3.1.29 \(\int \frac {d+e x}{x^3 (d^2-e^2 x^2)^{7/2}} \, dx\) [29]

Optimal. Leaf size=184 \[ \frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}-\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^8} \]

[Out]

1/5*(e*x+d)/d^2/x^2/(-e^2*x^2+d^2)^(5/2)+1/15*(6*e*x+7*d)/d^4/x^2/(-e^2*x^2+d^2)^(3/2)-7/2*e^2*arctanh((-e^2*x
^2+d^2)^(1/2)/d)/d^8+1/15*(24*e*x+35*d)/d^6/x^2/(-e^2*x^2+d^2)^(1/2)-7/2*(-e^2*x^2+d^2)^(1/2)/d^7/x^2-16/5*e*(
-e^2*x^2+d^2)^(1/2)/d^8/x

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Rubi [A]
time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {837, 849, 821, 272, 65, 214} \begin {gather*} \frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^8}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(d + e*x)/(5*d^2*x^2*(d^2 - e^2*x^2)^(5/2)) + (7*d + 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + (35*d + 24*e*
x)/(15*d^6*x^2*Sqrt[d^2 - e^2*x^2]) - (7*Sqrt[d^2 - e^2*x^2])/(2*d^7*x^2) - (16*e*Sqrt[d^2 - e^2*x^2])/(5*d^8*
x) - (7*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^8)

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])

Rubi steps

\begin {align*} \int \frac {d+e x}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {7 d^3 e^2+6 d^2 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^4 e^2}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {35 d^5 e^4+24 d^4 e^5 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^8 e^4}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {105 d^7 e^6+48 d^6 e^7 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{12} e^6}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}-\frac {\int \frac {-96 d^8 e^7-105 d^7 e^8 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^{14} e^6}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}-\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}+\frac {\left (7 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^7}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}-\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}+\frac {\left (7 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^7}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}-\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 \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^7}\\ &=\frac {d+e x}{5 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {7 d+6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {35 d+24 e x}{15 d^6 x^2 \sqrt {d^2-e^2 x^2}}-\frac {7 \sqrt {d^2-e^2 x^2}}{2 d^7 x^2}-\frac {16 e \sqrt {d^2-e^2 x^2}}{5 d^8 x}-\frac {7 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^8}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 147, normalized size = 0.80 \begin {gather*} \frac {\frac {\sqrt {d^2-e^2 x^2} \left (15 d^6+15 d^5 e x-176 d^4 e^2 x^2-4 d^3 e^3 x^3+249 d^2 e^4 x^4-9 d e^5 x^5-96 e^6 x^6\right )}{x^2 (-d+e x)^3 (d+e x)^2}+210 e^2 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{30 d^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(15*d^6 + 15*d^5*e*x - 176*d^4*e^2*x^2 - 4*d^3*e^3*x^3 + 249*d^2*e^4*x^4 - 9*d*e^5*x^5 -
96*e^6*x^6))/(x^2*(-d + e*x)^3*(d + e*x)^2) + 210*e^2*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(30*d^
8)

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Maple [A]
time = 0.09, size = 244, normalized size = 1.33

method	result	size
default	\(d \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )}{2 d^{2}}\right )+e \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )\)	\(244\)
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (2 e x +d \right )}{2 d^{8} x^{2}}+\frac {29 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{48 d^{8} \left (x +\frac {d}{e}\right )}-\frac {673 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{240 d^{8} \left (x -\frac {d}{e}\right )}-\frac {7 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{7} \sqrt {d^{2}}}+\frac {11 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{30 d^{7} \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d \left (x -\frac {d}{e}\right ) e}}{20 d^{6} e \left (x -\frac {d}{e}\right )^{3}}+\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{24 d^{7} \left (x +\frac {d}{e}\right )^{2}}\)	\(299\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

d*(-1/2/d^2/x^2/(-e^2*x^2+d^2)^(5/2)+7/2*e^2/d^2*(1/5/d^2/(-e^2*x^2+d^2)^(5/2)+1/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*(-1/d^2/x/(-e^2*x^2+d^2)^(5/2)+6*e^2/d^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/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))))

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Maxima [A]
time = 0.27, size = 206, normalized size = 1.12 \begin {gather*} \frac {6 \, x e^{3}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}} + \frac {7 \, e^{2}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} - \frac {e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x} - \frac {1}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{2}} + \frac {8 \, x e^{3}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}} + \frac {7 \, e^{2}}{6 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} - \frac {7 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{8}} + \frac {16 \, x e^{3}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{8}} + \frac {7 \, e^{2}}{2 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

