\(\int x^3 (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\) [67]

Optimal result

Integrand size = 27, antiderivative size = 252 \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {41 d^{12} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4} \]

[Out]

41/1536*d^8*x*(-e^2*x^2+d^2)^(3/2)/e^3+41/1920*d^6*x*(-e^2*x^2+d^2)^(5/2)/e^3-23/99*d^3*x^2*(-e^2*x^2+d^2)^(7/
2)/e^2-41/120*d^2*x^3*(-e^2*x^2+d^2)^(7/2)/e-3/11*d*x^4*(-e^2*x^2+d^2)^(7/2)-1/12*e*x^5*(-e^2*x^2+d^2)^(7/2)-1
/221760*d^4*(28413*e*x+14720*d)*(-e^2*x^2+d^2)^(7/2)/e^4+41/1024*d^12*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4+41/
1024*d^10*x*(-e^2*x^2+d^2)^(1/2)/e^3

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1823, 847, 794, 201, 223, 209} \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {41 d^{12} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}+\frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2} \]

[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 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*} \text {integral}& = -\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 ) \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^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] (verified)

Time = 0.66 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.71 \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-235520 d^{11}-142065 d^{10} e x-117760 d^9 e^2 x^2-94710 d^8 e^3 x^3+798720 d^7 e^4 x^4+2053128 d^6 e^5 x^5+665600 d^5 e^6 x^6-2295216 d^4 e^7 x^7-2078720 d^3 e^8 x^8+325248 d^2 e^9 x^9+967680 d e^{10} x^{10}+295680 e^{11} x^{11}\right )-284130 d^{12} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{3548160 e^4} \]

[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]*(-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) - 284130*d^12*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(35
48160*e^4)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {\left (-295680 e^{11} x^{11}-967680 d \,e^{10} x^{10}-325248 d^{2} e^{9} x^{9}+2078720 d^{3} e^{8} x^{8}+2295216 d^{4} e^{7} x^{7}-665600 d^{5} e^{6} x^{6}-2053128 d^{6} e^{5} x^{5}-798720 d^{7} e^{4} x^{4}+94710 d^{8} e^{3} x^{3}+117760 d^{9} e^{2} x^{2}+142065 d^{10} e x +235520 d^{11}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3548160 e^{4}}+\frac {41 d^{12} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 e^{3} \sqrt {e^{2}}}\)	\(174\)
default	\(e^{3} \left (-\frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )}{12 e^{2}}\right )+d^{3} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+3 d \,e^{2} \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )+3 d^{2} e \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )\)	\(486\)

[In]

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

[Out]

-1/3548160*(-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^7-66
5600*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+2
35520*d^11)/e^4*(-e^2*x^2+d^2)^(1/2)+41/1024*d^12/e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.68 \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {284130 \, d^{12} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (295680 \, e^{11} x^{11} + 967680 \, d e^{10} x^{10} + 325248 \, d^{2} e^{9} x^{9} - 2078720 \, d^{3} e^{8} x^{8} - 2295216 \, d^{4} e^{7} x^{7} + 665600 \, d^{5} e^{6} x^{6} + 2053128 \, d^{6} e^{5} x^{5} + 798720 \, d^{7} e^{4} x^{4} - 94710 \, d^{8} e^{3} x^{3} - 117760 \, d^{9} e^{2} x^{2} - 142065 \, d^{10} e x - 235520 \, d^{11}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3548160 \, e^{4}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.09 \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {41 d^{12} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{1024 e^{3}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {46 d^{11}}{693 e^{4}} - \frac {41 d^{10} x}{1024 e^{3}} - \frac {23 d^{9} x^{2}}{693 e^{2}} - \frac {41 d^{8} x^{3}}{1536 e} + \frac {52 d^{7} x^{4}}{231} + \frac {1111 d^{6} e x^{5}}{1920} + \frac {130 d^{5} e^{2} x^{6}}{693} - \frac {207 d^{4} e^{3} x^{7}}{320} - \frac {58 d^{3} e^{4} x^{8}}{99} + \frac {11 d^{2} e^{5} x^{9}}{120} + \frac {3 d e^{6} x^{10}}{11} + \frac {e^{7} x^{11}}{12}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d^{3} x^{4}}{4} + \frac {3 d^{2} e x^{5}}{5} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{7}}{7}\right ) \left (d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((41*d**12*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0))
, (x*log(x)/sqrt(-e**2*x**2), True))/(1024*e**3) + sqrt(d**2 - e**2*x**2)*(-46*d**11/(693*e**4) - 41*d**10*x/(
1024*e**3) - 23*d**9*x**2/(693*e**2) - 41*d**8*x**3/(1536*e) + 52*d**7*x**4/231 + 1111*d**6*e*x**5/1920 + 130*
d**5*e**2*x**6/693 - 207*d**4*e**3*x**7/320 - 58*d**3*e**4*x**8/99 + 11*d**2*e**5*x**9/120 + 3*d*e**6*x**10/11
+ e**7*x**11/12), Ne(e**2, 0)), ((d**3*x**4/4 + 3*d**2*e*x**5/5 + d*e**2*x**6/2 + e**3*x**7/7)*(d**2)**(5/2),
True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.92 \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{5} + \frac {41 \, d^{12} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{1024 \, \sqrt {e^{2}} e^{3}} + \frac {41 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{10} x}{1024 \, e^{3}} - \frac {3}{11} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{4} + \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8} x}{1536 \, e^{3}} - \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{3}}{120 \, e} + \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} x}{1920 \, e^{3}} - \frac {23 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{2}}{99 \, e^{2}} - \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x}{320 \, e^{3}} - \frac {46 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5}}{693 \, e^{4}} \]

[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/(d*sqrt(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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.65 \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {41 \, d^{12} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{1024 \, e^{3} {\left | e \right |}} - \frac {1}{3548160} \, {\left (\frac {235520 \, d^{11}}{e^{4}} + {\left (\frac {142065 \, d^{10}}{e^{3}} + 2 \, {\left (\frac {58880 \, d^{9}}{e^{2}} + {\left (\frac {47355 \, d^{8}}{e} - 4 \, {\left (99840 \, d^{7} + {\left (256641 \, d^{6} e + 2 \, {\left (41600 \, d^{5} e^{2} - 7 \, {\left (20493 \, d^{4} e^{3} + 8 \, {\left (2320 \, d^{3} e^{4} - 3 \, {\left (121 \, d^{2} e^{5} + 10 \, {\left (11 \, e^{7} x + 36 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[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(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) - 1/3548160*(235520*d^11/e^4 + (142065*d^10/e^3 + 2*(588
80*d^9/e^2 + (47355*d^8/e - 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*e^7*x + 36*d*e^6)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

Timed out. \[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int x^3\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

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

[Out]

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