3.1.69 \(\int x (d+e x)^3 (d^2-e^2 x^2)^{5/2} \, dx\) [69]

Optimal. Leaf size=230 \[ \frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2} \]

[Out]

11/128*d^6*x*(-e^2*x^2+d^2)^(3/2)/e+11/160*d^4*x*(-e^2*x^2+d^2)^(5/2)/e-33/560*d^3*(-e^2*x^2+d^2)^(7/2)/e^2-11
/240*d^2*(e*x+d)*(-e^2*x^2+d^2)^(7/2)/e^2-1/30*d*(e*x+d)^2*(-e^2*x^2+d^2)^(7/2)/e^2-1/10*(e*x+d)^3*(-e^2*x^2+d
^2)^(7/2)/e^2+33/256*d^10*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^2+33/256*d^8*x*(-e^2*x^2+d^2)^(1/2)/e

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Rubi [A]
time = 0.07, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {809, 685, 655, 201, 223, 209} \begin {gather*} \frac {33 d^{10} \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(33*d^8*x*Sqrt[d^2 - e^2*x^2])/(256*e) + (11*d^6*x*(d^2 - e^2*x^2)^(3/2))/(128*e) + (11*d^4*x*(d^2 - e^2*x^2)^
(5/2))/(160*e) - (33*d^3*(d^2 - e^2*x^2)^(7/2))/(560*e^2) - (11*d^2*(d + e*x)*(d^2 - e^2*x^2)^(7/2))/(240*e^2)
- (d*(d + e*x)^2*(d^2 - e^2*x^2)^(7/2))/(30*e^2) - ((d + e*x)^3*(d^2 - e^2*x^2)^(7/2))/(10*e^2) + (33*d^10*Ar
cTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(256*e^2)

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[2*c*d*((m + p)/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 809

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[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rubi steps

\begin {align*} \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {(3 d) \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx}{10 e}\\ &=-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (11 d^2\right ) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx}{30 e}\\ &=-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^3\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e}\\ &=-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e}\\ &=\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (11 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{32 e}\\ &=\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^8\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{128 e}\\ &=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^{10}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{256 e}\\ &=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}\\ &=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 166, normalized size = 0.72 \begin {gather*} \frac {e \sqrt {d^2-e^2 x^2} \left (-6400 d^9-3465 d^8 e x+10240 d^7 e^2 x^2+24570 d^6 e^3 x^3+7680 d^5 e^4 x^4-23352 d^4 e^5 x^5-20480 d^3 e^6 x^6+3024 d^2 e^7 x^7+8960 d e^8 x^8+2688 e^9 x^9\right )+3465 d^{10} \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{26880 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[d^2 - e^2*x^2]*(-6400*d^9 - 3465*d^8*e*x + 10240*d^7*e^2*x^2 + 24570*d^6*e^3*x^3 + 7680*d^5*e^4*x^4 -
23352*d^4*e^5*x^5 - 20480*d^3*e^6*x^6 + 3024*d^2*e^7*x^7 + 8960*d*e^8*x^8 + 2688*e^9*x^9) + 3465*d^10*Sqrt[-e^
2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(26880*e^3)

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Maple [A]
time = 0.06, size = 366, normalized size = 1.59

method	result	size
risch	\(-\frac {\left (-2688 e^{9} x^{9}-8960 d \,e^{8} x^{8}-3024 d^{2} e^{7} x^{7}+20480 d^{3} e^{6} x^{6}+23352 d^{4} e^{5} x^{5}-7680 d^{5} e^{4} x^{4}-24570 d^{6} e^{3} x^{3}-10240 x^{2} d^{7} e^{2}+3465 d^{8} x e +6400 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{26880 e^{2}}+\frac {33 d^{10} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 e \sqrt {e^{2}}}\)	\(152\)
default	\(e^{3} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )+3 e^{2} d \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+3 e \,d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )-\frac {d^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{2}}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^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)*ar
ctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))))+3*e^2*d*(-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*(-1/8*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))))))-1/7*d^3*(-e^2*x^2+d^2)^(7/2)/e^2

