3.178 \(\int \frac{1}{x^3 (d+e x)^2 (d^2-e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=183 \[ \frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{9 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^7} \]

[Out]

(2*e^2*(d - e*x))/(5*d^3*(d^2 - e^2*x^2)^(5/2)) + (e^2*(5*d - 6*e*x))/(5*d^5*(d^2 - e^2*x^2)^(3/2)) + (2*e^2*(
10*d - 11*e*x))/(5*d^7*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(2*d^6*x^2) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^7
*x) - (9*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^7)

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Rubi [A]  time = 0.374615, antiderivative size = 183, normalized size of antiderivative = 1., 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 = {852, 1805, 1807, 807, 266, 63, 208} \[ \frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}-\frac{9 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

(2*e^2*(d - e*x))/(5*d^3*(d^2 - e^2*x^2)^(5/2)) + (e^2*(5*d - 6*e*x))/(5*d^5*(d^2 - e^2*x^2)^(3/2)) + (2*e^2*(
10*d - 11*e*x))/(5*d^7*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(2*d^6*x^2) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^7
*x) - (9*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^7)

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])

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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{x^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\int \frac{(d-e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{\int \frac{-5 d^2+10 d e x-10 e^2 x^2+\frac{8 e^3 x^3}{d}}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{15 d^2-30 d e x+45 e^2 x^2-\frac{36 e^3 x^3}{d}}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-15 d^2+30 d e x-60 e^2 x^2}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{\int \frac{-60 d^3 e+135 d^2 e^2 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^8}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}+\frac{\left (9 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^6}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^6}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{9 \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 )}{2 d^6}\\ &=\frac{2 e^2 (d-e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{e^2 (5 d-6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{2 e^2 (10 d-11 e x)}{5 d^7 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{2 d^6 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^7 x}-\frac{9 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^7}\\ \end{align*}

Mathematica [A]  time = 0.148093, size = 127, normalized size = 0.69 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-94 d^3 e^2 x^2-58 d^2 e^3 x^3-10 d^4 e x+5 d^5+83 d e^4 x^4+64 e^5 x^5\right )}{x^2 (e x-d) (d+e x)^3}-45 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+45 e^2 \log (x)}{10 d^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(5*d^5 - 10*d^4*e*x - 94*d^3*e^2*x^2 - 58*d^2*e^3*x^3 + 83*d*e^4*x^4 + 64*e^5*x^5))/(x^2
*(-d + e*x)*(d + e*x)^3) + 45*e^2*Log[x] - 45*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(10*d^7)

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Maple [A]  time = 0.068, size = 259, normalized size = 1.4 \begin{align*}{\frac{9\,{e}^{2}}{2\,{d}^{6}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{9\,{e}^{2}}{2\,{d}^{6}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{6\,e}{5\,{d}^{5}} \left ({\frac{d}{e}}+x \right ) ^{-1}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{12\,{e}^{3}x}{5\,{d}^{7}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}+{\frac{1}{5\,{d}^{4}} \left ({\frac{d}{e}}+x \right ) ^{-2}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{1}{2\,{d}^{4}{x}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+2\,{\frac{e}{{d}^{5}x\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}-4\,{\frac{{e}^{3}x}{{d}^{7}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x)

[Out]

9/2*e^2/d^6/(-e^2*x^2+d^2)^(1/2)-9/2*e^2/d^6/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+6/5/
d^5*e/(d/e+x)/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)-12/5/d^7*e^3/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x+1/5/d^4
/(d/e+x)^2/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)-1/2/d^4/x^2/(-e^2*x^2+d^2)^(1/2)+2/d^5*e/x/(-e^2*x^2+d^2)^(1/2
)-4/d^7*e^3*x/(-e^2*x^2+d^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)^2*x^3), x)

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Fricas [A]  time = 1.75807, size = 440, normalized size = 2.4 \begin{align*} \frac{54 \, e^{6} x^{6} + 108 \, d e^{5} x^{5} - 108 \, d^{3} e^{3} x^{3} - 54 \, d^{4} e^{2} x^{2} + 45 \,{\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - 2 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (64 \, e^{5} x^{5} + 83 \, d e^{4} x^{4} - 58 \, d^{2} e^{3} x^{3} - 94 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x + 5 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (d^{7} e^{4} x^{6} + 2 \, d^{8} e^{3} x^{5} - 2 \, d^{10} e x^{3} - d^{11} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")

[Out]

1/10*(54*e^6*x^6 + 108*d*e^5*x^5 - 108*d^3*e^3*x^3 - 54*d^4*e^2*x^2 + 45*(e^6*x^6 + 2*d*e^5*x^5 - 2*d^3*e^3*x^
3 - d^4*e^2*x^2)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (64*e^5*x^5 + 83*d*e^4*x^4 - 58*d^2*e^3*x^3 - 94*d^3*e^2
*x^2 - 10*d^4*e*x + 5*d^5)*sqrt(-e^2*x^2 + d^2))/(d^7*e^4*x^6 + 2*d^8*e^3*x^5 - 2*d^10*e*x^3 - d^11*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(e*x+d)**2/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral(1/(x**3*(-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**2), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError