3.69 \(\int x (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=230 \[ \frac{33 d^8 x \sqrt{d^2-e^2 x^2}}{256 e}+\frac{11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{33 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^2} \]

[Out]

(33*d^8*x*Sqrt[d^2 - e^2*x^2])/(256*e) + (11*d^6*x*(d^2 - e^2*x^2)^(3/2))/(128*e) + (11*d^4*x*(d^2 - e^2*x^2)^
(5/2))/(160*e) - (33*d^3*(d^2 - e^2*x^2)^(7/2))/(560*e^2) - (11*d^2*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(240*e^2)
- (d*(d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(30*e^2) - ((d + e*x)^3*(d^2 - e^2*x^2)^(7/2))/(10*e^2) + (33*d^10*Ar
cTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^2)

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Rubi [A]  time = 0.121367, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {795, 671, 641, 195, 217, 203} \[ \frac{33 d^8 x \sqrt{d^2-e^2 x^2}}{256 e}+\frac{11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{33 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(33*d^8*x*Sqrt[d^2 - e^2*x^2])/(256*e) + (11*d^6*x*(d^2 - e^2*x^2)^(3/2))/(128*e) + (11*d^4*x*(d^2 - e^2*x^2)^
(5/2))/(160*e) - (33*d^3*(d^2 - e^2*x^2)^(7/2))/(560*e^2) - (11*d^2*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(240*e^2)
- (d*(d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(30*e^2) - ((d + e*x)^3*(d^2 - e^2*x^2)^(7/2))/(10*e^2) + (33*d^10*Ar
cTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^2)

Rule 795

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[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 671

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*(m + p))/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

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 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{(3 d) \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx}{10 e}\\ &=-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (11 d^2\right ) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx}{30 e}\\ &=-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (33 d^3\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e}\\ &=-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (33 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e}\\ &=\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (11 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{32 e}\\ &=\frac{11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (33 d^8\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{128 e}\\ &=\frac{33 d^8 x \sqrt{d^2-e^2 x^2}}{256 e}+\frac{11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (33 d^{10}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{256 e}\\ &=\frac{33 d^8 x \sqrt{d^2-e^2 x^2}}{256 e}+\frac{11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{\left (33 d^{10}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e}\\ &=\frac{33 d^8 x \sqrt{d^2-e^2 x^2}}{256 e}+\frac{11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac{11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac{33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac{11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac{d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac{33 d^{10} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^2}\\ \end{align*}

Mathematica [A]  time = 0.41101, size = 167, normalized size = 0.73 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (10240 d^7 e^2 x^2+24570 d^6 e^3 x^3+7680 d^5 e^4 x^4-23352 d^4 e^5 x^5-20480 d^3 e^6 x^6+3024 d^2 e^7 x^7-3465 d^8 e x-6400 d^9+8960 d e^8 x^8+2688 e^9 x^9\right )+3465 d^9 \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{26880 e^2 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(Sqrt[1 - (e^2*x^2)/d^2]*(-6400*d^9 - 3465*d^8*e*x + 10240*d^7*e^2*x^2 + 24570*d^6*e^3*x^
3 + 7680*d^5*e^4*x^4 - 23352*d^4*e^5*x^5 - 20480*d^3*e^6*x^6 + 3024*d^2*e^7*x^7 + 8960*d*e^8*x^8 + 2688*e^9*x^
9) + 3465*d^9*ArcSin[(e*x)/d]))/(26880*e^2*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.067, size = 191, normalized size = 0.8 \begin{align*} -{\frac{e{x}^{3}}{10} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{33\,{d}^{2}x}{80\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{d}^{4}x}{160\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{11\,{d}^{6}x}{128\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{33\,{d}^{8}x}{256\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{33\,{d}^{10}}{256\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d{x}^{2}}{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{5\,{d}^{3}}{21\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/10*e*x^3*(-e^2*x^2+d^2)^(7/2)-33/80/e*d^2*x*(-e^2*x^2+d^2)^(7/2)+11/160*d^4*x*(-e^2*x^2+d^2)^(5/2)/e+11/128
*d^6*x*(-e^2*x^2+d^2)^(3/2)/e+33/256*d^8*x*(-e^2*x^2+d^2)^(1/2)/e+33/256/e*d^10/(e^2)^(1/2)*arctan((e^2)^(1/2)
*x/(-e^2*x^2+d^2)^(1/2))-1/3*d*x^2*(-e^2*x^2+d^2)^(7/2)-5/21*d^3*(-e^2*x^2+d^2)^(7/2)/e^2

