(2*e^8)

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Rubi [A]  time = 0.139297, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {819, 780, 217, 203} \[ \frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^6*(d + e*x))/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (x^4*(6*d + 7*e*x))/(15*e^4*(d^2 - e^2*x^2)^(3/2)) + (x^2*(24*
d + 35*e*x))/(15*e^6*Sqrt[d^2 - e^2*x^2]) + ((32*d + 35*e*x)*Sqrt[d^2 - e^2*x^2])/(10*e^8) - (7*d^2*ArcTan[(e*
x)/Sqrt[d^2 - e^2*x^2]])/(2*e^8)

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 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^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{x^5 \left (6 d^3+7 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{x^3 \left (24 d^5+35 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{x \left (48 d^7+105 d^6 e x\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{\left (7 d^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^7}\\ &=\frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{\left (7 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^7}\\ &=\frac{x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{x^2 (24 d+35 e x)}{15 e^6 \sqrt{d^2-e^2 x^2}}+\frac{(32 d+35 e x) \sqrt{d^2-e^2 x^2}}{10 e^8}-\frac{7 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^8}\\ \end{align*}

Mathematica [A]  time = 0.109221, size = 155, normalized size = 0.96 \[ \frac{-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-105 d^2 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+9 d^5 e x+96 d^6-15 d e^5 x^5-15 e^6 x^6}{30 e^8 (d-e x)^2 (d+e x) \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(96*d^6 + 9*d^5*e*x - 249*d^4*e^2*x^2 + 4*d^3*e^3*x^3 + 176*d^2*e^4*x^4 - 15*d*e^5*x^5 - 15*e^6*x^6 - 105*d^2*
(d - e*x)^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(30*e^8*(d - e*x)^2*(d + e*x)*Sqr
t[d^2 - e^2*x^2])

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Maple [A]  time = 0.117, size = 227, normalized size = 1.4 \begin{align*} -{\frac{{x}^{7}}{2\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{7\,{d}^{2}{x}^{5}}{10\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{7\,{d}^{2}{x}^{3}}{6\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{d}^{2}x}{2\,{e}^{7}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{7\,{d}^{2}}{2\,{e}^{7}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d{x}^{6}}{{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{5}{2}}}}+6\,{\frac{{d}^{3}{x}^{4}}{{e}^{4} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}}-8\,{\frac{{d}^{5}{x}^{2}}{{e}^{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{5/2}}}+{\frac{16\,{d}^{7}}{5\,{e}^{8}} \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^7*(e*x+d)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

-1/2*x^7/e/(-e^2*x^2+d^2)^(5/2)+7/10*d^2/e^3*x^5/(-e^2*x^2+d^2)^(5/2)-7/6*d^2/e^5*x^3/(-e^2*x^2+d^2)^(3/2)+7/2
*d^2/e^7*x/(-e^2*x^2+d^2)^(1/2)-7/2*d^2/e^7/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-d*x^6/e^2/(
-e^2*x^2+d^2)^(5/2)+6*d^3/e^4*x^4/(-e^2*x^2+d^2)^(5/2)-8*d^5/e^6*x^2/(-e^2*x^2+d^2)^(5/2)+16/5*d^7/e^8/(-e^2*x
^2+d^2)^(5/2)

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Maxima [B]  time = 1.48889, size = 439, normalized size = 2.73 \begin{align*} -\frac{x^{7}}{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e} + \frac{7 \, d^{2} 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 )}}{30 \, e} - \frac{d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{2}} - \frac{7 \, d^{2} 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 )}}{6 \, e^{3}} + \frac{6 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{4}} - \frac{8 \, d^{5} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{6}} + \frac{16 \, d^{7}}{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} e^{8}} + \frac{14 \, d^{4} x}{15 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} e^{7}} - \frac{49 \, d^{2} x}{30 \, \sqrt{-e^{2} x^{2} + d^{2}} e^{7}} - \frac{7 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^7/((-e^2*x^2 + d^2)^(5/2)*e) + 7/30*d^2*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))/e - d*x^6/((-e^2*x^2 + d^2)^(5/2)*e^2) - 7/6*d^2*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^3 + 6*d^3*x^4/((-e^2*x^2 + d^2)^(5/2)
*e^4) - 8*d^5*x^2/((-e^2*x^2 + d^2)^(5/2)*e^6) + 16/5*d^7/((-e^2*x^2 + d^2)^(5/2)*e^8) + 14/15*d^4*x/((-e^2*x^
2 + d^2)^(3/2)*e^7) - 49/30*d^2*x/(sqrt(-e^2*x^2 + d^2)*e^7) - 7/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*
e^7)

