3.1.66 \(\int x^4 (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\) [66]

Optimal. Leaf size=281 \[ \frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {27 d^{13} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5} \]

[Out]

9/512*d^9*x*(-e^2*x^2+d^2)^(3/2)/e^4+9/640*d^7*x*(-e^2*x^2+d^2)^(5/2)/e^4-20/143*d^4*x^2*(-e^2*x^2+d^2)^(7/2)/
e^3-9/40*d^3*x^3*(-e^2*x^2+d^2)^(7/2)/e^2-45/143*d^2*x^4*(-e^2*x^2+d^2)^(7/2)/e-1/4*d*x^5*(-e^2*x^2+d^2)^(7/2)
-1/13*e*x^6*(-e^2*x^2+d^2)^(7/2)-1/320320*d^5*(27027*e*x+12800*d)*(-e^2*x^2+d^2)^(7/2)/e^5+27/1024*d^13*arctan
(e*x/(-e^2*x^2+d^2)^(1/2))/e^5+27/1024*d^11*x*(-e^2*x^2+d^2)^(1/2)/e^4

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Rubi [A]
time = 0.25, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 11, 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 {27 d^{13} \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}+\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(27*d^11*x*Sqrt[d^2 - e^2*x^2])/(1024*e^4) + (9*d^9*x*(d^2 - e^2*x^2)^(3/2))/(512*e^4) + (9*d^7*x*(d^2 - e^2*x
^2)^(5/2))/(640*e^4) - (20*d^4*x^2*(d^2 - e^2*x^2)^(7/2))/(143*e^3) - (9*d^3*x^3*(d^2 - e^2*x^2)^(7/2))/(40*e^
2) - (45*d^2*x^4*(d^2 - e^2*x^2)^(7/2))/(143*e) - (d*x^5*(d^2 - e^2*x^2)^(7/2))/4 - (e*x^6*(d^2 - e^2*x^2)^(7/
2))/13 - (d^5*(12800*d + 27027*e*x)*(d^2 - e^2*x^2)^(7/2))/(320320*e^5) + (27*d^13*ArcTan[(e*x)/Sqrt[d^2 - e^2
*x^2]])/(1024*e^5)

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^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^4 \left (d^2-e^2 x^2\right )^{5/2} \left (-13 d^3 e^2-45 d^2 e^3 x-39 d e^4 x^2\right ) \, dx}{13 e^2}\\ &=-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^4 \left (351 d^3 e^4+540 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{156 e^4}\\ &=-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^3 \left (-2160 d^4 e^5-3861 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{1716 e^6}\\ &=-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^2 \left (11583 d^5 e^6+21600 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{17160 e^8}\\ &=-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x \left (-43200 d^6 e^7-104247 d^5 e^8 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{154440 e^{10}}\\ &=-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^7\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{320 e^4}\\ &=\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (9 d^9\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{128 e^4}\\ &=\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{11}\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{512 e^4}\\ &=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{13}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{1024 e^4}\\ &=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{13}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4}\\ &=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {27 d^{13} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 199, normalized size = 0.71 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (-204800 d^{12}-135135 d^{11} e x-102400 d^{10} e^2 x^2-90090 d^9 e^3 x^3-76800 d^8 e^4 x^4+952952 d^7 e^5 x^5+2498560 d^6 e^6 x^6+816816 d^5 e^7 x^7-2938880 d^4 e^8 x^8-2690688 d^3 e^9 x^9+430080 d^2 e^{10} x^{10}+1281280 d e^{11} x^{11}+394240 e^{12} x^{12}\right )+135135 d^{13} \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{5125120 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(e*Sqrt[d^2 - e^2*x^2]*(-204800*d^12 - 135135*d^11*e*x - 102400*d^10*e^2*x^2 - 90090*d^9*e^3*x^3 - 76800*d^8*e
^4*x^4 + 952952*d^7*e^5*x^5 + 2498560*d^6*e^6*x^6 + 816816*d^5*e^7*x^7 - 2938880*d^4*e^8*x^8 - 2690688*d^3*e^9
*x^9 + 430080*d^2*e^10*x^10 + 1281280*d*e^11*x^11 + 394240*e^12*x^12) + 135135*d^13*Sqrt[-e^2]*Log[-(Sqrt[-e^2
]*x) + Sqrt[d^2 - e^2*x^2]])/(5125120*e^6)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(547\) vs. \(2(241)=482\).
time = 0.06, size = 548, normalized size = 1.95

method	result	size
risch	\(-\frac {\left (-394240 e^{12} x^{12}-1281280 d \,e^{11} x^{11}-430080 d^{2} e^{10} x^{10}+2690688 d^{3} e^{9} x^{9}+2938880 d^{4} e^{8} x^{8}-816816 d^{5} e^{7} x^{7}-2498560 d^{6} e^{6} x^{6}-952952 d^{7} e^{5} x^{5}+76800 d^{8} e^{4} x^{4}+90090 d^{9} e^{3} x^{3}+102400 d^{10} e^{2} x^{2}+135135 d^{11} e x +204800 d^{12}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5125120 e^{5}}+\frac {27 d^{13} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 e^{4} \sqrt {e^{2}}}\)	\(185\)
default	\(e^{3} \left (-\frac {x^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{13 e^{2}}+\frac {6 d^{2} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )}{13 e^{2}}\right )+3 e^{2} d \left (-\frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )}{12 e^{2}}\right )+3 e \,d^{2} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )+d^{3} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )\)	\(548\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

e^3*(-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^2*d*(-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*d^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)))+d^3*(-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
)))))))

