3.114 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^7 (d+e x)} \, dx\)

Optimal. Leaf size=143 \[ -\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2}+\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2} \]

[Out]

-1/24*e^2*(-e^2*x^2+d^2)^(3/2)/d/x^4-1/6*(-e^2*x^2+d^2)^(5/2)/d/x^6+1/5*e*(-e^2*x^2+d^2)^(5/2)/d^2/x^5-1/16*e^
6*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^2+1/16*e^4*(-e^2*x^2+d^2)^(1/2)/d/x^2

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {850, 835, 807, 266, 47, 63, 208} \[ \frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

(e^4*Sqrt[d^2 - e^2*x^2])/(16*d*x^2) - (e^2*(d^2 - e^2*x^2)^(3/2))/(24*d*x^4) - (d^2 - e^2*x^2)^(5/2)/(6*d*x^6
) + (e*(d^2 - e^2*x^2)^(5/2))/(5*d^2*x^5) - (e^6*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(16*d^2)

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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 {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx &=\int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}-\frac {\int \frac {\left (6 d^2 e-d e^2 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^6} \, dx}{6 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac {e^2 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{6 d}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac {e^2 \operatorname {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{12 d}\\ &=-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^4 \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d}\\ &=\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}+\frac {e^6 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{32 d}\\ &=\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^4 \operatorname {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{16 d}\\ &=\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 117, normalized size = 0.82 \[ \frac {-15 e^6 x^6 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-40 d^5+48 d^4 e x+70 d^3 e^2 x^2-96 d^2 e^3 x^3-15 d e^4 x^4+48 e^5 x^5\right )+15 e^6 x^6 \log (x)}{240 d^2 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-40*d^5 + 48*d^4*e*x + 70*d^3*e^2*x^2 - 96*d^2*e^3*x^3 - 15*d*e^4*x^4 + 48*e^5*x^5) + 15
*e^6*x^6*Log[x] - 15*e^6*x^6*Log[d + Sqrt[d^2 - e^2*x^2]])/(240*d^2*x^6)

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 108, normalized size = 0.76 \[ \frac {15 \, e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (48 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} - 96 \, d^{2} e^{3} x^{3} + 70 \, d^{3} e^{2} x^{2} + 48 \, d^{4} e x - 40 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, d^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d),x, algorithm="fricas")

[Out]

1/240*(15*e^6*x^6*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (48*e^5*x^5 - 15*d*e^4*x^4 - 96*d^2*e^3*x^3 + 70*d^3*e^
2*x^2 + 48*d^4*e*x - 40*d^5)*sqrt(-e^2*x^2 + d^2))/(d^2*x^6)

________________________________________________________________________________________

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((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 1/1920*((-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^
2*exp(2))*exp(1))/x/exp(2))^5*(-960*exp(1)^10*exp(2)^2+2880*exp(1)^8*exp(2)^3-3600*exp(1)^6*exp(2)^4+2160*exp(
1)^4*exp(2)^5-600*exp(2)^7)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*(240*exp(1)^8*exp(2)
^3-720*exp(1)^6*exp(2)^4+960*exp(1)^4*exp(2)^5-495*exp(2)^7)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))
/x/exp(2))^3*(-80*exp(1)^6*exp(2)^4+240*exp(1)^4*exp(2)^5-100*exp(2)^7)+(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(
2))*exp(1))/x/exp(2))^2*(30*exp(1)^4*exp(2)^5-45*exp(2)^7)+5*exp(2)^7+6*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*ex
p(1))*exp(2)^7/x/exp(2))/d^2/(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6/exp(1)^8+1/68719476
736*(-8589934592*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^48*exp(2)^9+8589934
592/3*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^46*exp(2)^10-1073741824*d^10*(
-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^44*exp(2)^11+2147483648/5*d^10*(-1/2*(-2*d
*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^5*exp(1)^42*exp(2)^12-536870912/3*d^10*(-1/2*(-2*d*exp(1)-2*s
qrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^6*exp(1)^40*exp(2)^13+25769803776*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2
*exp(2))*exp(1))/x/exp(2))^2*exp(1)^46*exp(2)^10-8589934592*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp
(1))/x/exp(2))^3*exp(1)^44*exp(2)^11+3221225472*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2
))^4*exp(1)^42*exp(2)^12-34359738368*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)
^44*exp(2)^11+10737418240/3*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^3*exp(1)^42*exp(2
)^12-1610612736*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^4*exp(1)^40*exp(2)^13+2576980
3776*d^10*(-1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^42*exp(2)^12-8053063680*d^10*(-
1/2*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x/exp(2))^2*exp(1)^40*exp(2)^13-10737418240*d^10*(-2*d*exp(1)-
2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^42*exp(2)^12/x/exp(2)[phone]*d^10*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2
))*exp(1))*exp(1)^44*exp(2)^11/x/exp(2)-64424509440*d^10*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^46
*exp(2)^10/x/exp(2)[phone]*d^10*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^48*exp(2)^9/x/exp(2)-1
7179869184*d^10*(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))*exp(1)^50*exp(2)^8/x/exp(2))/d^12/exp(1)^48/exp(2)
^6+1/2*(-12*exp(1)^5*exp(2)^2+12*exp(1)^3*exp(2)^3+4*exp(1)^7*exp(2)-4*exp(1)*exp(2)^4)*atan((-1/2*(-2*d*exp(1
)-2*sqrt(d^2-x^2*exp(2))*exp(1))/x+exp(2))/sqrt(-exp(1)^4+exp(2)^2))/d^2/sqrt(-exp(1)^4+exp(2)^2)/exp(1)+1/16*
(48*exp(1)^10*exp(2)^2-56*exp(1)^8*exp(2)^3+40*exp(1)^6*exp(2)^4-30*exp(1)^4*exp(2)^5+13*exp(2)^7-16*exp(1)^12
*exp(2))*ln(1/2*abs(-2*d*exp(1)-2*sqrt(d^2-x^2*exp(2))*exp(1))/abs(x)/exp(2))/d^2/exp(1)^7/exp(1)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 521, normalized size = 3.64 \[ -\frac {e^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 \sqrt {d^{2}}\, d}-\frac {3 e^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{2}}+\frac {3 e^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{2}}-\frac {3 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{7} x}{8 d^{4}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7} x}{8 d^{4}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{6}}{16 d^{3}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{7} x}{4 d^{6}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7} x}{4 d^{6}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6}}{48 d^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7} x}{5 d^{8}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{6}}{5 d^{7}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}{80 d^{7}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{5 d^{8} x}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{16 d^{7} x^{2}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{5 d^{6} x^{3}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{24 d^{5} x^{4}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{5 d^{4} x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d),x)

