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

Optimal. Leaf size=252 \[ \frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac{41 d^{12} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^4} \]

[Out]

(41*d^10*x*Sqrt[d^2 - e^2*x^2])/(1024*e^3) + (41*d^8*x*(d^2 - e^2*x^2)^(3/2))/(1536*e^3) + (41*d^6*x*(d^2 - e^
2*x^2)^(5/2))/(1920*e^3) - (23*d^3*x^2*(d^2 - e^2*x^2)^(7/2))/(99*e^2) - (41*d^2*x^3*(d^2 - e^2*x^2)^(7/2))/(1
20*e) - (3*d*x^4*(d^2 - e^2*x^2)^(7/2))/11 - (e*x^5*(d^2 - e^2*x^2)^(7/2))/12 - (d^4*(14720*d + 28413*e*x)*(d^
2 - e^2*x^2)^(7/2))/(221760*e^4) + (41*d^12*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1024*e^4)

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Rubi [A]  time = 0.364037, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac{41 d^{12} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(41*d^10*x*Sqrt[d^2 - e^2*x^2])/(1024*e^3) + (41*d^8*x*(d^2 - e^2*x^2)^(3/2))/(1536*e^3) + (41*d^6*x*(d^2 - e^
2*x^2)^(5/2))/(1920*e^3) - (23*d^3*x^2*(d^2 - e^2*x^2)^(7/2))/(99*e^2) - (41*d^2*x^3*(d^2 - e^2*x^2)^(7/2))/(1
20*e) - (3*d*x^4*(d^2 - e^2*x^2)^(7/2))/11 - (e*x^5*(d^2 - e^2*x^2)^(7/2))/12 - (d^4*(14720*d + 28413*e*x)*(d^
2 - e^2*x^2)^(7/2))/(221760*e^4) + (41*d^12*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(1024*e^4)

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^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{\int x^3 \left (d^2-e^2 x^2\right )^{5/2} \left (-12 d^3 e^2-41 d^2 e^3 x-36 d e^4 x^2\right ) \, dx}{12 e^2}\\ &=-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac{\int x^3 \left (276 d^3 e^4+451 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{132 e^4}\\ &=-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{\int x^2 \left (-1353 d^4 e^5-2760 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{1320 e^6}\\ &=-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac{\int x \left (5520 d^5 e^6+12177 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{11880 e^8}\\ &=-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac{\left (41 d^6\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{320 e^3}\\ &=\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac{\left (41 d^8\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{384 e^3}\\ &=\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac{\left (41 d^{10}\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{512 e^3}\\ &=\frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac{\left (41 d^{12}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{1024 e^3}\\ &=\frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac{\left (41 d^{12}\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^3}\\ &=\frac{41 d^{10} x \sqrt{d^2-e^2 x^2}}{1024 e^3}+\frac{41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac{41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac{23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac{41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac{3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac{1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac{d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac{41 d^{12} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{1024 e^4}\\ \end{align*}

Mathematica [A]  time = 0.31831, size = 189, normalized size = 0.75 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (-117760 d^9 e^2 x^2-94710 d^8 e^3 x^3+798720 d^7 e^4 x^4+2053128 d^6 e^5 x^5+665600 d^5 e^6 x^6-2295216 d^4 e^7 x^7-2078720 d^3 e^8 x^8+325248 d^2 e^9 x^9-142065 d^{10} e x-235520 d^{11}+967680 d e^{10} x^{10}+295680 e^{11} x^{11}\right )+142065 d^{11} \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{3548160 e^4 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(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]*(-235520*d^11 - 142065*d^10*e*x - 117760*d^9*e^2*x^2 - 94710*d^8
*e^3*x^3 + 798720*d^7*e^4*x^4 + 2053128*d^6*e^5*x^5 + 665600*d^5*e^6*x^6 - 2295216*d^4*e^7*x^7 - 2078720*d^3*e
^8*x^8 + 325248*d^2*e^9*x^9 + 967680*d*e^10*x^10 + 295680*e^11*x^11) + 142065*d^11*ArcSin[(e*x)/d]))/(3548160*
e^4*Sqrt[1 - (e^2*x^2)/d^2])

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Maple [A]  time = 0.092, size = 241, normalized size = 1. \begin{align*} -{\frac{e{x}^{5}}{12} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{41\,{d}^{2}{x}^{3}}{120\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{41\,{d}^{4}x}{320\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{41\,{d}^{6}x}{1920\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{41\,{d}^{8}x}{1536\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{41\,{d}^{10}x}{1024\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{41\,{d}^{12}}{1024\,{e}^{3}}\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}^{4}}{11} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{23\,{d}^{3}{x}^{2}}{99\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{46\,{d}^{5}}{693\,{e}^{4}} \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^3*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x)

[Out]

-1/12*e*x^5*(-e^2*x^2+d^2)^(7/2)-41/120*d^2*x^3*(-e^2*x^2+d^2)^(7/2)/e-41/320/e^3*d^4*x*(-e^2*x^2+d^2)^(7/2)+4
1/1920*d^6*x*(-e^2*x^2+d^2)^(5/2)/e^3+41/1536*d^8*x*(-e^2*x^2+d^2)^(3/2)/e^3+41/1024*d^10*x*(-e^2*x^2+d^2)^(1/
2)/e^3+41/1024/e^3*d^12/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-3/11*d*x^4*(-e^2*x^2+d^2)^(7/2)
-23/99*d^3*x^2*(-e^2*x^2+d^2)^(7/2)/e^2-46/693*d^5/e^4*(-e^2*x^2+d^2)^(7/2)

