3.1.79 \(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^9} \, dx\) [79]

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

[Out]

1/192*e^4*(64*e*x+125*d)*(-e^2*x^2+d^2)^(3/2)/x^4-1/240*e^2*(48*e*x+125*d)*(-e^2*x^2+d^2)^(5/2)/x^6-1/8*d*(-e^
2*x^2+d^2)^(7/2)/x^8-3/7*e*(-e^2*x^2+d^2)^(7/2)/x^7-e^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+125/128*e^8*arctanh((
-e^2*x^2+d^2)^(1/2)/d)-1/128*e^6*(128*e*x+125*d)*(-e^2*x^2+d^2)^(1/2)/x^2

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Rubi [A]
time = 0.18, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 = {1821, 825, 858, 223, 209, 272, 65, 214} \begin {gather*} e^8 \left (-\text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-\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 {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}+\frac {125}{128} e^8 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*(e^6*(125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/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 65

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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 223

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 272

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 825

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/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((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), 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 858

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 1821

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 {(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 ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )-e^9 \text {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 ) \text {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*}

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Mathematica [A]
time = 0.80, size = 185, normalized size = 0.91 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-1680 d^7-5760 d^6 e x-1960 d^5 e^2 x^2+14592 d^4 e^3 x^3+17710 d^3 e^4 x^4-7424 d^2 e^5 x^5-27195 d e^6 x^6-14848 e^7 x^7\right )}{13440 x^8}-\frac {125}{64} e^8 \tanh ^{-1}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )+\frac {e^9 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{\sqrt {-e^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-1680*d^7 - 5760*d^6*e*x - 1960*d^5*e^2*x^2 + 14592*d^4*e^3*x^3 + 17710*d^3*e^4*x^4 - 74
24*d^2*e^5*x^5 - 27195*d*e^6*x^6 - 14848*e^7*x^7))/(13440*x^8) - (125*e^8*ArcTanh[(Sqrt[-e^2]*x)/d - Sqrt[d^2
- e^2*x^2]/d])/64 + (e^9*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/Sqrt[-e^2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(639\) vs. \(2(178)=356\).
time = 0.08, size = 640, normalized size = 3.14

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (14848 e^{7} x^{7}+27195 d \,e^{6} x^{6}+7424 d^{2} e^{5} x^{5}-17710 d^{3} e^{4} x^{4}-14592 d^{4} e^{3} x^{3}+1960 d^{5} e^{2} x^{2}+5760 d^{6} e x +1680 d^{7}\right )}{13440 x^{8}}-\frac {e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {125 e^{8} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}\)	\(170\)
default	\(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )-\frac {3 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 x^{7}}\)	\(640\)

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,method=_RETURNVERBOSE)

[Out]

e^3*(-1/5/d^2/x^5*(-e^2*x^2+d^2)^(7/2)-2/5*e^2/d^2*(-1/3/d^2/x^3*(-e^2*x^2+d^2)^(7/2)-4/3*e^2/d^2*(-1/d^2/x*(-
e^2*x^2+d^2)^(7/2)-6*e^2/d^2*(1/6*x*(-e^2*x^2+d^2)^(5/2)+5/6*d^2*(1/4*x*(-e^2*x^2+d^2)^(3/2)+3/4*d^2*(1/2*x*(-
e^2*x^2+d^2)^(1/2)+1/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))))+d^3*(-1/8/d^2/x^8*(-e^
2*x^2+d^2)^(7/2)+1/8*e^2/d^2*(-1/6/d^2/x^6*(-e^2*x^2+d^2)^(7/2)-1/6*e^2/d^2*(-1/4/d^2/x^4*(-e^2*x^2+d^2)^(7/2)
-3/4*e^2/d^2*(-1/2/d^2/x^2*(-e^2*x^2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^2)^
(3/2)+d^2*((-e^2*x^2+d^2)^(1/2)-d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x))))))))+3*d*e^
2*(-1/6/d^2/x^6*(-e^2*x^2+d^2)^(7/2)-1/6*e^2/d^2*(-1/4/d^2/x^4*(-e^2*x^2+d^2)^(7/2)-3/4*e^2/d^2*(-1/2/d^2/x^2*
(-e^2*x^2+d^2)^(7/2)-5/2*e^2/d^2*(1/5*(-e^2*x^2+d^2)^(5/2)+d^2*(1/3*(-e^2*x^2+d^2)^(3/2)+d^2*((-e^2*x^2+d^2)^(
1/2)-d^2/(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 [A]
time = 0.50, size = 327, normalized size = 1.60 \begin {gather*} -\arcsin \left (\frac {x e}{d}\right ) e^{8} + \frac {125}{128} \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {\sqrt {-x^{2} e^{2} + d^{2}} x e^{9}}{d^{2}} - \frac {125 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{8}}{128 \, d} - \frac {2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} x e^{9}}{3 \, d^{4}} - \frac {125 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{384 \, d^{3}} - \frac {25 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}}{128 \, d^{5}} - \frac {8 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{15 \, d^{4} x} - \frac {25 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{128 \, d^{5} x^{2}} + \frac {2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{15 \, d^{4} x^{3}} + \frac {25 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{192 \, d^{3} x^{4}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{5 \, d^{2} x^{5}} - \frac {25 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{48 \, d x^{6}} - \frac {3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{7 \, x^{7}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{8 \, x^{8}} \end {gather*}