[Out]

1/48*(-e^2*x^2+d^2)^(3/2)/d^5*e^6+1/16*(-e^2*x^2+d^2)^(1/2)/d^3*e^6+1/5*e/d^4/x^5*(-e^2*x^2+d^2)^(7/2)-1/4/d^6
*e^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x-3/8/d^4*e^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x-3/8/d^2*e^7/(e^2)
^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x)+1/5/d^6*e^3/x^3*(-e^2*x^2+d^2)^(7/2)+1/5/d^8*
e^5/x*(-e^2*x^2+d^2)^(7/2)+1/5/d^8*e^7*x*(-e^2*x^2+d^2)^(5/2)+1/4/d^6*e^7*x*(-e^2*x^2+d^2)^(3/2)+3/8/d^4*e^7*x
*(-e^2*x^2+d^2)^(1/2)+3/8/d^2*e^7/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)-5/24/d^5*e^2/x^4*(-e^
2*x^2+d^2)^(7/2)-3/16/d^7*e^4/x^2*(-e^2*x^2+d^2)^(7/2)-1/16/(d^2)^(1/2)/d*e^6*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^
2+d^2)^(1/2))/x)-1/5/d^7*e^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)-1/6/d^3/x^6*(-e^2*x^2+d^2)^(7/2)+1/80/d^7*e^6
*(-e^2*x^2+d^2)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 178, normalized size = 1.24 \[ -\frac {e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{6}}{16 \, d^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{16 \, d^{3} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{5 \, d^{2} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{8 \, d x^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{5 \, x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d),x, algorithm="maxima")

[Out]

-1/16*e^6*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^2 + 1/16*sqrt(-e^2*x^2 + d^2)*e^6/d^3 + 1/16*(
-e^2*x^2 + d^2)^(3/2)*e^4/(d^3*x^2) - 1/5*(-e^2*x^2 + d^2)^(3/2)*e^3/(d^2*x^3) + 1/8*(-e^2*x^2 + d^2)^(3/2)*e^
2/(d*x^4) + 1/5*(-e^2*x^2 + d^2)^(3/2)*e/x^5 - 1/6*(-e^2*x^2 + d^2)^(3/2)*d/x^6

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^7\,\left (d+e\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)),x)

[Out]

int((d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)), x)

________________________________________________________________________________________

sympy [C]  time = 18.69, size = 918, normalized size = 6.42 \[ d^{3} \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e**2*x**2+d**2)**(5/2)/x**7/(e*x+d),x)

[Out]

d**3*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/
(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d
**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2
*x**2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) - d**2*e*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(
-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I
*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e**4*x**4*sqrt(-1 + e**2*x**2/d**
2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e**2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5
+ 15*e**2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*e**6*x**6*sqrt(1
- e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15
*d*e**2*x**7), True)) - d*e**2*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3*sqrt(d**2/
(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**2*
x**2)) > 1), (I*d**2/(4*e*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e
**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/(8*d**3), True)) + e**3*Piecewise((-e*sqrt(d
**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sq
rt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True))

________________________________________________________________________________________