3.66 \(\int x^4 (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\)

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

[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)

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Rubi [A]  time = 0.405587, antiderivative size = 281, normalized size of antiderivative = 1., 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 = {1809, 833, 780, 195, 217, 203} \[ \frac{27 d^{11} x \sqrt{d^2-e^2 x^2}}{1024 e^4}+\frac{9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac{9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac{d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}-\frac{20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac{9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac{45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac{1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac{27 d^{13} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^5} \]

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 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^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 ) \operatorname{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*}

Mathematica [A]  time = 0.360975, size = 200, normalized size = 0.71 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (-102400 d^{10} e^2 x^2-90090 d^9 e^3 x^3-76800 d^8 e^4 x^4+952952 d^7 e^5 x^5+2498560 d^6 e^6 x^6+816816 d^5 e^7 x^7-2938880 d^4 e^8 x^8-2690688 d^3 e^9 x^9+430080 d^2 e^{10} x^{10}-135135 d^{11} e x-204800 d^{12}+1281280 d e^{11} x^{11}+394240 e^{12} x^{12}\right )+135135 d^{12} \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{5125120 e^5 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(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]*(-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^12*Arc
Sin[(e*x)/d]))/(5125120*e^5*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.112, size = 266, normalized size = 1. \begin{align*} -{\frac{e{x}^{6}}{13} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{45\,{d}^{2}{x}^{4}}{143\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{20\,{d}^{4}{x}^{2}}{143\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{40\,{d}^{6}}{1001\,{e}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{d{x}^{5}}{4} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{9\,{d}^{3}{x}^{3}}{40\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{27\,{d}^{5}x}{320\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{9\,{d}^{7}x}{640\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{9\,{d}^{9}x}{512\,{e}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{27\,{d}^{11}x}{1024\,{e}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{27\,{d}^{13}}{1024\,{e}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

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)

[Out]

-1/13*e*x^6*(-e^2*x^2+d^2)^(7/2)-45/143*d^2*x^4*(-e^2*x^2+d^2)^(7/2)/e-20/143*d^4*x^2*(-e^2*x^2+d^2)^(7/2)/e^3
-40/1001/e^5*d^6*(-e^2*x^2+d^2)^(7/2)-1/4*d*x^5*(-e^2*x^2+d^2)^(7/2)-9/40*d^3*x^3*(-e^2*x^2+d^2)^(7/2)/e^2-27/
320*d^5/e^4*x*(-e^2*x^2+d^2)^(7/2)+9/640*d^7*x*(-e^2*x^2+d^2)^(5/2)/e^4+9/512*d^9*x*(-e^2*x^2+d^2)^(3/2)/e^4+2
7/1024*d^11*x*(-e^2*x^2+d^2)^(1/2)/e^4+27/1024*d^13/e^4/(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 = 1.52526, size = 348, normalized size = 1.24 \begin{align*} -\frac{1}{13} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{6} - \frac{1}{4} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x^{5} + \frac{27 \, d^{13} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{1024 \, \sqrt{e^{2}} e^{4}} + \frac{27 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{11} x}{1024 \, e^{4}} - \frac{45 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x^{4}}{143 \, e} + \frac{9 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{9} x}{512 \, e^{4}} - \frac{9 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x^{3}}{40 \, e^{2}} + \frac{9 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{7} x}{640 \, e^{4}} - \frac{20 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{4} x^{2}}{143 \, e^{3}} - \frac{27 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{5} x}{320 \, e^{4}} - \frac{40 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{6}}{1001 \, e^{5}} \end{align*}

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]

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

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Fricas [A]  time = 1.92356, size = 487, normalized size = 1.73 \begin{align*} -\frac{270270 \, d^{13} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\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}} \end{align*}

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(-e^2*x^2 + d^2))/(e*x)) - (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)*sqrt(-e^2*x^2 + d^2))/e^5

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Sympy [C]  time = 83.5954, size = 2035, normalized size = 7.24 \begin{align*} \text{result too large to display} \end{align*}

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]

d**7*Piecewise((-I*d**6*acosh(e*x/d)/(16*e**5) + I*d**5*x/(16*e**4*sqrt(-1 + e**2*x**2/d**2)) - I*d**3*x**3/(4
8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 5*I*d*x**5/(24*sqrt(-1 + e**2*x**2/d**2)) + I*e**2*x**7/(6*d*sqrt(-1 + e**
2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (d**6*asin(e*x/d)/(16*e**5) - d**5*x/(16*e**4*sqrt(1 - e**2*x**2
/d**2)) + d**3*x**3/(48*e**2*sqrt(1 - e**2*x**2/d**2)) + 5*d*x**5/(24*sqrt(1 - e**2*x**2/d**2)) - e**2*x**7/(6
*d*sqrt(1 - e**2*x**2/d**2)), True)) + 3*d**6*e*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)) + d**5*e**2*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)) - 5*d**4*e**3*
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**3*e**4*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)) + d**2*e**
5*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)) +
3*d*e**6*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*s
qrt(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)) + e**7*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)/(300
3*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)/1
2, True))

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Giac [A]  time = 1.13118, size = 216, normalized size = 0.77 \begin{align*} \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{align*}

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)