\(\int x^5 (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\) [65]

Optimal result

Integrand size = 27, antiderivative size = 310 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {35 d^{14} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^6} \]

[Out]

35/3072*d^10*x*(-e^2*x^2+d^2)^(3/2)/e^5+7/768*d^8*x*(-e^2*x^2+d^2)^(5/2)/e^5-124/1287*d^5*x^2*(-e^2*x^2+d^2)^(
7/2)/e^4-7/48*d^4*x^3*(-e^2*x^2+d^2)^(7/2)/e^3-31/143*d^3*x^4*(-e^2*x^2+d^2)^(7/2)/e^2-7/24*d^2*x^5*(-e^2*x^2+
d^2)^(7/2)/e-3/13*d*x^6*(-e^2*x^2+d^2)^(7/2)-1/14*e*x^7*(-e^2*x^2+d^2)^(7/2)-1/1153152*d^6*(63063*e*x+31744*d)
*(-e^2*x^2+d^2)^(7/2)/e^6+35/2048*d^14*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6+35/2048*d^12*x*(-e^2*x^2+d^2)^(1/2
)/e^5

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1823, 847, 794, 201, 223, 209} \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {35 d^{14} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^6}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}+\frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2} \]

[In]

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

[Out]

(35*d^12*x*Sqrt[d^2 - e^2*x^2])/(2048*e^5) + (35*d^10*x*(d^2 - e^2*x^2)^(3/2))/(3072*e^5) + (7*d^8*x*(d^2 - e^
2*x^2)^(5/2))/(768*e^5) - (124*d^5*x^2*(d^2 - e^2*x^2)^(7/2))/(1287*e^4) - (7*d^4*x^3*(d^2 - e^2*x^2)^(7/2))/(
48*e^3) - (31*d^3*x^4*(d^2 - e^2*x^2)^(7/2))/(143*e^2) - (7*d^2*x^5*(d^2 - e^2*x^2)^(7/2))/(24*e) - (3*d*x^6*(
d^2 - e^2*x^2)^(7/2))/13 - (e*x^7*(d^2 - e^2*x^2)^(7/2))/14 - (d^6*(31744*d + 63063*e*x)*(d^2 - e^2*x^2)^(7/2)
)/(1153152*e^6) + (35*d^14*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(2048*e^6)

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}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^5 \left (d^2-e^2 x^2\right )^{5/2} \left (-14 d^3 e^2-49 d^2 e^3 x-42 d e^4 x^2\right ) \, dx}{14 e^2} \\ & = -\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^5 \left (434 d^3 e^4+637 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{182 e^4} \\ & = -\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^4 \left (-3185 d^4 e^5-5208 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{2184 e^6} \\ & = -\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^3 \left (20832 d^5 e^6+35035 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{24024 e^8} \\ & = -\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^2 \left (-105105 d^6 e^7-208320 d^5 e^8 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{240240 e^{10}} \\ & = -\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x \left (416640 d^7 e^8+945945 d^6 e^9 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{2162160 e^{12}} \\ & = -\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (7 d^8\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{128 e^5} \\ & = \frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{10}\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{768 e^5} \\ & = \frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{12}\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{1024 e^5} \\ & = \frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{14}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2048 e^5} \\ & = \frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{14}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^5} \\ & = \frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {35 d^{14} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.65 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (507904 d^{13}+315315 d^{12} e x+253952 d^{11} e^2 x^2+210210 d^{10} e^3 x^3+190464 d^9 e^4 x^4+168168 d^8 e^5 x^5-2916352 d^7 e^6 x^6-7763184 d^6 e^7 x^7-2551808 d^5 e^8 x^8+9499776 d^4 e^9 x^9+8773632 d^3 e^{10} x^{10}-1427712 d^2 e^{11} x^{11}-4257792 d e^{12} x^{12}-1317888 e^{13} x^{13}\right )+630630 d^{14} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{18450432 e^6} \]

[In]

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

[Out]

-1/18450432*(Sqrt[d^2 - e^2*x^2]*(507904*d^13 + 315315*d^12*e*x + 253952*d^11*e^2*x^2 + 210210*d^10*e^3*x^3 +
190464*d^9*e^4*x^4 + 168168*d^8*e^5*x^5 - 2916352*d^7*e^6*x^6 - 7763184*d^6*e^7*x^7 - 2551808*d^5*e^8*x^8 + 94
99776*d^4*e^9*x^9 + 8773632*d^3*e^10*x^10 - 1427712*d^2*e^11*x^11 - 4257792*d*e^12*x^12 - 1317888*e^13*x^13) +
630630*d^14*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e^6

