3.1.68 \(\int x^2 (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\) [68]

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

[Out]

19/384*d^7*x*(-e^2*x^2+d^2)^(3/2)/e^2+19/480*d^5*x*(-e^2*x^2+d^2)^(5/2)/e^2-37/99*d^2*x^2*(-e^2*x^2+d^2)^(7/2)
/e-3/10*d*x^3*(-e^2*x^2+d^2)^(7/2)-1/11*e*x^4*(-e^2*x^2+d^2)^(7/2)-1/55440*d^3*(13167*e*x+5920*d)*(-e^2*x^2+d^
2)^(7/2)/e^3+19/256*d^11*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^3+19/256*d^9*x*(-e^2*x^2+d^2)^(1/2)/e^2

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 = {1823, 847, 794, 201, 223, 209} \begin {gather*} \frac {19 d^{11} \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^3}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}+\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 {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3} \end {gather*}

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 201

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

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 794

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 847

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 1823

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])

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 ) \text {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 = 0.39, size = 177, normalized size = 0.79 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (-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}\right )+65835 d^{11} \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{887040 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[d^2 - e^2*x^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 + 80
640*e^10*x^10) + 65835*d^11*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(887040*e^4)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(423\) vs. \(2(191)=382\).
time = 0.07, size = 424, normalized size = 1.90

method	result	size
risch	\(-\frac {\left (-80640 e^{10} x^{10}-266112 d \,e^{9} x^{9}-89600 d^{2} e^{8} x^{8}+587664 d^{3} e^{7} x^{7}+657920 d^{4} e^{6} x^{6}-201432 d^{5} e^{5} x^{5}-629760 d^{6} e^{4} x^{4}-251790 d^{7} e^{3} x^{3}+47360 d^{8} e^{2} x^{2}+65835 d^{9} e x +94720 d^{10}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{887040 e^{3}}+\frac {19 d^{11} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 e^{2} \sqrt {e^{2}}}\)	\(163\)
default	\(e^{3} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )+3 e^{2} d \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )+3 e \,d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+d^{3} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )\)	\(424\)

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,method=_RETURNVERBOSE)

[Out]

e^3*(-1/11*x^4*(-e^2*x^2+d^2)^(7/2)/e^2+4/11*d^2/e^2*(-1/9*x^2*(-e^2*x^2+d^2)^(7/2)/e^2-2/63*d^2/e^4*(-e^2*x^2
+d^2)^(7/2)))+3*e^2*d*(-1/10*x^3*(-e^2*x^2+d^2)^(7/2)/e^2+3/10*d^2/e^2*(-1/8*x*(-e^2*x^2+d^2)^(7/2)/e^2+1/8*d^
2/e^2*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*
d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))))+3*e*d^2*(-1/9*x^2*(-e^2*x^2+d^2)^(7/2)/e^2-2/6
3*d^2/e^4*(-e^2*x^2+d^2)^(7/2))+d^3*(-1/8*x*(-e^2*x^2+d^2)^(7/2)/e^2+1/8*d^2/e^2*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5
/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-e^2*x^2+d^2)^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*
x/(-e^2*x^2+d^2)^(1/2))))))

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 182, normalized size = 0.82 \begin {gather*} \frac {19}{256} \, d^{11} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} + \frac {19}{256} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{9} x e^{\left (-2\right )} + \frac {19}{384} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} x e^{\left (-2\right )} + \frac {19}{480} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x e^{\left (-2\right )} - \frac {1}{11} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x^{4} e - \frac {37}{99} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{2} e^{\left (-1\right )} - \frac {19}{80} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x e^{\left (-2\right )} - \frac {74}{693} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} e^{\left (-3\right )} - \frac {3}{10} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{3} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 1.40, size = 149, normalized size = 0.67 \begin {gather*} -\frac {1}{887040} \, {\left (131670 \, d^{11} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (80640 \, x^{10} e^{10} + 266112 \, d x^{9} e^{9} + 89600 \, d^{2} x^{8} e^{8} - 587664 \, d^{3} x^{7} e^{7} - 657920 \, d^{4} x^{6} e^{6} + 201432 \, d^{5} x^{5} e^{5} + 629760 \, d^{6} x^{4} e^{4} + 251790 \, d^{7} x^{3} e^{3} - 47360 \, d^{8} x^{2} e^{2} - 65835 \, d^{9} x e - 94720 \, d^{10}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-3\right )} \end {gather*}

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(-x^2*e^2 + d^2))*e^(-1)/x) - (80640*x^10*e^10 + 266112*d*x^9*e^9 + 89
600*d^2*x^8*e^8 - 587664*d^3*x^7*e^7 - 657920*d^4*x^6*e^6 + 201432*d^5*x^5*e^5 + 629760*d^6*x^4*e^4 + 251790*d
^7*x^3*e^3 - 47360*d^8*x^2*e^2 - 65835*d^9*x*e - 94720*d^10)*sqrt(-x^2*e^2 + d^2))*e^(-3)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 138.75, size = 1681, normalized size = 7.54 \begin {gather*} \text {Too large to display} \end {gather*}

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/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)) + 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*sqrt(-1
+ e**2*x**2/d**2)), Abs(e**2*x**2/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/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*Pi
ecewise((-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/d**2) > 1), (7*d**10*asi
n(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((-1
28*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))

________________________________________________________________________________________

Giac [A]
time = 1.35, size = 139, normalized size = 0.62 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________