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

Optimal result

Integrand size = 27, antiderivative size = 223 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3}+\frac {19 d^{11} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^3} \]

[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] (verified)

Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1823, 847, 794, 201, 223, 209} \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {19 d^{11} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^3}-\frac {37 d^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e}-\frac {1}{11} e x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{10} d x^3 \left (d^2-e^2 x^2\right )^{7/2}+\frac {19 d^9 x \sqrt {d^2-e^2 x^2}}{256 e^2}+\frac {19 d^7 x \left (d^2-e^2 x^2\right )^{3/2}}{384 e^2}+\frac {19 d^5 x \left (d^2-e^2 x^2\right )^{5/2}}{480 e^2}-\frac {d^3 (5920 d+13167 e x) \left (d^2-e^2 x^2\right )^{7/2}}{55440 e^3} \]

[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*} \text {integral}& = -\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] (verified)

Time = 0.62 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.76 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-94720 d^{10}-65835 d^9 e x-47360 d^8 e^2 x^2+251790 d^7 e^3 x^3+629760 d^6 e^4 x^4+201432 d^5 e^5 x^5-657920 d^4 e^6 x^6-587664 d^3 e^7 x^7+89600 d^2 e^8 x^8+266112 d e^9 x^9+80640 e^{10} x^{10}\right )-131670 d^{11} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{887040 e^3} \]

[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]*(-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 + 8064
0*e^10*x^10) - 131670*d^11*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(887040*e^3)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.73

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	\(e^{3} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )+d^{3} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )+3 d \,e^{2} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )+3 d^{2} e \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )\)	\(424\)

[In]

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

[Out]

-1/887040*(-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)/e^3*(-e^2*x^2+d^2)
^(1/2)+19/256*d^11/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.72 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {131670 \, d^{11} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (80640 \, e^{10} x^{10} + 266112 \, d e^{9} x^{9} + 89600 \, d^{2} e^{8} x^{8} - 587664 \, d^{3} e^{7} x^{7} - 657920 \, d^{4} e^{6} x^{6} + 201432 \, d^{5} e^{5} x^{5} + 629760 \, d^{6} e^{4} x^{4} + 251790 \, d^{7} e^{3} x^{3} - 47360 \, d^{8} e^{2} x^{2} - 65835 \, d^{9} e x - 94720 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{887040 \, e^{3}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.17 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {19 d^{11} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{256 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {74 d^{10}}{693 e^{3}} - \frac {19 d^{9} x}{256 e^{2}} - \frac {37 d^{8} x^{2}}{693 e} + \frac {109 d^{7} x^{3}}{384} + \frac {164 d^{6} e x^{4}}{231} + \frac {109 d^{5} e^{2} x^{5}}{480} - \frac {514 d^{4} e^{3} x^{6}}{693} - \frac {53 d^{3} e^{4} x^{7}}{80} + \frac {10 d^{2} e^{5} x^{8}}{99} + \frac {3 d e^{6} x^{9}}{10} + \frac {e^{7} x^{10}}{11}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d^{3} x^{3}}{3} + \frac {3 d^{2} e x^{4}}{4} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{6}}{6}\right ) \left (d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((19*d**11*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0))
, (x*log(x)/sqrt(-e**2*x**2), True))/(256*e**2) + sqrt(d**2 - e**2*x**2)*(-74*d**10/(693*e**3) - 19*d**9*x/(25
6*e**2) - 37*d**8*x**2/(693*e) + 109*d**7*x**3/384 + 164*d**6*e*x**4/231 + 109*d**5*e**2*x**5/480 - 514*d**4*e
**3*x**6/693 - 53*d**3*e**4*x**7/80 + 10*d**2*e**5*x**8/99 + 3*d*e**6*x**9/10 + e**7*x**10/11), Ne(e**2, 0)),
((d**3*x**3/3 + 3*d**2*e*x**4/4 + 3*d*e**2*x**5/5 + e**3*x**6/6)*(d**2)**(5/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.93 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {19 \, d^{11} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{256 \, \sqrt {e^{2}} e^{2}} + \frac {19 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9} x}{256 \, e^{2}} - \frac {1}{11} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{4} + \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} x}{384 \, e^{2}} - \frac {3}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{3} + \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x}{480 \, e^{2}} - \frac {37 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{2}}{99 \, e} - \frac {19 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x}{80 \, e^{2}} - \frac {74 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4}}{693 \, e^{3}} \]

[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/(d*sqrt(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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.69 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {19 \, d^{11} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{256 \, e^{2} {\left | e \right |}} - \frac {1}{887040} \, {\left (\frac {94720 \, d^{10}}{e^{3}} + {\left (\frac {65835 \, d^{9}}{e^{2}} + 2 \, {\left (\frac {23680 \, d^{8}}{e} - {\left (125895 \, d^{7} + 4 \, {\left (78720 \, d^{6} e + {\left (25179 \, d^{5} e^{2} - 2 \, {\left (41120 \, d^{4} e^{3} + 7 \, {\left (5247 \, d^{3} e^{4} - 8 \, {\left (100 \, d^{2} e^{5} + 9 \, {\left (10 \, e^{7} x + 33 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[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(e*x/d)*sgn(d)*sgn(e)/(e^2*abs(e)) - 1/887040*(94720*d^10/e^3 + (65835*d^9/e^2 + 2*(23680*d^
8/e - (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*
(10*e^7*x + 33*d*e^6)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

Timed out. \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]

[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)