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Maxima [A]  time = 1.49361, size = 247, normalized size = 1.07 \begin{align*} \frac{33 \, d^{10} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}} e} + \frac{33 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{8} x}{256 \, e} + \frac{11 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{6} x}{128 \, e} - \frac{1}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{3} + \frac{11 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{4} x}{160 \, e} - \frac{1}{3} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x^{2} - \frac{33 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x}{80 \, e} - \frac{5 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3}}{21 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

33/256*d^10*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e) + 33/256*sqrt(-e^2*x^2 + d^2)*d^8*x/e + 11/128*(-e^2*x^2
+ d^2)^(3/2)*d^6*x/e - 1/10*(-e^2*x^2 + d^2)^(7/2)*e*x^3 + 11/160*(-e^2*x^2 + d^2)^(5/2)*d^4*x/e - 1/3*(-e^2*
x^2 + d^2)^(7/2)*d*x^2 - 33/80*(-e^2*x^2 + d^2)^(7/2)*d^2*x/e - 5/21*(-e^2*x^2 + d^2)^(7/2)*d^3/e^2

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Fricas [A]  time = 1.86996, size = 360, normalized size = 1.57 \begin{align*} -\frac{6930 \, d^{10} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (2688 \, e^{9} x^{9} + 8960 \, d e^{8} x^{8} + 3024 \, d^{2} e^{7} x^{7} - 20480 \, d^{3} e^{6} x^{6} - 23352 \, d^{4} e^{5} x^{5} + 7680 \, d^{5} e^{4} x^{4} + 24570 \, d^{6} e^{3} x^{3} + 10240 \, d^{7} e^{2} x^{2} - 3465 \, d^{8} e x - 6400 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{26880 \, e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/26880*(6930*d^10*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (2688*e^9*x^9 + 8960*d*e^8*x^8 + 3024*d^2*e^7*
x^7 - 20480*d^3*e^6*x^6 - 23352*d^4*e^5*x^5 + 7680*d^5*e^4*x^4 + 24570*d^6*e^3*x^3 + 10240*d^7*e^2*x^2 - 3465*
d^8*e*x - 6400*d^9)*sqrt(-e^2*x^2 + d^2))/e^2

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Sympy [A]  time = 53.3296, size = 1561, normalized size = 6.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)**3*(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**7*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) + 3*d**6*e*Piece
wise((-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*sqrt(-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*asin(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**5*e**2*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)) - 5*d*
*4*e**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
/(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)) - 5*d**3*e**4*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**2*e**5*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)) + 3*d*e*
*6*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)) + e**7*Piecewise((-7*I*d**10*acosh(e*x/d)/(256*e**9) + 7*I*d*
*9*x/(256*e**8*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**7*x**3/(768*e**6*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**5*x**5
/(1920*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**7/(480*e**2*sqrt(-1 + e**2*x**2/d**2)) - 9*I*d*x**9/(80*sqr
t(-1 + e**2*x**2/d**2)) + I*e**2*x**11/(10*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (7*d**
10*asin(e*x/d)/(256*e**9) - 7*d**9*x/(256*e**8*sqrt(1 - e**2*x**2/d**2)) + 7*d**7*x**3/(768*e**6*sqrt(1 - e**2
*x**2/d**2)) + 7*d**5*x**5/(1920*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**7/(480*e**2*sqrt(1 - e**2*x**2/d**2)
) + 9*d*x**9/(80*sqrt(1 - e**2*x**2/d**2)) - e**2*x**11/(10*d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [A]  time = 1.14265, size = 173, normalized size = 0.75 \begin{align*} \frac{33}{256} \, d^{10} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-2\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{26880} \,{\left (6400 \, d^{9} e^{\left (-2\right )} +{\left (3465 \, d^{8} e^{\left (-1\right )} - 2 \,{\left (5120 \, d^{7} +{\left (12285 \, d^{6} e + 4 \,{\left (960 \, d^{5} e^{2} -{\left (2919 \, d^{4} e^{3} + 2 \,{\left (1280 \, d^{3} e^{4} - 7 \,{\left (27 \, d^{2} e^{5} + 8 \,{\left (3 \, x e^{7} + 10 \, d e^{6}\right )} x\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*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

33/256*d^10*arcsin(x*e/d)*e^(-2)*sgn(d) - 1/26880*(6400*d^9*e^(-2) + (3465*d^8*e^(-1) - 2*(5120*d^7 + (12285*d
^6*e + 4*(960*d^5*e^2 - (2919*d^4*e^3 + 2*(1280*d^3*e^4 - 7*(27*d^2*e^5 + 8*(3*x*e^7 + 10*d*e^6)*x)*x)*x)*x)*x
)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)