3.2.70 \(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^8 (d+e x)^2} \, dx\)

Optimal. Leaf size=198 \[ -\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4} \]

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Rubi [A]  time = 0.24, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

(e^5*Sqrt[d^2 - e^2*x^2])/(8*d^3*x^2) - (d^2 - e^2*x^2)^(3/2)/(7*x^7) + (e*(d^2 - e^2*x^2)^(3/2))/(3*d*x^6) -
(11*e^2*(d^2 - e^2*x^2)^(3/2))/(35*d^2*x^5) + (e^3*(d^2 - e^2*x^2)^(3/2))/(4*d^3*x^4) - (22*e^4*(d^2 - e^2*x^2
)^(3/2))/(105*d^4*x^3) - (e^7*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8*d^4)

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^8 (d+e x)^2} \, dx &=\int \frac {(d-e x)^2 \sqrt {d^2-e^2 x^2}}{x^8} \, dx\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}-\frac {\int \frac {\left (14 d^3 e-11 d^2 e^2 x\right ) \sqrt {d^2-e^2 x^2}}{x^7} \, dx}{7 d^2}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}+\frac {\int \frac {\left (66 d^4 e^2-42 d^3 e^3 x\right ) \sqrt {d^2-e^2 x^2}}{x^6} \, dx}{42 d^4}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}-\frac {\int \frac {\left (210 d^5 e^3-132 d^4 e^4 x\right ) \sqrt {d^2-e^2 x^2}}{x^5} \, dx}{210 d^6}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}+\frac {\int \frac {\left (528 d^6 e^4-210 d^5 e^5 x\right ) \sqrt {d^2-e^2 x^2}}{x^4} \, dx}{840 d^8}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \int \frac {\sqrt {d^2-e^2 x^2}}{x^3} \, dx}{4 d^3}\\ &=-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \operatorname {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{8 d^3}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}+\frac {e^7 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^5 \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 )}{8 d^3}\\ &=\frac {e^5 \sqrt {d^2-e^2 x^2}}{8 d^3 x^2}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{7 x^7}+\frac {e \left (d^2-e^2 x^2\right )^{3/2}}{3 d x^6}-\frac {11 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 x^5}+\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{4 d^3 x^4}-\frac {22 e^4 \left (d^2-e^2 x^2\right )^{3/2}}{105 d^4 x^3}-\frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 128, normalized size = 0.65 \begin {gather*} \frac {-105 e^7 x^7 \log \left (\sqrt {d^2-e^2 x^2}+d\right )+\sqrt {d^2-e^2 x^2} \left (-120 d^6+280 d^5 e x-144 d^4 e^2 x^2-70 d^3 e^3 x^3+88 d^2 e^4 x^4-105 d e^5 x^5+176 e^6 x^6\right )+105 e^7 x^7 \log (x)}{840 d^4 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-120*d^6 + 280*d^5*e*x - 144*d^4*e^2*x^2 - 70*d^3*e^3*x^3 + 88*d^2*e^4*x^4 - 105*d*e^5*x
^5 + 176*e^6*x^6) + 105*e^7*x^7*Log[x] - 105*e^7*x^7*Log[d + Sqrt[d^2 - e^2*x^2]])/(840*d^4*x^7)

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IntegrateAlgebraic [A]  time = 0.85, size = 137, normalized size = 0.69 \begin {gather*} \frac {e^7 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^4}+\frac {\sqrt {d^2-e^2 x^2} \left (-120 d^6+280 d^5 e x-144 d^4 e^2 x^2-70 d^3 e^3 x^3+88 d^2 e^4 x^4-105 d e^5 x^5+176 e^6 x^6\right )}{840 d^4 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-120*d^6 + 280*d^5*e*x - 144*d^4*e^2*x^2 - 70*d^3*e^3*x^3 + 88*d^2*e^4*x^4 - 105*d*e^5*x
^5 + 176*e^6*x^6))/(840*d^4*x^7) + (e^7*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d])/(4*d^4)

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fricas [A]  time = 0.40, size = 119, normalized size = 0.60 \begin {gather*} \frac {105 \, e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (176 \, e^{6} x^{6} - 105 \, d e^{5} x^{5} + 88 \, d^{2} e^{4} x^{4} - 70 \, d^{3} e^{3} x^{3} - 144 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x - 120 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{840 \, d^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x, algorithm="fricas")

[Out]

1/840*(105*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (176*e^6*x^6 - 105*d*e^5*x^5 + 88*d^2*e^4*x^4 - 70*d^3
*e^3*x^3 - 144*d^4*e^2*x^2 + 280*d^5*e*x - 120*d^6)*sqrt(-e^2*x^2 + d^2))/(d^4*x^7)