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Fricas [A]  time = 2.67781, size = 575, normalized size = 3.57 \begin{align*} \frac{96 \, d^{2} e^{5} x^{5} - 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} + 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x - 96 \, d^{7} + 210 \,{\left (d^{2} e^{5} x^{5} - d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} + 2 \, d^{5} e^{2} x^{2} + d^{6} e x - d^{7}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (15 \, e^{6} x^{6} + 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} - 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (e^{13} x^{5} - d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} + 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x - d^{5} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(96*d^2*e^5*x^5 - 96*d^3*e^4*x^4 - 192*d^4*e^3*x^3 + 192*d^5*e^2*x^2 + 96*d^6*e*x - 96*d^7 + 210*(d^2*e^5
*x^5 - d^3*e^4*x^4 - 2*d^4*e^3*x^3 + 2*d^5*e^2*x^2 + d^6*e*x - d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x))
+ (15*e^6*x^6 + 15*d*e^5*x^5 - 176*d^2*e^4*x^4 - 4*d^3*e^3*x^3 + 249*d^4*e^2*x^2 - 9*d^5*e*x - 96*d^6)*sqrt(-e
^2*x^2 + d^2))/(e^13*x^5 - d*e^12*x^4 - 2*d^2*e^11*x^3 + 2*d^3*e^10*x^2 + d^4*e^9*x - d^5*e^8)

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Sympy [B]  time = 48.8218, size = 2006, normalized size = 12.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12
*x**4*sqrt(d**2 - e**2*x**2)) - 40*d**4*e**2*x**2/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqr
t(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) + 30*d**2*e**4*x**4/(5*d**4*e**8*sqrt(d**2 - e**2*x
**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 5*e**6*x**6/(5*d**4*
e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2))
, Ne(e, 0)), (x**8/(8*(d**2)**(7/2)), True)) + e*Piecewise((210*I*d**7*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/
(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt
(-1 + e**2*x**2/d**2)) - 105*pi*d**7*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d
**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 210*I*d**6*e*x/(60*d**
5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e
**2*x**2/d**2)) - 420*I*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1 + e**2*x**
2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 210*pi*
d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1
+ e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 490*I*d**4*e**3*x**3/(60*d**5*e**9*sqrt(-1 +
e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2))
+ 210*I*d**3*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d*
*3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 105*pi*d**3*e**4*x**4*s
qrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**
2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 322*I*d**2*e**5*x**5/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2)
- 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*I*e**7*x**7
/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqr
t(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (-105*d**7*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(30*d*
*5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2
*x**2/d**2)) + 105*d**6*e*x/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**
2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) + 210*d**5*e**2*x**2*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d)/(30*d
**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**
2*x**2/d**2)) - 245*d**4*e**3*x**3/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x
**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 105*d**3*e**4*x**4*sqrt(1 - e**2*x**2/d**2)*asin(e*x/d
)/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(
1 - e**2*x**2/d**2)) + 161*d**2*e**5*x**5/(30*d**5*e**9*sqrt(1 - e**2*x**2/d**2) - 60*d**3*e**11*x**2*sqrt(1 -
e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)) - 15*e**7*x**7/(30*d**5*e**9*sqrt(1 - e**2*x**2/d
**2) - 60*d**3*e**11*x**2*sqrt(1 - e**2*x**2/d**2) + 30*d*e**13*x**4*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [A]  time = 1.18818, size = 162, normalized size = 1.01 \begin{align*} -\frac{7}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-8\right )} \mathrm{sgn}\left (d\right ) - \frac{{\left (96 \, d^{7} e^{\left (-8\right )} +{\left (105 \, d^{6} e^{\left (-7\right )} -{\left (240 \, d^{5} e^{\left (-6\right )} +{\left (245 \, d^{4} e^{\left (-5\right )} -{\left (180 \, d^{3} e^{\left (-4\right )} +{\left (161 \, d^{2} e^{\left (-3\right )} - 15 \,{\left (x e^{\left (-1\right )} + 2 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}}}{30 \,{\left (x^{2} e^{2} - d^{2}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/2*d^2*arcsin(x*e/d)*e^(-8)*sgn(d) - 1/30*(96*d^7*e^(-8) + (105*d^6*e^(-7) - (240*d^5*e^(-6) + (245*d^4*e^(-
5) - (180*d^3*e^(-4) + (161*d^2*e^(-3) - 15*(x*e^(-1) + 2*d*e^(-2))*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)/(x^
2*e^2 - d^2)^3