\(\int \frac {x^3 (d^2-e^2 x^2)^{5/2}}{d+e x} \, dx\) [104]

Optimal result

Integrand size = 27, antiderivative size = 172 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}-\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {3 d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4} \]

[Out]

-1/64*d^4*x*(-e^2*x^2+d^2)^(3/2)/e^3-1/7*d*x^2*(-e^2*x^2+d^2)^(5/2)/e^2+1/8*x^3*(-e^2*x^2+d^2)^(5/2)/e-1/560*d
^2*(-35*e*x+32*d)*(-e^2*x^2+d^2)^(5/2)/e^4-3/128*d^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4-3/128*d^6*x*(-e^2*x^
2+d^2)^(1/2)/e^3

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {864, 847, 794, 201, 223, 209} \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}-\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3} \]

[In]

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

[Out]

(-3*d^6*x*Sqrt[d^2 - e^2*x^2])/(128*e^3) - (d^4*x*(d^2 - e^2*x^2)^(3/2))/(64*e^3) - (d*x^2*(d^2 - e^2*x^2)^(5/
2))/(7*e^2) + (x^3*(d^2 - e^2*x^2)^(5/2))/(8*e) - (d^2*(32*d - 35*e*x)*(d^2 - e^2*x^2)^(5/2))/(560*e^4) - (3*d
^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*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 864

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

Rubi steps \begin{align*} \text {integral}& = \int x^3 (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {\int x^2 \left (3 d^2 e-8 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{8 e^2} \\ & = -\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}+\frac {\int x \left (16 d^3 e^2-21 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{56 e^4} \\ & = -\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {d^4 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^3} \\ & = -\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {\left (3 d^6\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^3} \\ & = -\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}-\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {\left (3 d^8\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^3} \\ & = -\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}-\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {\left (3 d^8\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^3} \\ & = -\frac {3 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^3}-\frac {d^4 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^3}-\frac {d x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e^2}+\frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d^2 (32 d-35 e x) \left (d^2-e^2 x^2\right )^{5/2}}{560 e^4}-\frac {3 d^8 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-256 d^7+105 d^6 e x-128 d^5 e^2 x^2+70 d^4 e^3 x^3+1024 d^3 e^4 x^4-840 d^2 e^5 x^5-640 d e^6 x^6+560 e^7 x^7\right )+210 d^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{4480 e^4} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-256*d^7 + 105*d^6*e*x - 128*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 1024*d^3*e^4*x^4 - 840*d^2*e
^5*x^5 - 640*d*e^6*x^6 + 560*e^7*x^7) + 210*d^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(4480*e^4)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\left (-560 e^{7} x^{7}+640 d \,e^{6} x^{6}+840 d^{2} e^{5} x^{5}-1024 d^{3} e^{4} x^{4}-70 d^{4} e^{3} x^{3}+128 d^{5} e^{2} x^{2}-105 d^{6} e x +256 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{4480 e^{4}}-\frac {3 d^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{3} \sqrt {e^{2}}}\)	\(130\)
default	\(\frac {-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}}{e}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e^{3}}+\frac {d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{4}}-\frac {d^{3} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{4}}\)	\(449\)

[In]

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

[Out]

-1/4480*(-560*e^7*x^7+640*d*e^6*x^6+840*d^2*e^5*x^5-1024*d^3*e^4*x^4-70*d^4*e^3*x^3+128*d^5*e^2*x^2-105*d^6*e*
x+256*d^7)/e^4*(-e^2*x^2+d^2)^(1/2)-3/128*d^8/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 = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.74 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {210 \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (560 \, e^{7} x^{7} - 640 \, d e^{6} x^{6} - 840 \, d^{2} e^{5} x^{5} + 1024 \, d^{3} e^{4} x^{4} + 70 \, d^{4} e^{3} x^{3} - 128 \, d^{5} e^{2} x^{2} + 105 \, d^{6} e x - 256 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4480 \, e^{4}} \]

[In]

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

[Out]

1/4480*(210*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (560*e^7*x^7 - 640*d*e^6*x^6 - 840*d^2*e^5*x^5 + 1
024*d^3*e^4*x^4 + 70*d^4*e^3*x^3 - 128*d^5*e^2*x^2 + 105*d^6*e*x - 256*d^7)*sqrt(-e^2*x^2 + d^2))/e^4

Sympy [A] (verification not implemented)

Time = 1.37 (sec) , antiderivative size = 400, normalized size of antiderivative = 2.33 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=d^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \frac {5 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{6}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{6} x}{128 e^{6}} - \frac {5 d^{4} x^{3}}{192 e^{4}} - \frac {d^{2} x^{5}}{48 e^{2}} + \frac {x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{7} \sqrt {d^{2}}}{7} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**3*Piecewise((sqrt(d**2 - e**2*x**2)*(-2*d**4/(15*e**4) - d**2*x**2/(15*e**2) + x**4/5), Ne(e**2, 0)), (x**4
*sqrt(d**2)/4, True)) - d**2*e*Piecewise((d**6*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))/(16*e**4) + sqrt(d**2 - e**2*x**2)*(-d**4*x/(1
6*e**4) - d**2*x**3/(24*e**2) + x**5/6), Ne(e**2, 0)), (x**5*sqrt(d**2)/5, True)) - d*e**2*Piecewise((sqrt(d**
2 - e**2*x**2)*(-8*d**6/(105*e**6) - 4*d**4*x**2/(105*e**4) - d**2*x**4/(35*e**2) + x**6/7), Ne(e**2, 0)), (x*
*6*sqrt(d**2)/6, True)) + e**3*Piecewise((5*d**8*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))/(128*e**6) + sqrt(d**2 - e**2*x**2)*(-5*d**6
*x/(128*e**6) - 5*d**4*x**3/(192*e**4) - d**2*x**5/(48*e**2) + x**7/8), Ne(e**2, 0)), (x**7*sqrt(d**2)/7, True
))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.28 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 i \, d^{8} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{4}} + \frac {45 \, d^{8} \arcsin \left (\frac {e x}{d}\right )}{128 \, e^{4}} - \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6} x}{8 \, e^{3}} + \frac {45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} x}{128 \, e^{3}} - \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7}}{4 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x}{64 \, e^{3}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x}{16 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{5 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x}{8 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{7 \, e^{4}} \]

[In]

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

[Out]

3/8*I*d^8*arcsin(e*x/d + 2)/e^4 + 45/128*d^8*arcsin(e*x/d)/e^4 - 3/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^6*x/e^3
+ 45/128*sqrt(-e^2*x^2 + d^2)*d^6*x/e^3 - 3/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^7/e^4 - 1/64*(-e^2*x^2 + d^2)
^(3/2)*d^4*x/e^3 + 3/16*(-e^2*x^2 + d^2)^(5/2)*d^2*x/e^3 - 1/5*(-e^2*x^2 + d^2)^(5/2)*d^3/e^4 - 1/8*(-e^2*x^2
+ d^2)^(7/2)*x/e^3 + 1/7*(-e^2*x^2 + d^2)^(7/2)*d/e^4

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 \, d^{8} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, e^{3} {\left | e \right |}} - \frac {1}{4480} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {256 \, d^{7}}{e^{4}} - {\left (\frac {105 \, d^{6}}{e^{3}} - 2 \, {\left (\frac {64 \, d^{5}}{e^{2}} - {\left (\frac {35 \, d^{4}}{e} + 4 \, {\left (128 \, d^{3} - 5 \, {\left (21 \, d^{2} e - 2 \, {\left (7 \, e^{3} x - 8 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]

[In]

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

[Out]

-3/128*d^8*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) - 1/4480*sqrt(-e^2*x^2 + d^2)*(256*d^7/e^4 - (105*d^6/e^3
- 2*(64*d^5/e^2 - (35*d^4/e + 4*(128*d^3 - 5*(21*d^2*e - 2*(7*e^3*x - 8*d*e^2)*x)*x)*x)*x)*x)*x)

Mupad [F(-1)]

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

[In]

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

[Out]

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