3.79 \(\int \frac{(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=204 \[ -\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+e^8 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{125}{128} e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

-(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (e^4*(125*d + 64*e*x)*(d^2 - e^2*x^2)^(3/2))/(192*x^4
) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e
^2*x^2)^(7/2))/(7*x^7) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (125*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/128

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Rubi [A]  time = 0.303815, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {1807, 811, 844, 217, 203, 266, 63, 208} \[ -\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}+e^8 \left (-\tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )\right )+\frac{125}{128} e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

-(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/(128*x^2) + (e^4*(125*d + 64*e*x)*(d^2 - e^2*x^2)^(3/2))/(192*x^4
) - (e^2*(125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/(240*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(8*x^8) - (3*e*(d^2 - e
^2*x^2)^(7/2))/(7*x^7) - e^8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] + (125*e^8*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/128

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 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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{(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{5/2} \left (-24 d^4 e-25 d^3 e^2 x-8 d^2 e^3 x^2\right )}{x^8} \, dx}{8 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac{\int \frac{\left (175 d^5 e^2+56 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{56 d^4}\\ &=-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{\int \frac{\left (1750 d^7 e^4+672 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx}{672 d^6}\\ &=\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac{\int \frac{\left (10500 d^9 e^6+5376 d^8 e^7 x\right ) \sqrt{d^2-e^2 x^2}}{x^3} \, dx}{5376 d^8}\\ &=-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{\int \frac{21000 d^{11} e^8+21504 d^{10} e^9 x}{x \sqrt{d^2-e^2 x^2}} \, dx}{21504 d^{10}}\\ &=-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{1}{128} \left (125 d e^8\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx-e^9 \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-\frac{1}{256} \left (125 d e^8\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )-e^9 \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{128} \left (125 d e^6\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 )\\ &=-\frac{e^6 (125 d+128 e x) \sqrt{d^2-e^2 x^2}}{128 x^2}+\frac{e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac{e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{125}{128} e^8 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )\\ \end{align*}

Mathematica [C]  time = 0.165984, size = 245, normalized size = 1.2 \[ -\frac{e^8 \left (d^2-e^2 x^2\right )^{7/2} \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )}{7 d^7}-\frac{d^4 e^3 \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{5 x^5 \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}+\frac{-59 d^3 e^6 x^4+34 d^5 e^4 x^2+15 d e^8 x^6 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )-8 d^7 e^2+33 d e^8 x^6}{16 x^6 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^9,x]

[Out]

(-3*e*(d^2 - e^2*x^2)^(7/2))/(7*x^7) + (-8*d^7*e^2 + 34*d^5*e^4*x^2 - 59*d^3*e^6*x^4 + 33*d*e^8*x^6 + 15*d*e^8
*x^6*Sqrt[1 - (e^2*x^2)/d^2]*ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]])/(16*x^6*Sqrt[d^2 - e^2*x^2]) - (d^4*e^3*Sqrt[d^
2 - e^2*x^2]*Hypergeometric2F1[-5/2, -5/2, -3/2, (e^2*x^2)/d^2])/(5*x^5*Sqrt[1 - (e^2*x^2)/d^2]) - (e^8*(d^2 -
e^2*x^2)^(7/2)*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2])/(7*d^7)

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Maple [B]  time = 0.176, size = 402, normalized size = 2. \begin{align*} -{\frac{{e}^{3}}{5\,{d}^{2}{x}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{2\,{e}^{5}}{15\,{d}^{4}{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{7}}{15\,{d}^{6}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{e}^{9}x}{15\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{e}^{9}x}{3\,{d}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{9}x}{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{{e}^{9}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d}{8\,{x}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{25\,{e}^{2}}{48\,d{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{25\,{e}^{4}}{192\,{d}^{3}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{25\,{e}^{6}}{128\,{d}^{5}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{25\,{e}^{8}}{128\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{125\,{e}^{8}}{384\,{d}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{125\,{e}^{8}}{128\,d}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{125\,d{e}^{8}}{128}\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{3\,e}{7\,{x}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x)

[Out]

-1/5*e^3/d^2/x^5*(-e^2*x^2+d^2)^(7/2)+2/15*e^5/d^4/x^3*(-e^2*x^2+d^2)^(7/2)-8/15*e^7/d^6/x*(-e^2*x^2+d^2)^(7/2
)-8/15*e^9/d^6*x*(-e^2*x^2+d^2)^(5/2)-2/3*e^9/d^4*x*(-e^2*x^2+d^2)^(3/2)-e^9/d^2*x*(-e^2*x^2+d^2)^(1/2)-e^9/(e
^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/8*d*(-e^2*x^2+d^2)^(7/2)/x^8-25/48/d*e^2/x^6*(-e^2*x^2+
d^2)^(7/2)+25/192/d^3*e^4/x^4*(-e^2*x^2+d^2)^(7/2)-25/128/d^5*e^6/x^2*(-e^2*x^2+d^2)^(7/2)-25/128/d^5*e^8*(-e^
2*x^2+d^2)^(5/2)-125/384/d^3*e^8*(-e^2*x^2+d^2)^(3/2)-125/128/d*e^8*(-e^2*x^2+d^2)^(1/2)+125/128*d*e^8/(d^2)^(
1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-3/7*e*(-e^2*x^2+d^2)^(7/2)/x^7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.27045, size = 382, normalized size = 1.87 \begin{align*} \frac{26880 \, e^{8} x^{8} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13125 \, e^{8} x^{8} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (14848 \, e^{7} x^{7} + 27195 \, d e^{6} x^{6} + 7424 \, d^{2} e^{5} x^{5} - 17710 \, d^{3} e^{4} x^{4} - 14592 \, d^{4} e^{3} x^{3} + 1960 \, d^{5} e^{2} x^{2} + 5760 \, d^{6} e x + 1680 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{13440 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="fricas")