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {\left (-1317888 e^{13} x^{13}-4257792 d \,e^{12} x^{12}-1427712 d^{2} e^{11} x^{11}+8773632 d^{3} e^{10} x^{10}+9499776 d^{4} e^{9} x^{9}-2551808 d^{5} e^{8} x^{8}-7763184 d^{6} e^{7} x^{7}-2916352 d^{7} e^{6} x^{6}+168168 d^{8} e^{5} x^{5}+190464 d^{9} e^{4} x^{4}+210210 d^{10} e^{3} x^{3}+253952 d^{11} e^{2} x^{2}+315315 d^{12} e x +507904 d^{13}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{18450432 e^{6}}+\frac {35 d^{14} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2048 e^{5} \sqrt {e^{2}}}\)	\(196\)
default	\(e^{3} \left (-\frac {x^{7} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{14 e^{2}}+\frac {d^{2} \left (-\frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12 e^{2}}+\frac {5 d^{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 )}{12 e^{2}}\right )}{2 e^{2}}\right )+d^{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 d \,e^{2} \left (-\frac {x^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{13 e^{2}}+\frac {6 d^{2} \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 )}{13 e^{2}}\right )+3 d^{2} e \left (-\frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12 e^{2}}+\frac {5 d^{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 )}{12 e^{2}}\right )\)	\(610\)

[In]

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

[Out]

-1/18450432*(-1317888*e^13*x^13-4257792*d*e^12*x^12-1427712*d^2*e^11*x^11+8773632*d^3*e^10*x^10+9499776*d^4*e^
9*x^9-2551808*d^5*e^8*x^8-7763184*d^6*e^7*x^7-2916352*d^7*e^6*x^6+168168*d^8*e^5*x^5+190464*d^9*e^4*x^4+210210
*d^10*e^3*x^3+253952*d^11*e^2*x^2+315315*d^12*e*x+507904*d^13)/e^6*(-e^2*x^2+d^2)^(1/2)+35/2048*d^14/e^5/(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 = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.63 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {630630 \, d^{14} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (1317888 \, e^{13} x^{13} + 4257792 \, d e^{12} x^{12} + 1427712 \, d^{2} e^{11} x^{11} - 8773632 \, d^{3} e^{10} x^{10} - 9499776 \, d^{4} e^{9} x^{9} + 2551808 \, d^{5} e^{8} x^{8} + 7763184 \, d^{6} e^{7} x^{7} + 2916352 \, d^{7} e^{6} x^{6} - 168168 \, d^{8} e^{5} x^{5} - 190464 \, d^{9} e^{4} x^{4} - 210210 \, d^{10} e^{3} x^{3} - 253952 \, d^{11} e^{2} x^{2} - 315315 \, d^{12} e x - 507904 \, d^{13}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{18450432 \, e^{6}} \]

[In]

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

[Out]

-1/18450432*(630630*d^14*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (1317888*e^13*x^13 + 4257792*d*e^12*x^12
+ 1427712*d^2*e^11*x^11 - 8773632*d^3*e^10*x^10 - 9499776*d^4*e^9*x^9 + 2551808*d^5*e^8*x^8 + 7763184*d^6*e^7*
x^7 + 2916352*d^7*e^6*x^6 - 168168*d^8*e^5*x^5 - 190464*d^9*e^4*x^4 - 210210*d^10*e^3*x^3 - 253952*d^11*e^2*x^
2 - 315315*d^12*e*x - 507904*d^13)*sqrt(-e^2*x^2 + d^2))/e^6

