3.103 \(\int \frac {x^4 (d^2-e^2 x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=201 \[ \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}+\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5} \]

[Out]

1/64*d^5*x*(-e^2*x^2+d^2)^(3/2)/e^4+4/63*d^2*x^2*(-e^2*x^2+d^2)^(5/2)/e^3-1/8*d*x^3*(-e^2*x^2+d^2)^(5/2)/e^2+1
/9*x^4*(-e^2*x^2+d^2)^(5/2)/e+1/5040*d^3*(-315*e*x+128*d)*(-e^2*x^2+d^2)^(5/2)/e^5+3/128*d^9*arctan(e*x/(-e^2*
x^2+d^2)^(1/2))/e^5+3/128*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^4

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {850, 833, 780, 195, 217, 203} \[ \frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]

[Out]

(3*d^7*x*Sqrt[d^2 - e^2*x^2])/(128*e^4) + (d^5*x*(d^2 - e^2*x^2)^(3/2))/(64*e^4) + (4*d^2*x^2*(d^2 - e^2*x^2)^
(5/2))/(63*e^3) - (d*x^3*(d^2 - e^2*x^2)^(5/2))/(8*e^2) + (x^4*(d^2 - e^2*x^2)^(5/2))/(9*e) + (d^3*(128*d - 31
5*e*x)*(d^2 - e^2*x^2)^(5/2))/(5040*e^5) + (3*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(128*e^5)

Rule 195

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 217

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 780

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 833

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 850

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*} \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx &=\int x^4 (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x^3 \left (4 d^2 e-9 d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{9 e^2}\\ &=-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {\int x^2 \left (27 d^3 e^2-32 d^2 e^3 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{72 e^4}\\ &=\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int x \left (64 d^4 e^3-189 d^3 e^4 x\right ) \left (d^2-e^2 x^2\right )^{3/2} \, dx}{504 e^6}\\ &=\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {d^5 \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{16 e^4}\\ &=\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{64 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{128 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {\left (3 d^9\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^4}\\ &=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 135, normalized size = 0.67 \[ \frac {945 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\sqrt {d^2-e^2 x^2} \left (1024 d^8-945 d^7 e x+512 d^6 e^2 x^2-630 d^5 e^3 x^3+384 d^4 e^4 x^4+7560 d^3 e^5 x^5-6400 d^2 e^6 x^6-5040 d e^7 x^7+4480 e^8 x^8\right )}{40320 e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(1024*d^8 - 945*d^7*e*x + 512*d^6*e^2*x^2 - 630*d^5*e^3*x^3 + 384*d^4*e^4*x^4 + 7560*d^3*
e^5*x^5 - 6400*d^2*e^6*x^6 - 5040*d*e^7*x^7 + 4480*e^8*x^8) + 945*d^9*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(4032
0*e^5)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 139, normalized size = 0.69 \[ -\frac {1890 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (4480 \, e^{8} x^{8} - 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} + 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} - 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} - 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x, algorithm="fricas")

[Out]

-1/40320*(1890*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (4480*e^8*x^8 - 5040*d*e^7*x^7 - 6400*d^2*e^6*x
^6 + 7560*d^3*e^5*x^5 + 384*d^4*e^4*x^4 - 630*d^5*e^3*x^3 + 512*d^6*e^2*x^2 - 945*d^7*e*x + 1024*d^8)*sqrt(-e^
2*x^2 + d^2))/e^5

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 1/2*(12*d^9*exp(1)^4*exp(2)^2-8*d^9*exp(
2)^4-4*d^9*exp(1)^6*exp(2))*atan((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+ex
p(2)^2))/sqrt(-exp(1)^4+exp(2)^2)/exp(1)^10/exp(1)+3/128*d^9*sign(d)*asin(x*exp(2)/d/exp(1))/exp(1)^5+2*((((((
((322560*exp(1)^17*1/5806080/exp(1)^14*x-362880*exp(1)^16*d*1/5806080/exp(1)^14)*x-460800*exp(1)^15*d^2*1/5806
080/exp(1)^14)*x+544320*exp(1)^14*d^3*1/5806080/exp(1)^14)*x+27648*exp(1)^13*d^4*1/5806080/exp(1)^14)*x-45360*
exp(1)^12*d^5*1/5806080/exp(1)^14)*x+36864*exp(1)^11*d^6*1/5806080/exp(1)^14)*x-68040*exp(1)^10*d^7*1/5806080/
exp(1)^14)*x+73728*exp(1)^9*d^8*1/5806080/exp(1)^14)*sqrt(d^2-x^2*exp(2))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 330, normalized size = 1.64 \[ \frac {3 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, e^{4}}-\frac {45 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}\, e^{4}}-\frac {45 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} x}{128 e^{4}}+\frac {3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{7} x}{8 e^{4}}-\frac {15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5} x}{64 e^{4}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{5} x}{4 e^{4}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} x}{16 e^{4}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{4}}{5 e^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} x^{2}}{9 e^{3}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d x}{8 e^{4}}-\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2}}{63 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x)

[Out]

-1/9/e^3*x^2*(-e^2*x^2+d^2)^(7/2)-11/63*d^2/e^5*(-e^2*x^2+d^2)^(7/2)+1/8*d/e^4*x*(-e^2*x^2+d^2)^(7/2)-3/16*(-e
^2*x^2+d^2)^(5/2)*d^3/e^4*x-15/64*(-e^2*x^2+d^2)^(3/2)*d^5/e^4*x-45/128*(-e^2*x^2+d^2)^(1/2)*d^7/e^4*x-45/128/
(e^2)^(1/2)*d^9/e^4*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)+1/5*d^4/e^5*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)
+1/4*d^5/e^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x+3/8*d^7/e^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x+3/8*d^9/e
^4/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x)

________________________________________________________________________________________

maxima [C]  time = 1.04, size = 246, normalized size = 1.22 \[ -\frac {3 i \, d^{9} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{5}} - \frac {45 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{128 \, e^{5}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7} x}{8 \, e^{4}} - \frac {45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{8}}{4 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{64 \, e^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{16 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{5 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x}{8 \, e^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x, algorithm="maxima")

[Out]

-3/8*I*d^9*arcsin(e*x/d + 2)/e^5 - 45/128*d^9*arcsin(e*x/d)/e^5 + 3/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^7*x/e^
4 - 45/128*sqrt(-e^2*x^2 + d^2)*d^7*x/e^4 + 3/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^8/e^5 + 1/64*(-e^2*x^2 + d^2
)^(3/2)*d^5*x/e^4 - 3/16*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^4 - 1/9*(-e^2*x^2 + d^2)^(7/2)*x^2/e^3 + 1/5*(-e^2*x^2
+ d^2)^(5/2)*d^4/e^5 + 1/8*(-e^2*x^2 + d^2)^(7/2)*d*x/e^4 - 11/63*(-e^2*x^2 + d^2)^(7/2)*d^2/e^5

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x),x)

[Out]

int((x^4*(d^2 - e^2*x^2)^(5/2))/(d + e*x), x)

________________________________________________________________________________________

sympy [C]  time = 25.15, size = 830, normalized size = 4.13 \[ d^{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 ) - d^{2} e \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 e^{2} \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 ) + e^{3} \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 ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-e**2*x**2+d**2)**(5/2)/(e*x+d),x)

[Out]

d**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/(4
8*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*sq
rt(1 - e**2*x**2/d**2)), True)) - d**2*e*Piecewise((-8*d**6*sqrt(d**2 - e**2*x**2)/(105*e**6) - 4*d**4*x**2*sq
rt(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*e**2*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*s
qrt(-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)) + e**3*Piecewise((-16*d**8*sq
rt(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))

________________________________________________________________________________________