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Maxima [A]
time = 0.49, size = 159, normalized size = 0.69 \begin {gather*} \frac {33}{256} \, d^{10} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} + \frac {33}{256} \, \sqrt {-x^{2} e^{2} + d^{2}} d^{8} x e^{\left (-1\right )} + \frac {11}{128} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x e^{\left (-1\right )} + \frac {11}{160} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x e^{\left (-1\right )} - \frac {1}{10} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} x^{3} e - \frac {33}{80} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x e^{\left (-1\right )} - \frac {5}{21} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} e^{\left (-2\right )} - \frac {1}{3} \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

33/256*d^10*arcsin(x*e/d)*e^(-2) + 33/256*sqrt(-x^2*e^2 + d^2)*d^8*x*e^(-1) + 11/128*(-x^2*e^2 + d^2)^(3/2)*d^
6*x*e^(-1) + 11/160*(-x^2*e^2 + d^2)^(5/2)*d^4*x*e^(-1) - 1/10*(-x^2*e^2 + d^2)^(7/2)*x^3*e - 33/80*(-x^2*e^2
+ d^2)^(7/2)*d^2*x*e^(-1) - 5/21*(-x^2*e^2 + d^2)^(7/2)*d^3*e^(-2) - 1/3*(-x^2*e^2 + d^2)^(7/2)*d*x^2

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Fricas [A]
time = 1.86, size = 139, normalized size = 0.60 \begin {gather*} -\frac {1}{26880} \, {\left (6930 \, d^{10} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) - {\left (2688 \, x^{9} e^{9} + 8960 \, d x^{8} e^{8} + 3024 \, d^{2} x^{7} e^{7} - 20480 \, d^{3} x^{6} e^{6} - 23352 \, d^{4} x^{5} e^{5} + 7680 \, d^{5} x^{4} e^{4} + 24570 \, d^{6} x^{3} e^{3} + 10240 \, d^{7} x^{2} e^{2} - 3465 \, d^{8} x e - 6400 \, d^{9}\right )} \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/26880*(6930*d^10*arctan(-(d - sqrt(-x^2*e^2 + d^2))*e^(-1)/x) - (2688*x^9*e^9 + 8960*d*x^8*e^8 + 3024*d^2*x
^7*e^7 - 20480*d^3*x^6*e^6 - 23352*d^4*x^5*e^5 + 7680*d^5*x^4*e^4 + 24570*d^6*x^3*e^3 + 10240*d^7*x^2*e^2 - 34
65*d^8*x*e - 6400*d^9)*sqrt(-x^2*e^2 + d^2))*e^(-2)

