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

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

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Rubi [A]  time = 0.49, 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 = {1809, 833, 780, 195, 217, 203} \begin {gather*} \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} \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 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 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])

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 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 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 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])

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*}

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Mathematica [A]  time = 0.36, size = 212, normalized size = 0.68 \begin {gather*} \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 (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 )\right )}{18450432 e^6 \sqrt {1-\frac {e^2 x^2}{d^2}}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.88, size = 213, normalized size = 0.69 \begin {gather*} \frac {35 d^{14} \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{2048 e^7}+\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 )}{18450432 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))/(18450432*e
^6) + (35*d^14*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(2048*e^7)

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fricas [A]  time = 0.43, size = 194, normalized size = 0.63 \begin {gather*} -\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 {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(-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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giac [A]  time = 0.26, 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}\relax (d) - \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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maple [A]  time = 0.11, size = 291, normalized size = 0.94 \begin {gather*} \frac {35 d^{14} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2048 \sqrt {e^{2}}\, e^{5}}+\frac {35 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{12} x}{2048 e^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e \,x^{7}}{14}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,x^{6}}{13}+\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^{2} x^{5}}{24 e}-\frac {31 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} x^{4}}{143 e^{2}}+\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{8} x}{768 e^{5}}-\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{4} x^{3}}{48 e^{3}}-\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 {gather*}

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)/(-e^2*x^2+d^2)^(1/2)*
x)-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/e^6*d^7*(-e^2*x^2+d^2)^(7/2)

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maxima [A]  time = 1.00, size = 270, normalized size = 0.87 \begin {gather*} -\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 x}{d}\right )}{2048 \, e^{6}} + \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 {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]

-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*x/d)/e^6 + 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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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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sympy [A]  time = 101.43, size = 2273, normalized size = 7.33

result too large to display

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/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)/(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)) - 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*sq
rt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/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*sqrt(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/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*s
qrt(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)), Ab
s(e**2*x**2/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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