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

Optimal. Leaf size=223 \[ \frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}+\frac{19 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^3} \]

[Out]

(19*d^9*x*Sqrt[d^2 - e^2*x^2])/(256*e^2) + (19*d^7*x*(d^2 - e^2*x^2)^(3/2))/(384*e^2) + (19*d^5*x*(d^2 - e^2*x
^2)^(5/2))/(480*e^2) - (37*d^2*x^2*(d^2 - e^2*x^2)^(7/2))/(99*e) - (3*d*x^3*(d^2 - e^2*x^2)^(7/2))/10 - (e*x^4
*(d^2 - e^2*x^2)^(7/2))/11 - (d^3*(5920*d + 13167*e*x)*(d^2 - e^2*x^2)^(7/2))/(55440*e^3) + (19*d^11*ArcTan[(e
*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^3)

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Rubi [A]  time = 0.305661, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1809, 833, 780, 195, 217, 203} \[ \frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}+\frac{19 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(19*d^9*x*Sqrt[d^2 - e^2*x^2])/(256*e^2) + (19*d^7*x*(d^2 - e^2*x^2)^(3/2))/(384*e^2) + (19*d^5*x*(d^2 - e^2*x
^2)^(5/2))/(480*e^2) - (37*d^2*x^2*(d^2 - e^2*x^2)^(7/2))/(99*e) - (3*d*x^3*(d^2 - e^2*x^2)^(7/2))/10 - (e*x^4
*(d^2 - e^2*x^2)^(7/2))/11 - (d^3*(5920*d + 13167*e*x)*(d^2 - e^2*x^2)^(7/2))/(55440*e^3) + (19*d^11*ArcTan[(e
*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^3)

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 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^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{\int x^2 \left (d^2-e^2 x^2\right )^{5/2} \left (-11 d^3 e^2-37 d^2 e^3 x-33 d e^4 x^2\right ) \, dx}{11 e^2}\\ &=-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}+\frac{\int x^2 \left (209 d^3 e^4+370 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{110 e^4}\\ &=-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{\int x \left (-740 d^4 e^5-1881 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{990 e^6}\\ &=-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac{\left (19 d^5\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e^2}\\ &=\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac{\left (19 d^7\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{96 e^2}\\ &=\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac{\left (19 d^9\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{128 e^2}\\ &=\frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac{\left (19 d^{11}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{256 e^2}\\ &=\frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac{\left (19 d^{11}\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^2}\\ &=\frac{19 d^9 x \sqrt{d^2-e^2 x^2}}{256 e^2}+\frac{19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac{19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac{37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac{3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac{19 d^{11} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{256 e^3}\\ \end{align*}

Mathematica [A]  time = 0.283597, size = 178, normalized size = 0.8 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (-47360 d^8 e^2 x^2+251790 d^7 e^3 x^3+629760 d^6 e^4 x^4+201432 d^5 e^5 x^5-657920 d^4 e^6 x^6-587664 d^3 e^7 x^7+89600 d^2 e^8 x^8-65835 d^9 e x-94720 d^{10}+266112 d e^9 x^9+80640 e^{10} x^{10}\right )+65835 d^{10} \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{887040 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(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]*(-94720*d^10 - 65835*d^9*e*x - 47360*d^8*e^2*x^2 + 251790*d^7*e^
3*x^3 + 629760*d^6*e^4*x^4 + 201432*d^5*e^5*x^5 - 657920*d^4*e^6*x^6 - 587664*d^3*e^7*x^7 + 89600*d^2*e^8*x^8
+ 266112*d*e^9*x^9 + 80640*e^10*x^10) + 65835*d^10*ArcSin[(e*x)/d]))/(887040*e^3*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.079, size = 216, normalized size = 1. \begin{align*} -{\frac{e{x}^{4}}{11} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{37\,{d}^{2}{x}^{2}}{99\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{74\,{d}^{4}}{693\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,d{x}^{3}}{10} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{d}^{3}x}{80\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{19\,{d}^{5}x}{480\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{19\,{d}^{7}x}{384\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{19\,{d}^{9}x}{256\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{19\,{d}^{11}}{256\,{e}^{2}}\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^2*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/11*e*x^4*(-e^2*x^2+d^2)^(7/2)-37/99*d^2*x^2*(-e^2*x^2+d^2)^(7/2)/e-74/693/e^3*d^4*(-e^2*x^2+d^2)^(7/2)-3/10
*d*x^3*(-e^2*x^2+d^2)^(7/2)-19/80*d^3/e^2*x*(-e^2*x^2+d^2)^(7/2)+19/480*d^5*x*(-e^2*x^2+d^2)^(5/2)/e^2+19/384*
d^7*x*(-e^2*x^2+d^2)^(3/2)/e^2+19/256*d^9*x*(-e^2*x^2+d^2)^(1/2)/e^2+19/256*d^11/e^2/(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 = 1.47916, size = 281, normalized size = 1.26 \begin{align*} \frac{19 \, d^{11} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{256 \, \sqrt{e^{2}} e^{2}} + \frac{19 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{9} x}{256 \, e^{2}} - \frac{1}{11} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{4} + \frac{19 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{7} x}{384 \, e^{2}} - \frac{3}{10} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x^{3} + \frac{19 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{5} x}{480 \, e^{2}} - \frac{37 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x^{2}}{99 \, e} - \frac{19 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x}{80 \, e^{2}} - \frac{74 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{4}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

