3.146 \(\int \frac{1}{x^3 (d+e x) (d^2-e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8} \]

[Out]

(7*d - 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^2*(d + e*x)*(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)

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Rubi [A]  time = 0.168233, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {857, 823, 835, 807, 266, 63, 208} \[ \frac{16 e \sqrt{d^2-e^2 x^2}}{5 d^8 x}-\frac{7 \sqrt{d^2-e^2 x^2}}{2 d^7 x^2}+\frac{35 d-24 e x}{15 d^6 x^2 \sqrt{d^2-e^2 x^2}}+\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{7 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),x]

[Out]

(7*d - 6*e*x)/(15*d^4*x^2*(d^2 - e^2*x^2)^(3/2)) + 1/(5*d^2*x^2*(d + e*x)*(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 857

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

Rule 823

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] + Di
st[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 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 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) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-7 d e^2+6 e^3 x}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac{\int \frac{-35 d^3 e^4+24 d^2 e^5 x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^4}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \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^5 e^6+48 d^4 e^7 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^{10} e^6}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \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^6 e^7+105 d^5 e^8 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{30 d^{12} e^6}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \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{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \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 ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^7}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \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 \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^7}\\ &=\frac{7 d-6 e x}{15 d^4 x^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{1}{5 d^2 x^2 (d+e x) \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*}

Mathematica [A]  time = 0.143633, size = 137, normalized size = 0.74 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (176 d^4 e^2 x^2-4 d^3 e^3 x^3-249 d^2 e^4 x^4+15 d^5 e x-15 d^6-9 d e^5 x^5+96 e^6 x^6\right )}{x^2 (d-e x)^2 (d+e x)^3}-105 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+105 e^2 \log (x)}{30 d^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(5/2)),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)^2*(d + e*x)^3) + 105*e^2*Log[x] - 105*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(30*d^8)

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Maple [A]  time = 0.065, size = 298, normalized size = 1.6 \begin{align*}{\frac{7\,{e}^{2}}{6\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{e}^{2}}{2\,{d}^{7}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}-{\frac{7\,{e}^{2}}{2\,{d}^{7}}\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{e}{5\,{d}^{4}} \left ({\frac{d}{e}}+x \right ) ^{-1} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{e}^{3}x}{15\,{d}^{6}} \left ( - \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{e}^{3}x}{15\,{d}^{8}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{1}{2\,{d}^{3}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{e}{{d}^{4}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,{e}^{3}x}{3\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,{e}^{3}x}{3\,{d}^{8}}{\frac{1}{\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)/(-e^2*x^2+d^2)^(5/2),x)

[Out]

7/6*e^2/d^5/(-e^2*x^2+d^2)^(3/2)+7/2*e^2/d^7/(-e^2*x^2+d^2)^(1/2)-7/2*e^2/d^7/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1
/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/5*e/d^4/(d/e+x)/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(3/2)-4/15*e^3/d^6/(-(d/e+x)^2*e
^2+2*d*e*(d/e+x))^(3/2)*x-8/15*e^3/d^8/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-1/2/d^3/x^2/(-e^2*x^2+d^2)^(3/2)
+e/d^4/x/(-e^2*x^2+d^2)^(3/2)-4/3*e^3/d^6*x/(-e^2*x^2+d^2)^(3/2)-8/3*e^3/d^8*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{5}{2}}{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(5/2)*(e*x + d)*x^3), x)

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Fricas [A]  time = 2.17023, size = 595, normalized size = 3.2 \begin{align*} \frac{116 \, e^{7} x^{7} + 116 \, d e^{6} x^{6} - 232 \, d^{2} e^{5} x^{5} - 232 \, d^{3} e^{4} x^{4} + 116 \, d^{4} e^{3} x^{3} + 116 \, d^{5} e^{2} x^{2} + 105 \,{\left (e^{7} x^{7} + d e^{6} x^{6} - 2 \, d^{2} e^{5} x^{5} - 2 \, d^{3} e^{4} x^{4} + d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (96 \, e^{6} x^{6} - 9 \, d e^{5} x^{5} - 249 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 176 \, d^{4} e^{2} x^{2} + 15 \, d^{5} e x - 15 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{30 \,{\left (d^{8} e^{5} x^{7} + d^{9} e^{4} x^{6} - 2 \, d^{10} e^{3} x^{5} - 2 \, d^{11} e^{2} x^{4} + d^{12} e x^{3} + d^{13} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

1/30*(116*e^7*x^7 + 116*d*e^6*x^6 - 232*d^2*e^5*x^5 - 232*d^3*e^4*x^4 + 116*d^4*e^3*x^3 + 116*d^5*e^2*x^2 + 10
5*(e^7*x^7 + d*e^6*x^6 - 2*d^2*e^5*x^5 - 2*d^3*e^4*x^4 + d^4*e^3*x^3 + d^5*e^2*x^2)*log(-(d - sqrt(-e^2*x^2 +
d^2))/x) + (96*e^6*x^6 - 9*d*e^5*x^5 - 249*d^2*e^4*x^4 - 4*d^3*e^3*x^3 + 176*d^4*e^2*x^2 + 15*d^5*e*x - 15*d^6
)*sqrt(-e^2*x^2 + d^2))/(d^8*e^5*x^7 + d^9*e^4*x^6 - 2*d^10*e^3*x^5 - 2*d^11*e^2*x^4 + d^12*e*x^3 + d^13*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{5}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(e*x+d)/(-e**2*x**2+d**2)**(5/2),x)

[Out]

Integral(1/(x**3*(-(-d + e*x)*(d + e*x))**(5/2)*(d + e*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]