3.1.13 \(\int \frac {(d+e x) (d^2-e^2 x^2)^{3/2}}{x^7} \, dx\) [13]

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

[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

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Rubi [A]
time = 0.06, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {849, 821, 272, 43, 65, 214} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^7,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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 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) \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 \text {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 \text {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 \text {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 \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 )}{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*}

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Mathematica [A]
time = 0.41, size = 126, normalized size = 0.88 \begin {gather*} \frac {\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 )}{240 d^2 x^6}+\frac {e^6 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(d^2 - e^2*x^2)^(3/2))/x^7,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))/(24
0*d^2*x^6) + (e^6*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d])/(8*d^2)

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Maple [A]
time = 0.06, size = 198, normalized size = 1.38

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (48 e^{5} x^{5}+15 d \,e^{4} x^{4}-96 d^{2} e^{3} x^{3}-70 x^{2} d^{3} e^{2}+48 d^{4} x e +40 d^{5}\right )}{240 x^{6} d^{2}}-\frac {e^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 d \sqrt {d^{2}}}\)	\(121\)
default	\(-\frac {e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5 d^{2} x^{5}}+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d^{2} x^{6}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*e*(-e^2*x^2+d^2)^(5/2)/d^2/x^5+d*(-1/6/d^2/x^6*(-e^2*x^2+d^2)^(5/2)+1/6*e^2/d^2*(-1/4/d^2/x^4*(-e^2*x^2+d
^2)^(5/2)-1/4*e^2/d^2*(-1/2/d^2/x^2*(-e^2*x^2+d^2)^(5/2)-3/2*e^2/d^2*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+
d^2)^(1/2)-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 [A]
time = 0.48, size = 169, normalized size = 1.18 \begin {gather*} -\frac {e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{2}} + \frac {\sqrt {-x^{2} e^{2} + d^{2}} e^{6}}{16 \, d^{3}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}}{48 \, d^{5}} + \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}}{48 \, d^{5} x^{2}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{24 \, d^{3} x^{4}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e}{5 \, d^{2} x^{5}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}}{6 \, d x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*e^6*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x))/d^2 + 1/16*sqrt(-x^2*e^2 + d^2)*e^6/d^3 + 1/48*(
-x^2*e^2 + d^2)^(3/2)*e^6/d^5 + 1/48*(-x^2*e^2 + d^2)^(5/2)*e^4/(d^5*x^2) - 1/24*(-x^2*e^2 + d^2)^(5/2)*e^2/(d
^3*x^4) - 1/5*(-x^2*e^2 + d^2)^(5/2)*e/(d^2*x^5) - 1/6*(-x^2*e^2 + d^2)^(5/2)/(d*x^6)

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Fricas [A]
time = 2.36, size = 103, normalized size = 0.72 \begin {gather*} \frac {15 \, x^{6} e^{6} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (48 \, x^{5} e^{5} + 15 \, d x^{4} e^{4} - 96 \, d^{2} x^{3} e^{3} - 70 \, d^{3} x^{2} e^{2} + 48 \, d^{4} x e + 40 \, d^{5}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{240 \, d^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 8.96, size = 918, normalized size = 6.42 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (116) = 232\).
time = 1.01, size = 425, normalized size = 2.97 \begin {gather*} \frac {x^{6} {\left (\frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{4}}{x} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{2}}{x^{2}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{\left (-2\right )}}{x^{4}} + \frac {120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-4\right )}}{x^{5}} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3}}{x^{3}} + 5 \, e^{6}\right )} e^{12}}{1920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{2}} - \frac {e^{6} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \, {\left | x \right |}}\right )}{16 \, d^{2}} - \frac {\frac {120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{10} e^{4}}{x} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{10} e^{2}}{x^{2}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{10} e^{\left (-2\right )}}{x^{4}} + \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{10} e^{\left (-4\right )}}{x^{5}} + \frac {5 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{10} e^{\left (-6\right )}}{x^{6}} - \frac {60 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{10}}{x^{3}}}{1920 \, d^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*x^6*(12*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^4/x - 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^2/x^2 - 15*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^4*e^(-2)/x^4 + 120*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^(-4)/x^5 - 60*(d*e + sqrt(-x^2*e
^2 + d^2)*e)^3/x^3 + 5*e^6)*e^12/((d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^2) - 1/16*e^6*log(1/2*abs(-2*d*e - 2*sqrt
(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^2 - 1/1920*(120*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^10*e^4/x - 15*(d*e + sqr
t(-x^2*e^2 + d^2)*e)^2*d^10*e^2/x^2 - 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^10*e^(-2)/x^4 + 12*(d*e + sqrt(-x^
2*e^2 + d^2)*e)^5*d^10*e^(-4)/x^5 + 5*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^10*e^(-6)/x^6 - 60*(d*e + sqrt(-x^2*e
^2 + d^2)*e)^3*d^10/x^3)/d^12

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Mupad [B]
time = 4.66, size = 118, normalized size = 0.83 \begin {gather*} \frac {d^3\,\sqrt {d^2-e^2\,x^2}}{16\,x^6}-\frac {d\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{6\,x^6}-\frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{16\,d\,x^6}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{5\,d^2\,x^5}+\frac {e^6\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,1{}\mathrm {i}}{16\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^3*(d^2 - e^2*x^2)^(1/2))/(16*x^6) - (d*(d^2 - e^2*x^2)^(3/2))/(6*x^6) - (d^2 - e^2*x^2)^(5/2)/(16*d*x^6) +
(e^6*atan(((d^2 - e^2*x^2)^(1/2)*1i)/d)*1i)/(16*d^2) - (e*(d^2 - e^2*x^2)^(5/2))/(5*d^2*x^5)

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