3.1.81 \(\int \frac {(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^{11}} \, dx\) [81]

Optimal. Leaf size=225 \[ -\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2} \]

[Out]

11/128*e^6*(-e^2*x^2+d^2)^(3/2)/d/x^4-11/160*e^4*(-e^2*x^2+d^2)^(5/2)/d/x^6-1/10*d*(-e^2*x^2+d^2)^(7/2)/x^10-1
/3*e*(-e^2*x^2+d^2)^(7/2)/x^9-33/80*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^8-5/21*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^7+33/25
6*e^10*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^2-33/256*e^8*(-e^2*x^2+d^2)^(1/2)/d/x^2

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Rubi [A]
time = 0.18, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1821, 849, 821, 272, 43, 65, 214} \begin {gather*} -\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2}-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]

[Out]

(-33*e^8*Sqrt[d^2 - e^2*x^2])/(256*d*x^2) + (11*e^6*(d^2 - e^2*x^2)^(3/2))/(128*d*x^4) - (11*e^4*(d^2 - e^2*x^
2)^(5/2))/(160*d*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (e*(d^2 - e^2*x^2)^(7/2))/(3*x^9) - (33*e^2*(d^2
- e^2*x^2)^(7/2))/(80*d*x^8) - (5*e^3*(d^2 - e^2*x^2)^(7/2))/(21*d^2*x^7) + (33*e^10*ArcTanh[Sqrt[d^2 - e^2*x
^2]/d])/(256*d^2)

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 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 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 821

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 849

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 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^{11}} \, dx &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (-30 d^4 e-33 d^3 e^2 x-10 d^2 e^3 x^2\right )}{x^{10}} \, dx}{10 d^2}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}+\frac {\int \frac {\left (297 d^5 e^2+150 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx}{90 d^4}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {\int \frac {\left (-1200 d^6 e^3-297 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{720 d^6}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^4\right ) \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{80 d}\\ &=-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^4\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )}{160 d}\\ &=-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {\left (11 e^6\right ) \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{64 d}\\ &=\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^8\right ) \text {Subst}\left (\int \frac {\sqrt {d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{256 d}\\ &=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac {\left (33 e^{10}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{512 d}\\ &=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {\left (33 e^8\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 )}{256 d}\\ &=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {33 e^{10} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 167, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \left (2688 d^9+8960 d^8 e x+3024 d^7 e^2 x^2-20480 d^6 e^3 x^3-23352 d^5 e^4 x^4+7680 d^4 e^5 x^5+24570 d^3 e^6 x^6+10240 d^2 e^7 x^7-3465 d e^8 x^8-6400 e^9 x^9\right )+6930 e^{10} x^{10} \tanh ^{-1}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{26880 d^2 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^11,x]

[Out]

-1/26880*(Sqrt[d^2 - e^2*x^2]*(2688*d^9 + 8960*d^8*e*x + 3024*d^7*e^2*x^2 - 20480*d^6*e^3*x^3 - 23352*d^5*e^4*
x^4 + 7680*d^4*e^5*x^5 + 24570*d^3*e^6*x^6 + 10240*d^2*e^7*x^7 - 3465*d*e^8*x^8 - 6400*e^9*x^9) + 6930*e^10*x^
10*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(d^2*x^10)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(193)=386\).
time = 0.13, size = 568, normalized size = 2.52

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-6400 e^{9} x^{9}-3465 d \,e^{8} x^{8}+10240 d^{2} e^{7} x^{7}+24570 d^{3} e^{6} x^{6}+7680 d^{4} e^{5} x^{5}-23352 d^{5} e^{4} x^{4}-20480 d^{6} e^{3} x^{3}+3024 d^{7} e^{2} x^{2}+8960 d^{8} x e +2688 d^{9}\right )}{26880 x^{10} d^{2}}+\frac {33 e^{10} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 d \sqrt {d^{2}}}\)	\(165\)
default	\(d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 d^{2} x^{10}}+\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )}{10 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )-\frac {e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d^{2} x^{7}}\)	\(568\)

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^11,x,method=_RETURNVERBOSE)

[Out]

d^3*(-1/10/d^2/x^10*(-e^2*x^2+d^2)^(7/2)+3/10*e^2/d^2*(-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*(-1/8/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^2*e*(-1/9/d^2/x^9*(
-e^2*x^2+d^2)^(7/2)-2/63*e^2/d^4/x^7*(-e^2*x^2+d^2)^(7/2))-1/7*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^7