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Maxima [A]
time = 0.51, size = 228, normalized size = 0.81 \begin {gather*} \frac {27}{1024} \, d^{13} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} + \frac {27}{1024} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{11} x e^{\left (-4\right )} + \frac {9}{512} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9} x e^{\left (-4\right )} + \frac {9}{640} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7} x e^{\left (-4\right )} - \frac {1}{13} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x^{6} e - \frac {45}{143} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{4} e^{\left (-1\right )} - \frac {9}{40} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{3} e^{\left (-2\right )} - \frac {20}{143} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x^{2} e^{\left (-3\right )} - \frac {27}{320} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} x e^{\left (-4\right )} - \frac {40}{1001} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6} e^{\left (-5\right )} - \frac {1}{4} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

27/1024*d^13*arcsin(x*e/d)*e^(-5) + 27/1024*sqrt(-x^2*e^2 + d^2)*d^11*x*e^(-4) + 9/512*(-x^2*e^2 + d^2)^(3/2)*
d^9*x*e^(-4) + 9/640*(-x^2*e^2 + d^2)^(5/2)*d^7*x*e^(-4) - 1/13*(-x^2*e^2 + d^2)^(7/2)*x^6*e - 45/143*(-x^2*e^
2 + d^2)^(7/2)*d^2*x^4*e^(-1) - 9/40*(-x^2*e^2 + d^2)^(7/2)*d^3*x^3*e^(-2) - 20/143*(-x^2*e^2 + d^2)^(7/2)*d^4
*x^2*e^(-3) - 27/320*(-x^2*e^2 + d^2)^(7/2)*d^5*x*e^(-4) - 40/1001*(-x^2*e^2 + d^2)^(7/2)*d^6*e^(-5) - 1/4*(-x
^2*e^2 + d^2)^(7/2)*d*x^5

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Fricas [A]
time = 2.42, size = 169, normalized size = 0.60 \begin {gather*} -\frac {1}{5125120} \, {\left (270270 \, d^{13} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (394240 \, x^{12} e^{12} + 1281280 \, d x^{11} e^{11} + 430080 \, d^{2} x^{10} e^{10} - 2690688 \, d^{3} x^{9} e^{9} - 2938880 \, d^{4} x^{8} e^{8} + 816816 \, d^{5} x^{7} e^{7} + 2498560 \, d^{6} x^{6} e^{6} + 952952 \, d^{7} x^{5} e^{5} - 76800 \, d^{8} x^{4} e^{4} - 90090 \, d^{9} x^{3} e^{3} - 102400 \, d^{10} x^{2} e^{2} - 135135 \, d^{11} x e - 204800 \, d^{12}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/5125120*(270270*d^13*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) - (394240*x^12*e^12 + 1281280*d*x^11*e^11
+ 430080*d^2*x^10*e^10 - 2690688*d^3*x^9*e^9 - 2938880*d^4*x^8*e^8 + 816816*d^5*x^7*e^7 + 2498560*d^6*x^6*e^6
+ 952952*d^7*x^5*e^5 - 76800*d^8*x^4*e^4 - 90090*d^9*x^3*e^3 - 102400*d^10*x^2*e^2 - 135135*d^11*x*e - 204800
*d^12)*sqrt(-x^2*e^2 + d^2))*e^(-5)

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

[Out]

Timed out

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Giac [A]
time = 1.20, size = 160, normalized size = 0.57 \begin {gather*} \frac {27}{1024} \, d^{13} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{5125120} \, {\left (204800 \, d^{12} e^{\left (-5\right )} + {\left (135135 \, d^{11} e^{\left (-4\right )} + 2 \, {\left (51200 \, d^{10} e^{\left (-3\right )} + {\left (45045 \, d^{9} e^{\left (-2\right )} + 4 \, {\left (9600 \, d^{8} e^{\left (-1\right )} - {\left (119119 \, d^{7} + 2 \, {\left (156160 \, d^{6} e + 7 \, {\left (7293 \, d^{5} e^{2} - 8 \, {\left (3280 \, d^{4} e^{3} + {\left (3003 \, d^{3} e^{4} - 10 \, {\left (48 \, d^{2} e^{5} + 11 \, {\left (4 \, x e^{7} + 13 \, 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 )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

27/1024*d^13*arcsin(x*e/d)*e^(-5)*sgn(d) - 1/5125120*(204800*d^12*e^(-5) + (135135*d^11*e^(-4) + 2*(51200*d^10
*e^(-3) + (45045*d^9*e^(-2) + 4*(9600*d^8*e^(-1) - (119119*d^7 + 2*(156160*d^6*e + 7*(7293*d^5*e^2 - 8*(3280*d
^4*e^3 + (3003*d^3*e^4 - 10*(48*d^2*e^5 + 11*(4*x*e^7 + 13*d*e^6)*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^4\,{\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^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^3,x)

[Out]

int(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^3, x)

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