3.2.47 \(\int \frac {1}{x^4 (d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\) [147]

Optimal. Leaf size=215 \[ \frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9} \]

[Out]

1/15*(-7*e*x+8*d)/d^4/x^3/(-e^2*x^2+d^2)^(3/2)+1/5/d^2/x^3/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+7/2*e^3*arctanh((-e^2*
x^2+d^2)^(1/2)/d)/d^9+1/15*(-35*e*x+48*d)/d^6/x^3/(-e^2*x^2+d^2)^(1/2)-64/15*(-e^2*x^2+d^2)^(1/2)/d^7/x^3+7/2*
e*(-e^2*x^2+d^2)^(1/2)/d^8/x^2-128/15*e^2*(-e^2*x^2+d^2)^(1/2)/d^9/x

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Rubi [A]
time = 0.13, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {871, 837, 849, 821, 272, 65, 214} \begin {gather*} \frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}+\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(8*d - 7*e*x)/(15*d^4*x^3*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (48*d - 35*
e*x)/(15*d^6*x^3*Sqrt[d^2 - e^2*x^2]) - (64*Sqrt[d^2 - e^2*x^2])/(15*d^7*x^3) + (7*e*Sqrt[d^2 - e^2*x^2])/(2*d
^8*x^2) - (128*e^2*Sqrt[d^2 - e^2*x^2])/(15*d^9*x) + (7*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^9)

Rule 65

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 214

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

Rule 272

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 821

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 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 849

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 871

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

Rubi steps

\begin {align*} \int \frac {1}{x^4 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-8 d e^2+7 e^3 x}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-48 d^3 e^4+35 d^2 e^5 x}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-192 d^5 e^6+105 d^4 e^7 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {\int \frac {-315 d^6 e^7+384 d^5 e^8 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^{12} e^6}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {\int \frac {-768 d^7 e^8+315 d^6 e^9 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{14} e^6}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}-\frac {\left (7 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^8}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}-\frac {\left (7 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^8}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {(7 e) \text {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 )}{2 d^8}\\ &=\frac {8 d-7 e x}{15 d^4 x^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {48 d-35 e x}{15 d^6 x^3 \sqrt {d^2-e^2 x^2}}-\frac {64 \sqrt {d^2-e^2 x^2}}{15 d^7 x^3}+\frac {7 e \sqrt {d^2-e^2 x^2}}{2 d^8 x^2}-\frac {128 e^2 \sqrt {d^2-e^2 x^2}}{15 d^9 x}+\frac {7 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^9}\\ \end {align*}

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Mathematica [A]
time = 0.59, size = 157, normalized size = 0.73 \begin {gather*} -\frac {\frac {\sqrt {d^2-e^2 x^2} \left (10 d^7-5 d^6 e x+75 d^5 e^2 x^2+236 d^4 e^3 x^3-244 d^3 e^4 x^4-489 d^2 e^5 x^5+151 d e^6 x^6+256 e^7 x^7\right )}{x^3 (d-e x)^2 (d+e x)^3}+210 e^3 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{30 d^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

-1/30*((Sqrt[d^2 - e^2*x^2]*(10*d^7 - 5*d^6*e*x + 75*d^5*e^2*x^2 + 236*d^4*e^3*x^3 - 244*d^3*e^4*x^4 - 489*d^2
*e^5*x^5 + 151*d*e^6*x^6 + 256*e^7*x^7))/(x^3*(d - e*x)^2*(d + e*x)^3) + 210*e^3*ArcTanh[(Sqrt[-e^2]*x - Sqrt[
d^2 - e^2*x^2])/d])/d^9

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(568\) vs. \(2(187)=374\).
time = 0.10, size = 569, normalized size = 2.65

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (22 e^{2} x^{2}-3 d e x +2 d^{2}\right )}{6 d^{9} x^{3}}-\frac {331 e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{80 d^{9} \left (x +\frac {d}{e}\right )}-\frac {35 e^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{48 d^{9} \left (x -\frac {d}{e}\right )}+\frac {e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{24 d^{8} \left (x -\frac {d}{e}\right )^{2}}+\frac {7 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{8} \sqrt {d^{2}}}-\frac {9 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{8} \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{7} \left (x +\frac {d}{e}\right )^{3}}\)	\(312\)
default	\(\frac {e^{3} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{d^{4}}-\frac {e \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 e^{2} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{2 d^{2}}\right )}{d^{2}}+\frac {e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{3}}+\frac {-\frac {1}{3 d^{2} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 e^{2} \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}\right )}{d^{2}}}{d}-\frac {e^{3} \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{d^{4}}\)	\(569\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((-x^2*e^2 + d^2)^(5/2)*(x*e + d)*x^4), x)

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Fricas [A]
time = 2.39, size = 275, normalized size = 1.28 \begin {gather*} -\frac {116 \, x^{8} e^{8} + 116 \, d x^{7} e^{7} - 232 \, d^{2} x^{6} e^{6} - 232 \, d^{3} x^{5} e^{5} + 116 \, d^{4} x^{4} e^{4} + 116 \, d^{5} x^{3} e^{3} + 105 \, {\left (x^{8} e^{8} + d x^{7} e^{7} - 2 \, d^{2} x^{6} e^{6} - 2 \, d^{3} x^{5} e^{5} + d^{4} x^{4} e^{4} + d^{5} x^{3} e^{3}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) + {\left (256 \, x^{7} e^{7} + 151 \, d x^{6} e^{6} - 489 \, d^{2} x^{5} e^{5} - 244 \, d^{3} x^{4} e^{4} + 236 \, d^{4} x^{3} e^{3} + 75 \, d^{5} x^{2} e^{2} - 5 \, d^{6} x e + 10 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (d^{9} x^{8} e^{5} + d^{10} x^{7} e^{4} - 2 \, d^{11} x^{6} e^{3} - 2 \, d^{12} x^{5} e^{2} + d^{13} x^{4} e + d^{14} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/30*(116*x^8*e^8 + 116*d*x^7*e^7 - 232*d^2*x^6*e^6 - 232*d^3*x^5*e^5 + 116*d^4*x^4*e^4 + 116*d^5*x^3*e^3 + 1
05*(x^8*e^8 + d*x^7*e^7 - 2*d^2*x^6*e^6 - 2*d^3*x^5*e^5 + d^4*x^4*e^4 + d^5*x^3*e^3)*log(-(d - sqrt(-x^2*e^2 +
d^2))/x) + (256*x^7*e^7 + 151*d*x^6*e^6 - 489*d^2*x^5*e^5 - 244*d^3*x^4*e^4 + 236*d^4*x^3*e^3 + 75*d^5*x^2*e^
2 - 5*d^6*x*e + 10*d^7)*sqrt(-x^2*e^2 + d^2))/(d^9*x^8*e^5 + d^10*x^7*e^4 - 2*d^11*x^6*e^3 - 2*d^12*x^5*e^2 +
d^13*x^4*e + d^14*x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/(x**4*(-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((-x^2*e^2 + d^2)^(5/2)*(x*e + d)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)),x)

[Out]

int(1/(x^4*(d^2 - e^2*x^2)^(5/2)*(d + e*x)), x)

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