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]

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

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Fricas [A]
time = 3.65, size = 152, normalized size = 0.75 \begin {gather*} \frac {26880 \, x^{8} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) e^{8} - 13125 \, x^{8} e^{8} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (14848 \, x^{7} e^{7} + 27195 \, d x^{6} e^{6} + 7424 \, d^{2} x^{5} e^{5} - 17710 \, d^{3} x^{4} e^{4} - 14592 \, d^{4} x^{3} e^{3} + 1960 \, d^{5} x^{2} e^{2} + 5760 \, d^{6} x e + 1680 \, d^{7}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{13440 \, x^{8}} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 31.67, size = 1719, normalized size = 8.43 \begin {gather*} \text {Too large to display} \end {gather*}

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/(e**2*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/(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)/(10
5*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) + d**5*e**2*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)) - 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/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), Abs(d**2/(e**2*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/(e**2*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((-e*sqrt(d**2/(e*
*2*x**2) - 1)/(2*x) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e**
2*x**2) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x**2) + 1)) - 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/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] Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (171) = 342\).
time = 0.74, size = 529, normalized size = 2.59 \begin {gather*} -\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^{6}}{x} + \frac {1120 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{x^{2}} - \frac {3696 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{2}}{x^{3}} - \frac {560 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-2\right )}}{x^{5}} + \frac {77280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-4\right )}}{x^{6}} + \frac {122640 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-6\right )}}{x^{7}} - \frac {14280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}}{x^{4}} + 105 \, e^{8}\right )} e^{16}}{215040 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8}} + \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 ) - \frac {73 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{6}}{128 \, x} - \frac {23 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{4}}{64 \, x^{2}} + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{2}}{384 \, x^{3}} + \frac {11 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} e^{\left (-2\right )}}{640 \, x^{5}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-4\right )}}{192 \, x^{6}} - \frac {3 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-6\right )}}{896 \, x^{7}} - \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-8\right )}}{2048 \, x^{8}} + \frac {17 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4}}{256 \, x^{4}} \end {gather*}

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^6/x + 1120*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^2*e^4/x^2 - 3696*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^2/x^3 - 560*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*e^
(-2)/x^5 + 77280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^(-4)/x^6 + 122640*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e^(-6)/
x^7 - 14280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4/x^4 + 105*e^8)*e^16/(d*e + sqrt(-x^2*e^2 + d^2)*e)^8 + 125/128*e^
8*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x)) - 73/128*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^6/x
- 23/64*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^4/x^2 + 1/384*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^2/x^3 + 11/640*(d*
e + sqrt(-x^2*e^2 + d^2)*e)^5*e^(-2)/x^5 - 1/192*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^(-4)/x^6 - 3/896*(d*e + sq
rt(-x^2*e^2 + d^2)*e)^7*e^(-6)/x^7 - 1/2048*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^(-8)/x^8 + 17/256*(d*e + sqrt(-
x^2*e^2 + d^2)*e)^4/x^4

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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