19/256*d^11*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^2) + 19/256*sqrt(-e^2*x^2 + d^2)*d^9*x/e^2 - 1/11*(-e^2*x
^2 + d^2)^(7/2)*e*x^4 + 19/384*(-e^2*x^2 + d^2)^(3/2)*d^7*x/e^2 - 3/10*(-e^2*x^2 + d^2)^(7/2)*d*x^3 + 19/480*(
-e^2*x^2 + d^2)^(5/2)*d^5*x/e^2 - 37/99*(-e^2*x^2 + d^2)^(7/2)*d^2*x^2/e - 19/80*(-e^2*x^2 + d^2)^(7/2)*d^3*x/
e^2 - 74/693*(-e^2*x^2 + d^2)^(7/2)*d^4/e^3

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Fricas [A]  time = 1.93752, size = 412, normalized size = 1.85 \begin{align*} -\frac{131670 \, d^{11} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (80640 \, e^{10} x^{10} + 266112 \, d e^{9} x^{9} + 89600 \, d^{2} e^{8} x^{8} - 587664 \, d^{3} e^{7} x^{7} - 657920 \, d^{4} e^{6} x^{6} + 201432 \, d^{5} e^{5} x^{5} + 629760 \, d^{6} e^{4} x^{4} + 251790 \, d^{7} e^{3} x^{3} - 47360 \, d^{8} e^{2} x^{2} - 65835 \, d^{9} e x - 94720 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{887040 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/887040*(131670*d^11*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (80640*e^10*x^10 + 266112*d*e^9*x^9 + 89600
*d^2*e^8*x^8 - 587664*d^3*e^7*x^7 - 657920*d^4*e^6*x^6 + 201432*d^5*e^5*x^5 + 629760*d^6*e^4*x^4 + 251790*d^7*
e^3*x^3 - 47360*d^8*e^2*x^2 - 65835*d^9*e*x - 94720*d^10)*sqrt(-e^2*x^2 + d^2))/e^3

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Sympy [C]  time = 51.3383, size = 1688, normalized size = 7.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**7*Piecewise((-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*sq
rt(-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*a
sin(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)) + 3*d**6*e*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
)) + d**5*e**2*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*sqr
t(-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**4*e**3*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)) - 5*d**3*e**4*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))
+ d**2*e**5*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)) + 3*d*e**6*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*sqrt(-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))
+ e**7*Piecewise((-128*d**10*sqrt(d**2 - e**2*x**2)/(3465*e**10) - 64*d**8*x**2*sqrt(d**2 - e**2*x**2)/(3465*e
**8) - 16*d**6*x**4*sqrt(d**2 - e**2*x**2)/(1155*e**6) - 8*d**4*x**6*sqrt(d**2 - e**2*x**2)/(693*e**4) - d**2*
x**8*sqrt(d**2 - e**2*x**2)/(99*e**2) + x**10*sqrt(d**2 - e**2*x**2)/11, Ne(e, 0)), (x**10*sqrt(d**2)/10, True
))

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Giac [A]  time = 1.13116, size = 188, normalized size = 0.84 \begin{align*} \frac{19}{256} \, d^{11} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{887040} \,{\left (94720 \, d^{10} e^{\left (-3\right )} +{\left (65835 \, d^{9} e^{\left (-2\right )} + 2 \,{\left (23680 \, d^{8} e^{\left (-1\right )} -{\left (125895 \, d^{7} + 4 \,{\left (78720 \, d^{6} e +{\left (25179 \, d^{5} e^{2} - 2 \,{\left (41120 \, d^{4} e^{3} + 7 \,{\left (5247 \, d^{3} e^{4} - 8 \,{\left (100 \, d^{2} e^{5} + 9 \,{\left (10 \, x e^{7} + 33 \, d e^{6}\right )} x\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^2*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

19/256*d^11*arcsin(x*e/d)*e^(-3)*sgn(d) - 1/887040*(94720*d^10*e^(-3) + (65835*d^9*e^(-2) + 2*(23680*d^8*e^(-1
) - (125895*d^7 + 4*(78720*d^6*e + (25179*d^5*e^2 - 2*(41120*d^4*e^3 + 7*(5247*d^3*e^4 - 8*(100*d^2*e^5 + 9*(1
0*x*e^7 + 33*d*e^6)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)