6/5*x*e^3/((-x^2*e^2 + d^2)^(5/2)*d^4) + 7/10*e^2/((-x^2*e^2 + d^2)^(5/2)*d^3) - e/((-x^2*e^2 + d^2)^(5/2)*d^2
*x) - 1/2/((-x^2*e^2 + d^2)^(5/2)*d*x^2) + 8/5*x*e^3/((-x^2*e^2 + d^2)^(3/2)*d^6) + 7/6*e^2/((-x^2*e^2 + d^2)^
(3/2)*d^5) - 7/2*e^2*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x))/d^8 + 16/5*x*e^3/(sqrt(-x^2*e^2 + d^2
)*d^8) + 7/2*e^2/(sqrt(-x^2*e^2 + d^2)*d^7)

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Fricas [A]
time = 2.38, size = 270, normalized size = 1.47 \begin {gather*} \frac {116 \, x^{7} e^{7} - 116 \, d x^{6} e^{6} - 232 \, d^{2} x^{5} e^{5} + 232 \, d^{3} x^{4} e^{4} + 116 \, d^{4} x^{3} e^{3} - 116 \, d^{5} x^{2} e^{2} + 105 \, {\left (x^{7} e^{7} - d x^{6} e^{6} - 2 \, d^{2} x^{5} e^{5} + 2 \, d^{3} x^{4} e^{4} + d^{4} x^{3} e^{3} - d^{5} x^{2} e^{2}\right )} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (96 \, x^{6} e^{6} + 9 \, d x^{5} e^{5} - 249 \, d^{2} x^{4} e^{4} + 4 \, d^{3} x^{3} e^{3} + 176 \, d^{4} x^{2} e^{2} - 15 \, d^{5} x e - 15 \, d^{6}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (d^{8} x^{7} e^{5} - d^{9} x^{6} e^{4} - 2 \, d^{10} x^{5} e^{3} + 2 \, d^{11} x^{4} e^{2} + d^{12} x^{3} e - d^{13} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(116*x^7*e^7 - 116*d*x^6*e^6 - 232*d^2*x^5*e^5 + 232*d^3*x^4*e^4 + 116*d^4*x^3*e^3 - 116*d^5*x^2*e^2 + 10
5*(x^7*e^7 - d*x^6*e^6 - 2*d^2*x^5*e^5 + 2*d^3*x^4*e^4 + d^4*x^3*e^3 - d^5*x^2*e^2)*log(-(d - sqrt(-x^2*e^2 +
d^2))/x) - (96*x^6*e^6 + 9*d*x^5*e^5 - 249*d^2*x^4*e^4 + 4*d^3*x^3*e^3 + 176*d^4*x^2*e^2 - 15*d^5*x*e - 15*d^6
)*sqrt(-x^2*e^2 + d^2))/(d^8*x^7*e^5 - d^9*x^6*e^4 - 2*d^10*x^5*e^3 + 2*d^11*x^4*e^2 + d^12*x^3*e - d^13*x^2)

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Sympy [C] Result contains complex when optimal does not.
time = 15.77, size = 2691, normalized size = 14.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**3/(-e**2*x**2+d**2)**(7/2),x)

[Out]