[Out]

1/13440*(26880*e^8*x^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 13125*e^8*x^8*log(-(d - sqrt(-e^2*x^2 + d^2
))/x) - (14848*e^7*x^7 + 27195*d*e^6*x^6 + 7424*d^2*e^5*x^5 - 17710*d^3*e^4*x^4 - 14592*d^4*e^3*x^3 + 1960*d^5
*e^2*x^2 + 5760*d^6*e*x + 1680*d^7)*sqrt(-e^2*x^2 + d^2))/x^8

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Sympy [C]  time = 34.5318, size = 1742, normalized size = 8.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(-e**2*x**2+d**2)**(5/2)/x**9,x)

[Out]

d**7*Piecewise((-d**2/(8*e*x**9*sqrt(d**2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/
(192*d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(384*d**4*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 5*e**7/(128*d
**6*x*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (I
*d**2/(8*e*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*
x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d**4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*
x*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) + 3*d**6*e*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)/(Abs(e**2)*Abs(x**2)) > 1), (-I*e*s
qrt(-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)) + d**5*e**2*Piecewi
se((-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), A
bs(d**2)/(Abs(e**2)*Abs(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)) - 5*d**4*e**3*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)/Abs(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)) - 5*d**3*e**4*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), Ab
s(d**2)/(Abs(e**2)*Abs(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)) + d*
*2*e**5*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**2), Abs(d**2
)/(Abs(e**2)*Abs(x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/
(3*d**2), True)) + 3*d*e**6*Piecewise((-d**2/(2*e*x**3*sqrt(d**2/(e**2*x**2) - 1)) + e/(2*x*sqrt(d**2/(e**2*x*
*2) - 1)) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2)/(Abs(e**2)*Abs(x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)
/(2*x) - I*e**2*asin(d/(e*x))/(2*d), True)) + e**7*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*
x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2)/Abs(d**2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2))
- e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True))

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Giac [B]  time = 1.27234, size = 726, normalized size = 3.56 \begin{align*} -\arcsin \left (\frac{x e}{d}\right ) e^{8} \mathrm{sgn}\left (d\right ) + \frac{x^{8}{\left (\frac{720 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{16}}{x} + \frac{1120 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{14}}{x^{2}} - \frac{3696 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{12}}{x^{3}} - \frac{14280 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{10}}{x^{4}} - \frac{560 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{8}}{x^{5}} + \frac{77280 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{6}}{x^{6}} + \frac{122640 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{4}}{x^{7}} + 105 \, e^{18}\right )} e^{6}}{215040 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{8}} - \frac{1}{215040} \,{\left (\frac{122640 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{86}}{x} + \frac{77280 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{84}}{x^{2}} - \frac{560 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{82}}{x^{3}} - \frac{14280 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{80}}{x^{4}} - \frac{3696 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{78}}{x^{5}} + \frac{1120 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{76}}{x^{6}} + \frac{720 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{74}}{x^{7}} + \frac{105 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{72}}{x^{8}}\right )} e^{\left (-80\right )} + \frac{125}{128} \, e^{8} \log \left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(-e^2*x^2+d^2)^(5/2)/x^9,x, algorithm="giac")

[Out]

-arcsin(x*e/d)*e^8*sgn(d) + 1/215040*x^8*(720*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^16/x + 1120*(d*e + sqrt(-x^2*e^
2 + d^2)*e)^2*e^14/x^2 - 3696*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^12/x^3 - 14280*(d*e + sqrt(-x^2*e^2 + d^2)*e)
^4*e^10/x^4 - 560*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^8/x^5 + 77280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^6/x^6 +
122640*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e^4/x^7 + 105*e^18)*e^6/(d*e + sqrt(-x^2*e^2 + d^2)*e)^8 - 1/215040*(1
22640*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^86/x + 77280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^84/x^2 - 560*(d*e + sqr
t(-x^2*e^2 + d^2)*e)^3*e^82/x^3 - 14280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^80/x^4 - 3696*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^5*e^78/x^5 + 1120*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^76/x^6 + 720*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e
^74/x^7 + 105*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^72/x^8)*e^(-80) + 125/128*e^8*log(1/2*abs(-2*d*e - 2*sqrt(-x^
2*e^2 + d^2)*e)*e^(-2)/abs(x))