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Maxima [A]
time = 0.49, size = 253, normalized size = 1.12 \begin {gather*} \frac {33 \, e^{10} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{2}} - \frac {33 \, \sqrt {-x^{2} e^{2} + d^{2}} e^{10}}{256 \, d^{3}} - \frac {11 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} e^{10}}{256 \, d^{5}} - \frac {33 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}} e^{10}}{1280 \, d^{7}} - \frac {33 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{8}}{1280 \, d^{7} x^{2}} + \frac {11 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{640 \, d^{5} x^{4}} - \frac {11 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{160 \, d^{3} x^{6}} - \frac {5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{21 \, d^{2} x^{7}} - \frac {33 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{80 \, d x^{8}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} e}{3 \, x^{9}} - \frac {{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {7}{2}} d}{10 \, x^{10}} \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^11,x, algorithm="maxima")

[Out]

33/256*e^10*log(2*d^2/abs(x) + 2*sqrt(-x^2*e^2 + d^2)*d/abs(x))/d^2 - 33/256*sqrt(-x^2*e^2 + d^2)*e^10/d^3 - 1
1/256*(-x^2*e^2 + d^2)^(3/2)*e^10/d^5 - 33/1280*(-x^2*e^2 + d^2)^(5/2)*e^10/d^7 - 33/1280*(-x^2*e^2 + d^2)^(7/
2)*e^8/(d^7*x^2) + 11/640*(-x^2*e^2 + d^2)^(7/2)*e^6/(d^5*x^4) - 11/160*(-x^2*e^2 + d^2)^(7/2)*e^4/(d^3*x^6) -
5/21*(-x^2*e^2 + d^2)^(7/2)*e^3/(d^2*x^7) - 33/80*(-x^2*e^2 + d^2)^(7/2)*e^2/(d*x^8) - 1/3*(-x^2*e^2 + d^2)^(
7/2)*e/x^9 - 1/10*(-x^2*e^2 + d^2)^(7/2)*d/x^10

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Fricas [A]
time = 3.46, size = 143, normalized size = 0.64 \begin {gather*} -\frac {3465 \, x^{10} e^{10} \log \left (-\frac {d - \sqrt {-x^{2} e^{2} + d^{2}}}{x}\right ) - {\left (6400 \, x^{9} e^{9} + 3465 \, d x^{8} e^{8} - 10240 \, d^{2} x^{7} e^{7} - 24570 \, d^{3} x^{6} e^{6} - 7680 \, d^{4} x^{5} e^{5} + 23352 \, d^{5} x^{4} e^{4} + 20480 \, d^{6} x^{3} e^{3} - 3024 \, d^{7} x^{2} e^{2} - 8960 \, d^{8} x e - 2688 \, d^{9}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{26880 \, d^{2} x^{10}} \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^11,x, algorithm="fricas")

[Out]

-1/26880*(3465*x^10*e^10*log(-(d - sqrt(-x^2*e^2 + d^2))/x) - (6400*x^9*e^9 + 3465*d*x^8*e^8 - 10240*d^2*x^7*e
^7 - 24570*d^3*x^6*e^6 - 7680*d^4*x^5*e^5 + 23352*d^5*x^4*e^4 + 20480*d^6*x^3*e^3 - 3024*d^7*x^2*e^2 - 8960*d^
8*x*e - 2688*d^9)*sqrt(-x^2*e^2 + d^2))/(d^2*x^10)

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Sympy [C] Result contains complex when optimal does not.
time = 143.03, size = 2159, normalized size = 9.60 \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**11,x)

