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

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

[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

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Rubi [A]
time = 0.22, antiderivative size = 252, normalized size of antiderivative = 1.00, 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 = {1823, 847, 794, 201, 223, 209} \begin {gather*} \frac {41 d^{12} \text {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} \end {gather*}

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 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^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 ) \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*}

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Mathematica [A]
time = 0.44, size = 188, normalized size = 0.75 \begin {gather*} \frac {e \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 )+142065 d^{12} \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{3548160 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*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) + 142065*d^12*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^
2]])/(3548160*e^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(485\) vs. \(2(216)=432\).
time = 0.07, size = 486, normalized size = 1.93

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	\(e^{3} \left (-\frac {x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )}{12 e^{2}}\right )+3 e^{2} d \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )}{11 e^{2}}\right )+3 e \,d^{2} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )+d^{3} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )\)	\(486\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 205, normalized size = 0.81 \begin {gather*} \frac {41}{1024} \, d^{12} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} + \frac {41}{1024} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{10} x e^{\left (-3\right )} + \frac {41}{1536} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8} x e^{\left (-3\right )} + \frac {41}{1920} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} x e^{\left (-3\right )} - \frac {1}{12} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x^{5} e - \frac {41}{120} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{3} e^{\left (-1\right )} - \frac {23}{99} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{2} e^{\left (-2\right )} - \frac {41}{320} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x e^{\left (-3\right )} - \frac {46}{693} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} e^{\left (-4\right )} - \frac {3}{11} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{4} \end {gather*}

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]

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

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Fricas [A]
time = 3.70, size = 159, normalized size = 0.63 \begin {gather*} -\frac {1}{3548160} \, {\left (284130 \, d^{12} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (295680 \, x^{11} e^{11} + 967680 \, d x^{10} e^{10} + 325248 \, d^{2} x^{9} e^{9} - 2078720 \, d^{3} x^{8} e^{8} - 2295216 \, d^{4} x^{7} e^{7} + 665600 \, d^{5} x^{6} e^{6} + 2053128 \, d^{6} x^{5} e^{5} + 798720 \, d^{7} x^{4} e^{4} - 94710 \, d^{8} x^{3} e^{3} - 117760 \, d^{9} x^{2} e^{2} - 142065 \, d^{10} x e - 235520 \, d^{11}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-4\right )} \end {gather*}

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

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

[Out]

Timed out

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Giac [A]
time = 1.25, size = 149, normalized size = 0.59 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\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^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)

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