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

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

[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

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Rubi [A]
time = 0.30, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 12, 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 {35 d^{14} \text {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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\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*}

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Mathematica [A]
time = 0.54, size = 210, normalized size = 0.68 \begin {gather*} \frac {e \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 )+315315 d^{14} \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{18450432 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*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 - 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) + 315315*d
^14*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(18450432*e^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(266)=532\).
time = 0.09, size = 610, normalized size = 1.97

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 )+3 e^{2} d \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 e \,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 )+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 )\)	\(610\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^3*(-1/14*x^7*(-e^2*x^2+d^2)^(7/2)/e^2+1/2*d^2/e^2*(-1/12*x^5*(-e^2*x^2+d^2)^(7/2)/e^2+5/12*d^2/e^2*(-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^2*d*(-1/13*x^6*(-e^2*x^2+d^2)^(7/2)/e^2+6/13*d^2/e^2*(-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*d^2
*(-1/12*x^5*(-e^2*x^2+d^2)^(7/2)/e^2+5/12*d^2/e^2*(-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))))))))+d^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)))

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Maxima [A]
time = 0.50, size = 251, normalized size = 0.81 \begin {gather*} \frac {35}{2048} \, d^{14} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} + \frac {35}{2048} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{12} x e^{\left (-5\right )} + \frac {35}{3072} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{10} x e^{\left (-5\right )} + \frac {7}{768} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{8} x e^{\left (-5\right )} - \frac {1}{14} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x^{7} e - \frac {7}{24} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{5} e^{\left (-1\right )} - \frac {31}{143} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{4} e^{\left (-2\right )} - \frac {7}{48} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x^{3} e^{\left (-3\right )} - \frac {124}{1287} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} x^{2} e^{\left (-4\right )} - \frac {7}{128} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6} x e^{\left (-5\right )} - \frac {248}{9009} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{7} e^{\left (-6\right )} - \frac {3}{13} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.00, size = 179, normalized size = 0.58 \begin {gather*} -\frac {1}{18450432} \, {\left (630630 \, d^{14} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (1317888 \, x^{13} e^{13} + 4257792 \, d x^{12} e^{12} + 1427712 \, d^{2} x^{11} e^{11} - 8773632 \, d^{3} x^{10} e^{10} - 9499776 \, d^{4} x^{9} e^{9} + 2551808 \, d^{5} x^{8} e^{8} + 7763184 \, d^{6} x^{7} e^{7} + 2916352 \, d^{7} x^{6} e^{6} - 168168 \, d^{8} x^{5} e^{5} - 190464 \, d^{9} x^{4} e^{4} - 210210 \, d^{10} x^{3} e^{3} - 253952 \, d^{11} x^{2} e^{2} - 315315 \, d^{12} x e - 507904 \, d^{13}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.22, size = 170, normalized size = 0.55 \begin {gather*} \frac {35}{2048} \, d^{14} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{18450432} \, {\left (507904 \, d^{13} e^{\left (-6\right )} + {\left (315315 \, d^{12} e^{\left (-5\right )} + 2 \, {\left (126976 \, d^{11} e^{\left (-4\right )} + {\left (105105 \, d^{10} e^{\left (-3\right )} + 4 \, {\left (23808 \, d^{9} e^{\left (-2\right )} + {\left (21021 \, d^{8} e^{\left (-1\right )} - 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 \, x e^{7} + 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 {-x^{2} e^{2} + d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(x*e/d)*e^(-6)*sgn(d) - 1/18450432*(507904*d^13*e^(-6) + (315315*d^12*e^(-5) + 2*(126976*d^
11*e^(-4) + (105105*d^10*e^(-3) + 4*(23808*d^9*e^(-2) + (21021*d^8*e^(-1) - 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*x*e^7 + 42*d*e^6)*x)*x)*x)*x)*
x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^5\,{\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^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)

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