d*Piecewise((30*I*d**8*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 +
60*d**9*e**6*x**8) - 322*I*d**6*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 1
80*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 105*d**6*e**2*x**2*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e
**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*d**6*e**2*x**2*log(e*x/d)/(-60*d**15*x**2 + 180*d**1
3*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*I*d**6*e**2*x**2*asin(d/(e*x))/(-60*d**15*x**2 +
180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 490*I*d**4*e**4*x**4*sqrt(-1 + e**2*x**2/d**2
)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*d**4*e**4*x**4*log(e*
*2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 630*d**4*e**4
*x**4*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 630*I*d**4
*e**4*x**4*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 21
0*I*d**2*e**6*x**6*sqrt(-1 + e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*
d**9*e**6*x**8) - 315*d**2*e**6*x**6*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**
4*x**6 + 60*d**9*e**6*x**8) + 630*d**2*e**6*x**6*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*
e**4*x**6 + 60*d**9*e**6*x**8) - 630*I*d**2*e**6*x**6*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 18
0*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 105*e**8*x**8*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x*
*4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*e**8*x**8*log(e*x/d)/(-60*d**15*x**2 + 180*d**13*e**2*x**4
- 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 210*I*e**8*x**8*asin(d/(e*x))/(-60*d**15*x**2 + 180*d**13*e**2*x
**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8), Abs(e**2*x**2/d**2) > 1), (30*d**8*sqrt(1 - e**2*x**2/d**2)/(-
60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 322*d**6*e**2*x**2*sqrt(1 - e
**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 105*d**6*e**
2*x**2*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) +
210*d**6*e**2*x**2*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x*
*6 + 60*d**9*e**6*x**8) - 105*I*pi*d**6*e**2*x**2/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6
+ 60*d**9*e**6*x**8) + 490*d**4*e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180
*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*d**4*e**4*x**4*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**
2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 630*d**4*e**4*x**4*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*
d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 315*I*pi*d**4*e**4*x**4/(-60*d**
15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 210*d**2*e**6*x**6*sqrt(1 - e**2*x*
*2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) - 315*d**2*e**6*x**6
*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 630*d*
*2*e**6*x**6*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 6
0*d**9*e**6*x**8) - 315*I*pi*d**2*e**6*x**6/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6 + 60*d
**9*e**6*x**8) + 105*e**8*x**8*log(e**2*x**2/d**2)/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d**11*e**4*x**6
+ 60*d**9*e**6*x**8) - 210*e**8*x**8*log(sqrt(1 - e**2*x**2/d**2) + 1)/(-60*d**15*x**2 + 180*d**13*e**2*x**4
- 180*d**11*e**4*x**6 + 60*d**9*e**6*x**8) + 105*I*pi*e**8*x**8/(-60*d**15*x**2 + 180*d**13*e**2*x**4 - 180*d*
*11*e**4*x**6 + 60*d**9*e**6*x**8), True)) + e*Piecewise((5*d**6*e*sqrt(d**2/(e**2*x**2) - 1)/(-5*d**14 + 15*d
**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) - 30*d**4*e**3*x**2*sqrt(d**2/(e**2*x**2) - 1)/(-5*d**
14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) + 40*d**2*e**5*x**4*sqrt(d**2/(e**2*x**2) - 1
)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) - 16*e**7*x**6*sqrt(d**2/(e**2*x**2)
- 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6), Abs(d**2/(e**2*x**2)) > 1), (5*
I*d**6*e*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*e**6*x**6) -
30*I*d**4*e**3*x**2*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x**4 + 5*d**8*
e**6*x**6) + 40*I*d**2*e**5*x**4*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*e**4*x*
*4 + 5*d**8*e**6*x**6) - 16*I*e**7*x**6*sqrt(-d**2/(e**2*x**2) + 1)/(-5*d**14 + 15*d**12*e**2*x**2 - 15*d**10*
e**4*x**4 + 5*d**8*e**6*x**6), True))

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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((e*x+d)/x^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

integrate((x*e + d)/((-x^2*e^2 + d^2)^(7/2)*x^3), x)

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Mupad [B]
time = 3.43, size = 181, normalized size = 0.98 \begin {gather*} \frac {161\,e^2}{30\,d^3\,{\left (d^2-e^2\,x^2\right )}^{5/2}}-\frac {1}{2\,d\,x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}}-\frac {7\,e^2\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )}{2\,d^8}-\frac {49\,e^4\,x^2}{6\,d^5\,{\left (d^2-e^2\,x^2\right )}^{5/2}}+\frac {7\,e^6\,x^4}{2\,d^7\,{\left (d^2-e^2\,x^2\right )}^{5/2}}-\frac {e\,\left (5\,d^6-30\,d^4\,e^2\,x^2+40\,d^2\,e^4\,x^4-16\,e^6\,x^6\right )}{5\,d^8\,x\,{\left (d^2-e^2\,x^2\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(x^3*(d^2 - e^2*x^2)^(7/2)),x)

[Out]

(161*e^2)/(30*d^3*(d^2 - e^2*x^2)^(5/2)) - 1/(2*d*x^2*(d^2 - e^2*x^2)^(5/2)) - (7*e^2*atanh((d^2 - e^2*x^2)^(1
/2)/d))/(2*d^8) - (49*e^4*x^2)/(6*d^5*(d^2 - e^2*x^2)^(5/2)) + (7*e^6*x^4)/(2*d^7*(d^2 - e^2*x^2)^(5/2)) - (e*
(5*d^6 - 16*e^6*x^6 - 30*d^4*e^2*x^2 + 40*d^2*e^4*x^4))/(5*d^8*x*(d^2 - e^2*x^2)^(5/2))

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