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

Optimal result

Integrand size = 25, antiderivative size = 230 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2} \]

[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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {809, 685, 655, 201, 223, 209} \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.69 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6400 d^9-3465 d^8 e x+10240 d^7 e^2 x^2+24570 d^6 e^3 x^3+7680 d^5 e^4 x^4-23352 d^4 e^5 x^5-20480 d^3 e^6 x^6+3024 d^2 e^7 x^7+8960 d e^8 x^8+2688 e^9 x^9\right )-6930 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{26880 e^2} \]

[In]

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

[Out]

(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 - 23
352*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) - 6930*d^10*ArcTan[(e*
x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(26880*e^2)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.66

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

[In]

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

[Out]

-1/26880*(-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^4-2
4570*d^6*e^3*x^3-10240*d^7*e^2*x^2+3465*d^8*e*x+6400*d^9)/e^2*(-e^2*x^2+d^2)^(1/2)+33/256*d^10/e/(e^2)^(1/2)*a
rctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.65 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {6930 \, d^{10} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (2688 \, e^{9} x^{9} + 8960 \, d e^{8} x^{8} + 3024 \, d^{2} e^{7} x^{7} - 20480 \, d^{3} e^{6} x^{6} - 23352 \, d^{4} e^{5} x^{5} + 7680 \, d^{5} e^{4} x^{4} + 24570 \, d^{6} e^{3} x^{3} + 10240 \, d^{7} e^{2} x^{2} - 3465 \, d^{8} e x - 6400 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{26880 \, e^{2}} \]

[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(-e^2*x^2 + d^2))/(e*x)) - (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*d^7*e^2*x^2 - 3465*
d^8*e*x - 6400*d^9)*sqrt(-e^2*x^2 + d^2))/e^2

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.05 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {33 d^{10} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{256 e} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{9}}{21 e^{2}} - \frac {33 d^{8} x}{256 e} + \frac {8 d^{7} x^{2}}{21} + \frac {117 d^{6} e x^{3}}{128} + \frac {2 d^{5} e^{2} x^{4}}{7} - \frac {139 d^{4} e^{3} x^{5}}{160} - \frac {16 d^{3} e^{4} x^{6}}{21} + \frac {9 d^{2} e^{5} x^{7}}{80} + \frac {d e^{6} x^{8}}{3} + \frac {e^{7} x^{9}}{10}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d^{3} x^{2}}{2} + d^{2} e x^{3} + \frac {3 d e^{2} x^{4}}{4} + \frac {e^{3} x^{5}}{5}\right ) \left (d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((33*d**10*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))/(256*e) + sqrt(d**2 - e**2*x**2)*(-5*d**9/(21*e**2) - 33*d**8*x/(256*e) +
8*d**7*x**2/21 + 117*d**6*e*x**3/128 + 2*d**5*e**2*x**4/7 - 139*d**4*e**3*x**5/160 - 16*d**3*e**4*x**6/21 + 9
*d**2*e**5*x**7/80 + d*e**6*x**8/3 + e**7*x**9/10), Ne(e**2, 0)), ((d**3*x**2/2 + d**2*e*x**3 + 3*d*e**2*x**4/
4 + e**3*x**5/5)*(d**2)**(5/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.79 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 \, d^{10} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{256 \, \sqrt {e^{2}} e} + \frac {33 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} x}{256 \, e} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x}{128 \, e} - \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{3} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x}{160 \, e} - \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{2} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x}{80 \, e} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3}}{21 \, e^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.62 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 \, d^{10} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{256 \, e {\left | e \right |}} - \frac {1}{26880} \, {\left (\frac {6400 \, d^{9}}{e^{2}} + {\left (\frac {3465 \, d^{8}}{e} - 2 \, {\left (5120 \, d^{7} + {\left (12285 \, d^{6} e + 4 \, {\left (960 \, d^{5} e^{2} - {\left (2919 \, d^{4} e^{3} + 2 \, {\left (1280 \, d^{3} e^{4} - 7 \, {\left (27 \, d^{2} e^{5} + 8 \, {\left (3 \, e^{7} x + 10 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

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

[Out]

33/256*d^10*arcsin(e*x/d)*sgn(d)*sgn(e)/(e*abs(e)) - 1/26880*(6400*d^9/e^2 + (3465*d^8/e - 2*(5120*d^7 + (1228
5*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*e^7*x + 10*d*e^6)*x)*x)*x)*x
)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

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

[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)