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Sympy [A]
time = 137.53, size = 1554, normalized size = 6.76 \begin {gather*} d^{7} \left (\begin {cases} \frac {x^{2} \sqrt {d^{2}}}{2} & \text {for}\: e^{2} = 0 \\- \frac {\left (d^{2} - e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e^{2}} & \text {otherwise} \end {cases}\right ) + 3 d^{6} e \left (\begin {cases} - \frac {i d^{4} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{8 e^{3}} + \frac {i d^{3} x}{8 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {3 i d x^{3}}{8 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{5}}{4 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{4} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{8 e^{3}} - \frac {d^{3} x}{8 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {3 d x^{3}}{8 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{5}}{4 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + d^{5} e^{2} \left (\begin {cases} - \frac {2 d^{4} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{4}} - \frac {d^{2} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{15 e^{2}} + \frac {x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - 5 d^{4} e^{3} \left (\begin {cases} - \frac {i d^{6} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{16 e^{5}} + \frac {i d^{5} x}{16 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{3}}{48 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d x^{5}}{24 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{7}}{6 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {d^{6} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{16 e^{5}} - \frac {d^{5} x}{16 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{3}}{48 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d x^{5}}{24 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{7}}{6 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - 5 d^{3} e^{4} \left (\begin {cases} - \frac {8 d^{6} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{6}} - \frac {4 d^{4} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{35 e^{2}} + \frac {x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) + d^{2} e^{5} \left (\begin {cases} - \frac {5 i d^{8} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{128 e^{7}} + \frac {5 i d^{7} x}{128 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {5 i d^{5} x^{3}}{384 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{5}}{192 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d x^{7}}{48 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{9}}{8 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {5 d^{8} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{128 e^{7}} - \frac {5 d^{7} x}{128 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {5 d^{5} x^{3}}{384 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{5}}{192 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d x^{7}}{48 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{9}}{8 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) + 3 d e^{6} \left (\begin {cases} - \frac {16 d^{8} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{8}} - \frac {8 d^{6} x^{2} \sqrt {d^{2} - e^{2} x^{2}}}{315 e^{6}} - \frac {2 d^{4} x^{4} \sqrt {d^{2} - e^{2} x^{2}}}{105 e^{4}} - \frac {d^{2} x^{6} \sqrt {d^{2} - e^{2} x^{2}}}{63 e^{2}} + \frac {x^{8} \sqrt {d^{2} - e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) + e^{7} \left (\begin {cases} - \frac {7 i d^{10} \operatorname {acosh}{\left (\frac {e x}{d} \right )}}{256 e^{9}} + \frac {7 i d^{9} x}{256 e^{8} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d^{7} x^{3}}{768 e^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {7 i d^{5} x^{5}}{1920 e^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {i d^{3} x^{7}}{480 e^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} - \frac {9 i d x^{9}}{80 \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + \frac {i e^{2} x^{11}}{10 d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {7 d^{10} \operatorname {asin}{\left (\frac {e x}{d} \right )}}{256 e^{9}} - \frac {7 d^{9} x}{256 e^{8} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d^{7} x^{3}}{768 e^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {7 d^{5} x^{5}}{1920 e^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {d^{3} x^{7}}{480 e^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} + \frac {9 d x^{9}}{80 \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - \frac {e^{2} x^{11}}{10 d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**7*Piecewise((x**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True)) + 3*d**6*e*Piece
wise((-I*d**4*acosh(e*x/d)/(8*e**3) + I*d**3*x/(8*e**2*sqrt(-1 + e**2*x**2/d**2)) - 3*I*d*x**3/(8*sqrt(-1 + e*
*2*x**2/d**2)) + I*e**2*x**5/(4*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**4*asin(e*x/d)/(8*e
**3) - d**3*x/(8*e**2*sqrt(1 - e**2*x**2/d**2)) + 3*d*x**3/(8*sqrt(1 - e**2*x**2/d**2)) - e**2*x**5/(4*d*sqrt(
1 - e**2*x**2/d**2)), True)) + d**5*e**2*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*sqrt(d**2 - e**2*x**2)/5, Ne(e, 0)), (x**4*sqrt(d**2)/4, True)) - 5*d**4*e*
*3*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/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)) - 5*d**3*e**4*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**2*e**5*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/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)) + 3*d*e**6*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)) + e**7*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*sqr
t(-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/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*sqr
t(1 - e**2*x**2/d**2)) - e**2*x**11/(10*d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [A]
time = 1.06, size = 128, normalized size = 0.56 \begin {gather*} \frac {33}{256} \, d^{10} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\left (d\right ) - \frac {1}{26880} \, {\left (6400 \, d^{9} e^{\left (-2\right )} + {\left (3465 \, d^{8} e^{\left (-1\right )} - 2 \, {\left (5120 \, d^{7} + {\left (12285 \, d^{6} e + 4 \, {\left (960 \, d^{5} e^{2} - {\left (2919 \, d^{4} e^{3} + 2 \, {\left (1280 \, d^{3} e^{4} - 7 \, {\left (27 \, d^{2} e^{5} + 8 \, {\left (3 \, x e^{7} + 10 \, d e^{6}\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*(e*x+d)^3*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

33/256*d^10*arcsin(x*e/d)*e^(-2)*sgn(d) - 1/26880*(6400*d^9*e^(-2) + (3465*d^8*e^(-1) - 2*(5120*d^7 + (12285*d
^6*e + 4*(960*d^5*e^2 - (2919*d^4*e^3 + 2*(1280*d^3*e^4 - 7*(27*d^2*e^5 + 8*(3*x*e^7 + 10*d*e^6)*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\,{\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*(d^2 - e^2*x^2)^(5/2)*(d + e*x)^3,x)

[Out]

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

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