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

Optimal. Leaf size=169 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3} \]

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Rubi [A]  time = 0.21, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {852, 1807, 835, 807, 266, 47, 63, 208} \begin {gather*} -\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 852

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

Rule 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps

\begin {align*} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^7} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}-\frac {\int \frac {\left (12 d^3 e-9 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^6} \, dx}{6 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}+\frac {\int \frac {\left (45 d^4 e^2-24 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{30 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}-\frac {\int \frac {\left (96 d^5 e^3-45 d^4 e^4 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{120 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {\left (3 e^4\right ) \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{8 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {\left (3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{16 d^2}\\ &=-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}-\frac {\left (3 e^6\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{32 d^2}\\ &=-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {\left (3 e^4\right ) \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^2}\\ &=-\frac {3 e^4 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac {2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac {3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}+\frac {4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3}+\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 117, normalized size = 0.69 \begin {gather*} -\frac {-45 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^5-96 d^4 e x+50 d^3 e^2 x^2+32 d^2 e^3 x^3-45 d e^4 x^4+64 e^5 x^5\right )+45 e^6 x^6 \log (x)}{240 d^3 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/240*(Sqrt[d^2 - e^2*x^2]*(40*d^5 - 96*d^4*e*x + 50*d^3*e^2*x^2 + 32*d^2*e^3*x^3 - 45*d*e^4*x^4 + 64*e^5*x^5
) + 45*e^6*x^6*Log[x] - 45*e^6*x^6*Log[d + Sqrt[d^2 - e^2*x^2]])/(d^3*x^6)

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IntegrateAlgebraic [A]  time = 0.75, size = 126, normalized size = 0.75 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-40 d^5+96 d^4 e x-50 d^3 e^2 x^2-32 d^2 e^3 x^3+45 d e^4 x^4-64 e^5 x^5\right )}{240 d^3 x^6}-\frac {3 e^6 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-40*d^5 + 96*d^4*e*x - 50*d^3*e^2*x^2 - 32*d^2*e^3*x^3 + 45*d*e^4*x^4 - 64*e^5*x^5))/(24
0*d^3*x^6) - (3*e^6*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d])/(8*d^3)

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fricas [A]  time = 0.42, size = 108, normalized size = 0.64 \begin {gather*} -\frac {45 \, e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (64 \, e^{5} x^{5} - 45 \, d e^{4} x^{4} + 32 \, d^{2} e^{3} x^{3} + 50 \, d^{3} e^{2} x^{2} - 96 \, d^{4} e x + 40 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, d^{3} x^{6}} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

-1/240*(45*e^6*x^6*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (64*e^5*x^5 - 45*d*e^4*x^4 + 32*d^2*e^3*x^3 + 50*d^3*e
^2*x^2 - 96*d^4*e*x + 40*d^5)*sqrt(-e^2*x^2 + d^2))/(d^3*x^6)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluat
ion time: 1.04Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.02, size = 566, normalized size = 3.35 \begin {gather*} \frac {3 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^{2}}-\frac {13 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 )}{4 \sqrt {e^{2}}\, d^{3}}+\frac {13 e^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 \sqrt {e^{2}}\, d^{3}}-\frac {13 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{7} x}{4 d^{5}}+\frac {13 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{7} x}{4 d^{5}}-\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6}}{16 d^{4}}-\frac {13 \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}{6 d^{7}}+\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7} x}{6 d^{7}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{6}}{16 d^{6}}+\frac {26 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7} x}{15 d^{9}}-\frac {26 \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}}{15 d^{8}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}{80 d^{8}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} e^{4}}{3 \left (x +\frac {d}{e}\right )^{2} d^{8}}+\frac {26 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{15 d^{9} x}-\frac {23 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{16 d^{8} x^{2}}+\frac {16 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{15 d^{7} x^{3}}-\frac {17 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{24 d^{6} x^{4}}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{5 d^{5} x^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{4} x^{6}} \end {gather*}

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)^2,x)

[Out]

-13/6/d^7*e^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x-13/4/d^5*e^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x-13/4/d^
3*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)-1/3/d^8*e^4/(x+d/e)^2*(2*(x+d/e)*d
*e-(x+d/e)^2*e^2)^(7/2)+2/5/d^5*e/x^5*(-e^2*x^2+d^2)^(7/2)+16/15/d^7*e^3/x^3*(-e^2*x^2+d^2)^(7/2)+13/4/d^5*e^7
*x*(-e^2*x^2+d^2)^(1/2)+13/4/d^3*e^7/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-e^2*x^2+d^2)^(1/2)*x)+26/15/d^9*e^5/x*(-
e^2*x^2+d^2)^(7/2)+26/15/d^9*e^7*x*(-e^2*x^2+d^2)^(5/2)+13/6/d^7*e^7*x*(-e^2*x^2+d^2)^(3/2)-17/24/d^6*e^2/x^4*
(-e^2*x^2+d^2)^(7/2)-23/16/d^8*e^4/x^2*(-e^2*x^2+d^2)^(7/2)+3/16/d^2*e^6/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(
-e^2*x^2+d^2)^(1/2))/x)-26/15/d^8*e^6*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2)-1/6/d^4/x^6*(-e^2*x^2+d^2)^(7/2)-3/8
0/d^8*e^6*(-e^2*x^2+d^2)^(5/2)-1/16/d^6*e^6*(-e^2*x^2+d^2)^(3/2)-3/16/d^4*e^6*(-e^2*x^2+d^2)^(1/2)

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maxima [A]  time = 1.01, size = 180, normalized size = 1.07 \begin {gather*} \frac {3 \, 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^{3}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}}{16 \, d^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{16 \, d^{4} x^{2}} + \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{15 \, d^{3} x^{3}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{8 \, d^{2} x^{4}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{5 \, d x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

3/16*e^6*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 - 3/16*sqrt(-e^2*x^2 + d^2)*e^6/d^4 - 3/16*(-
e^2*x^2 + d^2)^(3/2)*e^4/(d^4*x^2) + 4/15*(-e^2*x^2 + d^2)^(3/2)*e^3/(d^3*x^3) - 3/8*(-e^2*x^2 + d^2)^(3/2)*e^
2/(d^2*x^4) + 2/5*(-e^2*x^2 + d^2)^(3/2)*e/(d*x^5) - 1/6*(-e^2*x^2 + d^2)^(3/2)/x^6

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

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)^2),x)

[Out]

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

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sympy [C]  time = 19.72, size = 808, normalized size = 4.78 \begin {gather*} d^{2} \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 ) - 2 d 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 ) + 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 ) \end {gather*}

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)**2,x)

[Out]

d**2*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)) - 2*d*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)) + 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))

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