Sympy [A] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.98 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {35 d^{14} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2048 e^{5}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {248 d^{13}}{9009 e^{6}} - \frac {35 d^{12} x}{2048 e^{5}} - \frac {124 d^{11} x^{2}}{9009 e^{4}} - \frac {35 d^{10} x^{3}}{3072 e^{3}} - \frac {31 d^{9} x^{4}}{3003 e^{2}} - \frac {7 d^{8} x^{5}}{768 e} + \frac {1424 d^{7} x^{6}}{9009} + \frac {377 d^{6} e x^{7}}{896} + \frac {178 d^{5} e^{2} x^{8}}{1287} - \frac {173 d^{4} e^{3} x^{9}}{336} - \frac {68 d^{3} e^{4} x^{10}}{143} + \frac {13 d^{2} e^{5} x^{11}}{168} + \frac {3 d e^{6} x^{12}}{13} + \frac {e^{7} x^{13}}{14}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d^{3} x^{6}}{6} + \frac {3 d^{2} e x^{7}}{7} + \frac {3 d e^{2} x^{8}}{8} + \frac {e^{3} x^{9}}{9}\right ) \left (d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((35*d**14*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))/(2048*e**5) + sqrt(d**2 - e**2*x**2)*(-248*d**13/(9009*e**6) - 35*d**12*x
/(2048*e**5) - 124*d**11*x**2/(9009*e**4) - 35*d**10*x**3/(3072*e**3) - 31*d**9*x**4/(3003*e**2) - 7*d**8*x**5
/(768*e) + 1424*d**7*x**6/9009 + 377*d**6*e*x**7/896 + 178*d**5*e**2*x**8/1287 - 173*d**4*e**3*x**9/336 - 68*d
**3*e**4*x**10/143 + 13*d**2*e**5*x**11/168 + 3*d*e**6*x**12/13 + e**7*x**13/14), Ne(e**2, 0)), ((d**3*x**6/6
+ 3*d**2*e*x**7/7 + 3*d*e**2*x**8/8 + e**3*x**9/9)*(d**2)**(5/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.91 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {1}{14} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{7} - \frac {3}{13} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{6} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{5}}{24 \, e} + \frac {35 \, d^{14} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2048 \, \sqrt {e^{2}} e^{5}} + \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{12} x}{2048 \, e^{5}} - \frac {31 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{4}}{143 \, e^{2}} + \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{10} x}{3072 \, e^{5}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x^{3}}{48 \, e^{3}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{8} x}{768 \, e^{5}} - \frac {124 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} x^{2}}{1287 \, e^{4}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6} x}{128 \, e^{5}} - \frac {248 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{7}}{9009 \, e^{6}} \]

[In]

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

[Out]

-1/14*(-e^2*x^2 + d^2)^(7/2)*e*x^7 - 3/13*(-e^2*x^2 + d^2)^(7/2)*d*x^6 - 7/24*(-e^2*x^2 + d^2)^(7/2)*d^2*x^5/e
+ 35/2048*d^14*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^5) + 35/2048*sqrt(-e^2*x^2 + d^2)*d^12*x/e^5 - 31/143
*(-e^2*x^2 + d^2)^(7/2)*d^3*x^4/e^2 + 35/3072*(-e^2*x^2 + d^2)^(3/2)*d^10*x/e^5 - 7/48*(-e^2*x^2 + d^2)^(7/2)*
d^4*x^3/e^3 + 7/768*(-e^2*x^2 + d^2)^(5/2)*d^8*x/e^5 - 124/1287*(-e^2*x^2 + d^2)^(7/2)*d^5*x^2/e^4 - 7/128*(-e
^2*x^2 + d^2)^(7/2)*d^6*x/e^5 - 248/9009*(-e^2*x^2 + d^2)^(7/2)*d^7/e^6

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.61 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {35 \, d^{14} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2048 \, e^{5} {\left | e \right |}} - \frac {1}{18450432} \, {\left (\frac {507904 \, d^{13}}{e^{6}} + {\left (\frac {315315 \, d^{12}}{e^{5}} + 2 \, {\left (\frac {126976 \, d^{11}}{e^{4}} + {\left (\frac {105105 \, d^{10}}{e^{3}} + 4 \, {\left (\frac {23808 \, d^{9}}{e^{2}} + {\left (\frac {21021 \, d^{8}}{e} - 2 \, {\left (182272 \, d^{7} + {\left (485199 \, d^{6} e + 8 \, {\left (19936 \, d^{5} e^{2} - 3 \, {\left (24739 \, d^{4} e^{3} + 2 \, {\left (11424 \, d^{3} e^{4} - 11 \, {\left (169 \, d^{2} e^{5} + 12 \, {\left (13 \, e^{7} x + 42 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

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

[Out]

35/2048*d^14*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^5*abs(e)) - 1/18450432*(507904*d^13/e^6 + (315315*d^12/e^5 + 2*(12
6976*d^11/e^4 + (105105*d^10/e^3 + 4*(23808*d^9/e^2 + (21021*d^8/e - 2*(182272*d^7 + (485199*d^6*e + 8*(19936*
d^5*e^2 - 3*(24739*d^4*e^3 + 2*(11424*d^3*e^4 - 11*(169*d^2*e^5 + 12*(13*e^7*x + 42*d*e^6)*x)*x)*x)*x)*x)*x)*x
)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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