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Maxima [A]  time = 1.48652, size = 315, normalized size = 1.25 \begin{align*} -\frac{1}{12} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} e x^{5} + \frac{41 \, d^{12} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{1024 \, \sqrt{e^{2}} e^{3}} + \frac{41 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{10} x}{1024 \, e^{3}} - \frac{3}{11} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d x^{4} + \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} d^{8} x}{1536 \, e^{3}} - \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{2} x^{3}}{120 \, e} + \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}} d^{6} x}{1920 \, e^{3}} - \frac{23 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{3} x^{2}}{99 \, e^{2}} - \frac{41 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{4} x}{320 \, e^{3}} - \frac{46 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}} d^{5}}{693 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(-e^2*x^2 + d^2)^(7/2)*e*x^5 + 41/1024*d^12*arcsin(e^2*x/sqrt(d^2*e^2))/(sqrt(e^2)*e^3) + 41/1024*sqrt(-
e^2*x^2 + d^2)*d^10*x/e^3 - 3/11*(-e^2*x^2 + d^2)^(7/2)*d*x^4 + 41/1536*(-e^2*x^2 + d^2)^(3/2)*d^8*x/e^3 - 41/
120*(-e^2*x^2 + d^2)^(7/2)*d^2*x^3/e + 41/1920*(-e^2*x^2 + d^2)^(5/2)*d^6*x/e^3 - 23/99*(-e^2*x^2 + d^2)^(7/2)
*d^3*x^2/e^2 - 41/320*(-e^2*x^2 + d^2)^(7/2)*d^4*x/e^3 - 46/693*(-e^2*x^2 + d^2)^(7/2)*d^5/e^4

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Fricas [A]  time = 2.02802, size = 455, normalized size = 1.81 \begin{align*} -\frac{284130 \, d^{12} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (295680 \, e^{11} x^{11} + 967680 \, d e^{10} x^{10} + 325248 \, d^{2} e^{9} x^{9} - 2078720 \, d^{3} e^{8} x^{8} - 2295216 \, d^{4} e^{7} x^{7} + 665600 \, d^{5} e^{6} x^{6} + 2053128 \, d^{6} e^{5} x^{5} + 798720 \, d^{7} e^{4} x^{4} - 94710 \, d^{8} e^{3} x^{3} - 117760 \, d^{9} e^{2} x^{2} - 142065 \, d^{10} e x - 235520 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3548160 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3548160*(284130*d^12*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (295680*e^11*x^11 + 967680*d*e^10*x^10 + 3
25248*d^2*e^9*x^9 - 2078720*d^3*e^8*x^8 - 2295216*d^4*e^7*x^7 + 665600*d^5*e^6*x^6 + 2053128*d^6*e^5*x^5 + 798
720*d^7*e^4*x^4 - 94710*d^8*e^3*x^3 - 117760*d^9*e^2*x^2 - 142065*d^10*e*x - 235520*d^11)*sqrt(-e^2*x^2 + d^2)
)/e^4

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Sympy [A]  time = 66.9829, size = 1926, normalized size = 7.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**7*Piecewise((-2*d**4*sqrt(d**2 - e**2*x**2)/(15*e**4) - d**2*x**2*sqrt(d**2 - e**2*x**2)/(15*e**2) + x**4*s
qrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) + 3*d**6*e*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/(48*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)) + d*
*5*e**2*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, Tr
ue)) - 5*d**4*e**3*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**3*e**4*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)) + d**2*e**5*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)) + 3*d*e**6*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)/(11
55*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)) + e**7*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))

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Giac [A]  time = 1.12354, size = 201, normalized size = 0.8 \begin{align*} \frac{41}{1024} \, d^{12} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-4\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{3548160} \,{\left (235520 \, d^{11} e^{\left (-4\right )} +{\left (142065 \, d^{10} e^{\left (-3\right )} + 2 \,{\left (58880 \, d^{9} e^{\left (-2\right )} +{\left (47355 \, d^{8} e^{\left (-1\right )} - 4 \,{\left (99840 \, d^{7} +{\left (256641 \, d^{6} e + 2 \,{\left (41600 \, d^{5} e^{2} - 7 \,{\left (20493 \, d^{4} e^{3} + 8 \,{\left (2320 \, d^{3} e^{4} - 3 \,{\left (121 \, d^{2} e^{5} + 10 \,{\left (11 \, x e^{7} + 36 \, d e^{6}\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^3*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

41/1024*d^12*arcsin(x*e/d)*e^(-4)*sgn(d) - 1/3548160*(235520*d^11*e^(-4) + (142065*d^10*e^(-3) + 2*(58880*d^9*
e^(-2) + (47355*d^8*e^(-1) - 4*(99840*d^7 + (256641*d^6*e + 2*(41600*d^5*e^2 - 7*(20493*d^4*e^3 + 8*(2320*d^3*
e^4 - 3*(121*d^2*e^5 + 10*(11*x*e^7 + 36*d*e^6)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-x^2*e^2 + d^2)