3.197 \(\int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)^4} \, dx\)

Optimal. Leaf size=210 \[ -\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

-8/5*e^3*(-e*x+d)/d^2/(-e^2*x^2+d^2)^(5/2)-4/5*e^3*(-6*e*x+5*d)/d^4/(-e^2*x^2+d^2)^(3/2)+18*e^3*arctanh((-e^2*
x^2+d^2)^(1/2)/d)/d^6-1/5*e^3*(-93*e*x+80*d)/d^6/(-e^2*x^2+d^2)^(1/2)-1/3*(-e^2*x^2+d^2)^(1/2)/d^4/x^3+2*e*(-e
^2*x^2+d^2)^(1/2)/d^5/x^2-29/3*e^2*(-e^2*x^2+d^2)^(1/2)/d^6/x

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Rubi [A]  time = 0.49, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {852, 1805, 1807, 807, 266, 63, 208} \[ -\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*e^3*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) - (4*e^3*(5*d - 6*e*x))/(5*d^4*(d^2 - e^2*x^2)^(3/2)) - (e^3*
(80*d - 93*e*x))/(5*d^6*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(3*d^4*x^3) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^
5*x^2) - (29*e^2*Sqrt[d^2 - e^2*x^2])/(3*d^6*x) + (18*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^6

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 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 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

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 {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)^4} \, dx &=\int \frac {(d-e x)^4}{x^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^4+20 d^3 e x-35 d^2 e^2 x^2+40 d e^3 x^3-32 e^4 x^4}{x^4 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^4-60 d^3 e x+120 d^2 e^2 x^2-180 d e^3 x^3+144 e^4 x^4}{x^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^4+60 d^3 e x-135 d^2 e^2 x^2+240 d e^3 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {\int \frac {-180 d^5 e+435 d^4 e^2 x-720 d^3 e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{45 d^8}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {\int \frac {-870 d^6 e^2+1620 d^5 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{90 d^{10}}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {\left (18 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {\left (9 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {(18 e) \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 )}{d^5}\\ &=-\frac {8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {e^3 (80 d-93 e x)}{5 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 d^4 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d^5 x^2}-\frac {29 e^2 \sqrt {d^2-e^2 x^2}}{3 d^6 x}+\frac {18 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^6}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 118, normalized size = 0.56 \[ -\frac {-270 e^3 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\frac {\sqrt {d^2-e^2 x^2} \left (5 d^5-15 d^4 e x+70 d^3 e^2 x^2+674 d^2 e^3 x^3+1002 d e^4 x^4+424 e^5 x^5\right )}{x^3 (d+e x)^3}+270 e^3 \log (x)}{15 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*((Sqrt[d^2 - e^2*x^2]*(5*d^5 - 15*d^4*e*x + 70*d^3*e^2*x^2 + 674*d^2*e^3*x^3 + 1002*d*e^4*x^4 + 424*e^5*
x^5))/(x^3*(d + e*x)^3) + 270*e^3*Log[x] - 270*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/d^6

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fricas [A]  time = 0.82, size = 213, normalized size = 1.01 \[ -\frac {324 \, e^{6} x^{6} + 972 \, d e^{5} x^{5} + 972 \, d^{2} e^{4} x^{4} + 324 \, d^{3} e^{3} x^{3} + 270 \, {\left (e^{6} x^{6} + 3 \, d e^{5} x^{5} + 3 \, d^{2} e^{4} x^{4} + d^{3} e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (424 \, e^{5} x^{5} + 1002 \, d e^{4} x^{4} + 674 \, d^{2} e^{3} x^{3} + 70 \, d^{3} e^{2} x^{2} - 15 \, d^{4} e x + 5 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{6} e^{3} x^{6} + 3 \, d^{7} e^{2} x^{5} + 3 \, d^{8} e x^{4} + d^{9} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(324*e^6*x^6 + 972*d*e^5*x^5 + 972*d^2*e^4*x^4 + 324*d^3*e^3*x^3 + 270*(e^6*x^6 + 3*d*e^5*x^5 + 3*d^2*e^
4*x^4 + d^3*e^3*x^3)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (424*e^5*x^5 + 1002*d*e^4*x^4 + 674*d^2*e^3*x^3 + 70
*d^3*e^2*x^2 - 15*d^4*e*x + 5*d^5)*sqrt(-e^2*x^2 + d^2))/(d^6*e^3*x^6 + 3*d^7*e^2*x^5 + 3*d^8*e*x^4 + d^9*x^3)

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giac [A]  time = 0.37, size = 1, normalized size = 0.00 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity

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maple [B]  time = 0.02, size = 412, normalized size = 1.96 \[ \frac {18 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}\, d^{5}}+\frac {10 e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, d^{6}}-\frac {10 e^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, d^{6}}-\frac {10 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4} x}{d^{8}}-\frac {18 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}{d^{7}}+\frac {10 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{3}}{d^{7}}-\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{5 \left (x +\frac {d}{e}\right )^{4} d^{5} e}-\frac {7 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}}}{5 \left (x +\frac {d}{e}\right )^{3} d^{6}}-\frac {10 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e}{\left (x +\frac {d}{e}\right )^{2} d^{7}}-\frac {10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}{d^{8} x}+\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e}{d^{7} x^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/d^5/e/(x+d/e)^4*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)-7/5/d^6/(x+d/e)^3*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)
-10/d^8*e^2/x*(-e^2*x^2+d^2)^(3/2)-10/d^8*e^4*x*(-e^2*x^2+d^2)^(1/2)-10/d^6*e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)
/(-e^2*x^2+d^2)^(1/2)*x)+2/d^7*e/x^2*(-e^2*x^2+d^2)^(3/2)-18/d^7*e^3*(-e^2*x^2+d^2)^(1/2)+18/d^5*e^3/(d^2)^(1/
2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/3/d^6/x^3*(-e^2*x^2+d^2)^(3/2)+10/d^7*e^3*(2*(x+d/e)*d*e
-(x+d/e)^2*e^2)^(1/2)+10/d^6*e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x)-10/d^7*
e/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {d^2-e^2\,x^2}}{x^4\,{\left (d+e\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x**4*(d + e*x)**4), x)

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