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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^8/(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: 1.08Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.02, size = 591, normalized size = 2.98 \begin {gather*} -\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}\, d^{3}}+\frac {29 e^{8} \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 )}{8 \sqrt {e^{2}}\, d^{4}}-\frac {29 e^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}\, d^{4}}+\frac {29 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, e^{8} x}{8 d^{6}}-\frac {29 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{8} x}{8 d^{6}}+\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, e^{7}}{8 d^{5}}+\frac {29 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} e^{8} x}{12 d^{8}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{8} x}{12 d^{8}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{7}}{24 d^{7}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{8} x}{15 d^{10}}+\frac {29 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} e^{7}}{15 d^{9}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{7}}{40 d^{9}}+\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^{5}}{3 \left (x +\frac {d}{e}\right )^{2} d^{9}}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{6}}{15 d^{10} x}+\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{5}}{8 d^{9} x^{2}}-\frac {19 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{4}}{15 d^{8} x^{3}}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{3}}{12 d^{7} x^{4}}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e^{2}}{5 d^{6} x^{5}}+\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e}{3 d^{5} x^{6}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x)

[Out]

1/3/d^9*e^5/(x+d/e)^2*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(7/2)+29/12/d^8*e^8*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(3/2)*x+
29/8/d^6*e^8*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(1/2)*x+29/8/d^4*e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)/(2*(x+d/e)*d*e-
(x+d/e)^2*e^2)^(1/2)*x)-3/5/d^6*e^2/x^5*(-e^2*x^2+d^2)^(7/2)+1/3/d^5*e/x^6*(-e^2*x^2+d^2)^(7/2)-19/15/d^8*e^4/
x^3*(-e^2*x^2+d^2)^(7/2)-29/15/d^10*e^6/x*(-e^2*x^2+d^2)^(7/2)-29/15/d^10*e^8*x*(-e^2*x^2+d^2)^(5/2)-29/12/d^8
*e^8*x*(-e^2*x^2+d^2)^(3/2)-29/8/d^6*e^8*x*(-e^2*x^2+d^2)^(1/2)-29/8/d^4*e^8/(e^2)^(1/2)*arctan((e^2)^(1/2)/(-
e^2*x^2+d^2)^(1/2)*x)+11/12/d^7*e^3/x^4*(-e^2*x^2+d^2)^(7/2)+13/8/d^9*e^5/x^2*(-e^2*x^2+d^2)^(7/2)-1/8/d^3*e^7
/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+29/15/d^9*e^7*(2*(x+d/e)*d*e-(x+d/e)^2*e^2)^(5/2
)+1/40/d^9*e^7*(-e^2*x^2+d^2)^(5/2)+1/24/d^7*e^7*(-e^2*x^2+d^2)^(3/2)+1/8/d^5*e^7*(-e^2*x^2+d^2)^(1/2)-1/7/d^4
/x^7*(-e^2*x^2+d^2)^(7/2)

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maxima [A]  time = 0.99, size = 205, normalized size = 1.04 \begin {gather*} -\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{4}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{7}}{8 \, d^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}}{8 \, d^{5} x^{2}} - \frac {22 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{105 \, d^{4} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{4 \, d^{3} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{35 \, d^{2} x^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{3 \, d x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^8/(e*x+d)^2,x, algorithm="maxima")

[Out]

-1/8*e^7*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^4 + 1/8*sqrt(-e^2*x^2 + d^2)*e^7/d^5 + 1/8*(-e^
2*x^2 + d^2)^(3/2)*e^5/(d^5*x^2) - 22/105*(-e^2*x^2 + d^2)^(3/2)*e^4/(d^4*x^3) + 1/4*(-e^2*x^2 + d^2)^(3/2)*e^
3/(d^3*x^4) - 11/35*(-e^2*x^2 + d^2)^(3/2)*e^2/(d^2*x^5) + 1/3*(-e^2*x^2 + d^2)^(3/2)*e/(d*x^6) - 1/7*(-e^2*x^
2 + d^2)^(3/2)/x^7

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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^8\,{\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^8*(d + e*x)^2),x)

[Out]

int((d^2 - e^2*x^2)^(5/2)/(x^8*(d + e*x)^2), x)

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sympy [C]  time = 18.18, size = 835, normalized size = 4.22 \begin {gather*} d^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{7 x^{6}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{35 d^{2} x^{4}} + \frac {4 e^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{4} x^{2}} + \frac {8 e^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{105 d^{6}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{7 x^{6}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{35 d^{2} x^{4}} + \frac {4 i e^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{4} x^{2}} + \frac {8 i e^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{105 d^{6}} & \text {otherwise} \end {cases}\right ) - 2 d e \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 ) + e^{2} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e**2*x**2+d**2)**(5/2)/x**8/(e*x+d)**2,x)

[Out]

d**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e*
*5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x
**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4
*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True))
- 2*d*e*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)) + e**2*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 + 1
5*d*e**2*x**7), True))

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