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

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

[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)

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Rubi [A]  time = 0.48728, antiderivative size = 310, normalized size of antiderivative = 1., 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 = {1809, 833, 780, 195, 217, 203} \[ \frac{35 d^{12} x \sqrt{d^2-e^2 x^2}}{2048 e^5}+\frac{35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac{7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac{d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}-\frac{124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac{7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac{31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac{7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac{3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac{35 d^{14} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2048 e^6} \]

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 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^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 ) \operatorname{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]  time = 0.379081, size = 212, normalized size = 0.68 \[ \frac{\sqrt{d^2-e^2 x^2} \left (315315 d^{13} \sin ^{-1}\left (\frac{e x}{d}\right )-\sqrt{1-\frac{e^2 x^2}{d^2}} \left (253952 d^{11} e^2 x^2+210210 d^{10} e^3 x^3+190464 d^9 e^4 x^4+168168 d^8 e^5 x^5-2916352 d^7 e^6 x^6-7763184 d^6 e^7 x^7-2551808 d^5 e^8 x^8+9499776 d^4 e^9 x^9+8773632 d^3 e^{10} x^{10}-1427712 d^2 e^{11} x^{11}+315315 d^{12} e x+507904 d^{13}-4257792 d e^{12} x^{12}-1317888 e^{13} x^{13}\right )\right )}{18450432 e^6 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(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]*(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 - 13178
88*e^13*x^13)) + 315315*d^13*ArcSin[(e*x)/d]))/(18450432*e^6*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.154, size = 291, normalized size = 0.9 \begin{align*} -{\frac{e{x}^{7}}{14} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{d}^{2}{x}^{5}}{24\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{d}^{4}{x}^{3}}{48\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{d}^{6}x}{128\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{7\,{d}^{8}x}{768\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{35\,{d}^{10}x}{3072\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{d}^{12}x}{2048\,{e}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{35\,{d}^{14}}{2048\,{e}^{5}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{3\,d{x}^{6}}{13} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{31\,{d}^{3}{x}^{4}}{143\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{124\,{d}^{5}{x}^{2}}{1287\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{248\,{d}^{7}}{9009\,{e}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.44528, size = 382, normalized size = 1.23 \begin{align*} -\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}{\sqrt{d^{2} 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}} \end{align*}

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]

-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/sqrt(d^2*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

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Fricas [A]  time = 1.88778, size = 529, normalized size = 1.71 \begin{align*} -\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}} \end{align*}

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(-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

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Sympy [A]  time = 139.604, size = 2280, normalized size = 7.35 \begin{align*} \text{result too large to display} \end{align*}

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]

d**7*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)
) + 3*d**6*e*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**5*e**2*Piecewise((-16*d**8*sqrt(d**2 - e**2*x**2)/(31
5*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))
- 5*d**4*e**3*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)) - 5*d**3*e**4*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*sqr
t(d**2 - e**2*x**2)/11, Ne(e, 0)), (x**10*sqrt(d**2)/10, True)) + d**2*e**5*Piecewise((-21*I*d**12*acosh(e*x/d
)/(1024*e**11) + 21*I*d**11*x/(1024*e**10*sqrt(-1 + e**2*x**2/d**2)) - 7*I*d**9*x**3/(1024*e**8*sqrt(-1 + e**2
*x**2/d**2)) - 7*I*d**7*x**5/(2560*e**6*sqrt(-1 + e**2*x**2/d**2)) - I*d**5*x**7/(640*e**4*sqrt(-1 + e**2*x**2
/d**2)) - I*d**3*x**9/(960*e**2*sqrt(-1 + e**2*x**2/d**2)) - 11*I*d*x**11/(120*sqrt(-1 + e**2*x**2/d**2)) + I*
e**2*x**13/(12*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (21*d**12*asin(e*x/d)/(1024*e**11)
- 21*d**11*x/(1024*e**10*sqrt(1 - e**2*x**2/d**2)) + 7*d**9*x**3/(1024*e**8*sqrt(1 - e**2*x**2/d**2)) + 7*d**
7*x**5/(2560*e**6*sqrt(1 - e**2*x**2/d**2)) + d**5*x**7/(640*e**4*sqrt(1 - e**2*x**2/d**2)) + d**3*x**9/(960*e
**2*sqrt(1 - e**2*x**2/d**2)) + 11*d*x**11/(120*sqrt(1 - e**2*x**2/d**2)) - e**2*x**13/(12*d*sqrt(1 - e**2*x**
2/d**2)), True)) + 3*d*e**6*Piecewise((-256*d**12*sqrt(d**2 - e**2*x**2)/(9009*e**12) - 128*d**10*x**2*sqrt(d*
*2 - e**2*x**2)/(9009*e**10) - 32*d**8*x**4*sqrt(d**2 - e**2*x**2)/(3003*e**8) - 80*d**6*x**6*sqrt(d**2 - e**2
*x**2)/(9009*e**6) - 10*d**4*x**8*sqrt(d**2 - e**2*x**2)/(1287*e**4) - d**2*x**10*sqrt(d**2 - e**2*x**2)/(143*
e**2) + x**12*sqrt(d**2 - e**2*x**2)/13, Ne(e, 0)), (x**12*sqrt(d**2)/12, True)) + e**7*Piecewise((-33*I*d**14
*acosh(e*x/d)/(2048*e**13) + 33*I*d**13*x/(2048*e**12*sqrt(-1 + e**2*x**2/d**2)) - 11*I*d**11*x**3/(2048*e**10
*sqrt(-1 + e**2*x**2/d**2)) - 11*I*d**9*x**5/(5120*e**8*sqrt(-1 + e**2*x**2/d**2)) - 11*I*d**7*x**7/(8960*e**6
*sqrt(-1 + e**2*x**2/d**2)) - 11*I*d**5*x**9/(13440*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**11/(1680*e**2*
sqrt(-1 + e**2*x**2/d**2)) - 13*I*d*x**13/(168*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**15/(14*d*sqrt(-1 + e**2*
x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (33*d**14*asin(e*x/d)/(2048*e**13) - 33*d**13*x/(2048*e**12*sqrt(1
- e**2*x**2/d**2)) + 11*d**11*x**3/(2048*e**10*sqrt(1 - e**2*x**2/d**2)) + 11*d**9*x**5/(5120*e**8*sqrt(1 - e
**2*x**2/d**2)) + 11*d**7*x**7/(8960*e**6*sqrt(1 - e**2*x**2/d**2)) + 11*d**5*x**9/(13440*e**4*sqrt(1 - e**2*x
**2/d**2)) + d**3*x**11/(1680*e**2*sqrt(1 - e**2*x**2/d**2)) + 13*d*x**13/(168*sqrt(1 - e**2*x**2/d**2)) - e**
2*x**15/(14*d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [A]  time = 1.13629, size = 230, normalized size = 0.74 \begin{align*} \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{align*}

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)