[Out]

d**7*Piecewise((-d**2/(10*e*x**11*sqrt(d**2/(e**2*x**2) - 1)) + 9*e/(80*x**9*sqrt(d**2/(e**2*x**2) - 1)) + e**
3/(480*d**2*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**5/(1920*d**4*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**7/(76
8*d**6*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 7*e**9/(256*d**8*x*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**10*acosh(d/(e*
x))/(256*d**9), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(10*e*x**11*sqrt(-d**2/(e**2*x**2) + 1)) - 9*I*e/(80*x**9*
sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(480*d**2*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**5/(1920*d**4*x**5*s
qrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**7/(768*d**6*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 7*I*e**9/(256*d**8*x*sqrt
(-d**2/(e**2*x**2) + 1)) - 7*I*e**10*asin(d/(e*x))/(256*d**9), True)) + 3*d**6*e*Piecewise((-e*sqrt(d**2/(e**2
*x**2) - 1)/(9*x**8) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(63*d**2*x**6) + 2*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105
*d**4*x**4) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(315*d**6*x**2) + 16*e**9*sqrt(d**2/(e**2*x**2) - 1)/(315*d**8
), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(9*x**8) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)
/(63*d**2*x**6) + 2*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**4) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/
(315*d**6*x**2) + 16*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(315*d**8), True)) + d**5*e**2*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)) - 5*d**4*e**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sq
rt(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*s
qrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) - 5*d**3*e**4*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)) +
d**2*e**5*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)) + 3*d*e**6*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)) + e**7*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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (180) = 360\).
time = 0.58, size = 677, normalized size = 3.01 \begin {gather*} \frac {x^{10} {\left (\frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{8}}{x} + \frac {525 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} e^{6}}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} e^{4}}{x^{3}} - \frac {3570 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} e^{2}}{x^{4}} + \frac {5880 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} e^{\left (-2\right )}}{x^{6}} + \frac {16800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} e^{\left (-4\right )}}{x^{7}} + \frac {10500 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} e^{\left (-6\right )}}{x^{8}} - \frac {31920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} e^{\left (-8\right )}}{x^{9}} - \frac {3360 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5}}{x^{5}} + 42 \, e^{10}\right )} e^{20}}{430080 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} d^{2}} + \frac {33 \, e^{10} \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 )}{256 \, d^{2}} + \frac {\frac {31920 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{18} e^{8}}{x} - \frac {10500 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{18} e^{6}}{x^{2}} - \frac {16800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{18} e^{4}}{x^{3}} - \frac {5880 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{18} e^{2}}{x^{4}} + \frac {3570 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{6} d^{18} e^{\left (-2\right )}}{x^{6}} + \frac {600 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{7} d^{18} e^{\left (-4\right )}}{x^{7}} - \frac {525 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{8} d^{18} e^{\left (-6\right )}}{x^{8}} - \frac {280 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{9} d^{18} e^{\left (-8\right )}}{x^{9}} - \frac {42 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{10} d^{18} e^{\left (-10\right )}}{x^{10}} + \frac {3360 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{5} d^{18}}{x^{5}}}{430080 \, d^{20}} \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^11,x, algorithm="giac")

[Out]

1/430080*x^10*(280*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^8/x + 525*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^6/x^2 - 600*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^4/x^3 - 3570*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^2/x^4 + 5880*(d*e + sqrt(-x^
2*e^2 + d^2)*e)^6*e^(-2)/x^6 + 16800*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*e^(-4)/x^7 + 10500*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^8*e^(-6)/x^8 - 31920*(d*e + sqrt(-x^2*e^2 + d^2)*e)^9*e^(-8)/x^9 - 3360*(d*e + sqrt(-x^2*e^2 + d^2)*
e)^5/x^5 + 42*e^10)*e^20/((d*e + sqrt(-x^2*e^2 + d^2)*e)^10*d^2) + 33/256*e^10*log(1/2*abs(-2*d*e - 2*sqrt(-x^
2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^2 + 1/430080*(31920*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^18*e^8/x - 10500*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^2*d^18*e^6/x^2 - 16800*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^18*e^4/x^3 - 5880*(d*e + sqr
t(-x^2*e^2 + d^2)*e)^4*d^18*e^2/x^4 + 3570*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^18*e^(-2)/x^6 + 600*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^7*d^18*e^(-4)/x^7 - 525*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*d^18*e^(-6)/x^8 - 280*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^9*d^18*e^(-8)/x^9 - 42*(d*e + sqrt(-x^2*e^2 + d^2)*e)^10*d^18*e^(-10)/x^10 + 3360*(d*e + sq
rt(-x^2*e^2 + d^2)*e)^5*d^18/x^5)/d^20

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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^{11}} \,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^11,x)

[Out]

int(((d^2 - e^2*x^2)^(5/2)*(d + e*x)^3)/x^11, x)

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