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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.298169, antiderivative size = 225, normalized size of antiderivative = 1., 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 = {1807, 835, 807, 266, 47, 63, 208} \[ -\frac{33 e^8 \sqrt{d^2-e^2 x^2}}{256 d x^2}+\frac{11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac{11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac{5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}-\frac{33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac{e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}+\frac{33 e^{10} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^2} \]

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 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 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 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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \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 )}{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*}

Mathematica [C]  time = 0.074063, size = 102, normalized size = 0.45 \[ -\frac{e \left (d^2-e^2 x^2\right )^{7/2} \left (9 e^9 x^9 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )+3 e^9 x^9 \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )+5 d^7 e^2 x^2+7 d^9\right )}{21 d^9 x^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(d^2 - e^2*x^2)^(7/2)*(7*d^9 + 5*d^7*e^2*x^2 + 9*e^9*x^9*Hypergeometric2F1[7/2, 5, 9/2, 1 - (e^2*x^2)/d^2]
+ 3*e^9*x^9*Hypergeometric2F1[7/2, 6, 9/2, 1 - (e^2*x^2)/d^2]))/(21*d^9*x^9)

________________________________________________________________________________________

Maple [A]  time = 0.373, size = 278, normalized size = 1.2 \begin{align*} -{\frac{33\,{e}^{2}}{80\,d{x}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{11\,{e}^{4}}{160\,{d}^{3}{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{11\,{e}^{6}}{640\,{d}^{5}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{33\,{e}^{8}}{1280\,{d}^{7}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{33\,{e}^{10}}{1280\,{d}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{11\,{e}^{10}}{256\,{d}^{5}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{33\,{e}^{10}}{256\,{d}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{33\,{e}^{10}}{256\,d}\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{5\,{e}^{3}}{21\,{d}^{2}{x}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{e}{3\,{x}^{9}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{d}{10\,{x}^{10}} \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^11,x)

[Out]

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

________________________________________________________________________________________

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^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.09006, size = 367, normalized size = 1.63 \begin{align*} -\frac{3465 \, e^{10} x^{10} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\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} e x - 2688 \, d^{9}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{26880 \, d^{2} x^{10}} \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^11,x, algorithm="fricas")

[Out]

-1/26880*(3465*e^10*x^10*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (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*e*x - 2688*d^9)*sqrt(-e^2*x^2 + d^2))/(d^2*x^10)

________________________________________________________________________________________

Sympy [C]  time = 54.1913, size = 2184, normalized size = 9.71 \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**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)/(Abs(e**2)*Abs(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*sqrt(-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)/(Abs(e**2)*Abs(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*Pie
cewise((-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*sq
rt(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*sqr
t(-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*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*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)) - 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*s
qrt(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)/(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)) + d**2*e**5*Piecewise((3*I*d**3*sqrt(-1 + e**2*x**2/d**2)/(-1
5*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)) + 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)/
(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)) + e**7*Piece
wise((-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), Tru
e))

________________________________________________________________________________________

Giac [B]  time = 1.28323, size = 922, normalized size = 4.1 \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^11,x, algorithm="giac")

[Out]

1/430080*x^10*(280*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^20/x + 525*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^18/x^2 - 600
*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^16/x^3 - 3570*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^14/x^4 - 3360*(d*e + sqrt
(-x^2*e^2 + d^2)*e)^5*e^12/x^5 + 5880*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^10/x^6 + 16800*(d*e + sqrt(-x^2*e^2 +
d^2)*e)^7*e^8/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
^4/x^9 + 42*e^22)*e^8/((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^128/x - 10500*(d*e + s
qrt(-x^2*e^2 + d^2)*e)^2*d^18*e^126/x^2 - 16800*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^18*e^124/x^3 - 5880*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^4*d^18*e^122/x^4 + 3360*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^18*e^120/x^5 + 3570*(d*e +
sqrt(-x^2*e^2 + d^2)*e)^6*d^18*e^118/x^6 + 600*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*d^18*e^116/x^7 - 525*(d*e + sq
rt(-x^2*e^2 + d^2)*e)^8*d^18*e^114/x^8 - 280*(d*e + sqrt(-x^2*e^2 + d^2)*e)^9*d^18*e^112/x^9 - 42*(d*e + sqrt(
-x^2*e^2 + d^2)*e)^10*d^18*e^110/x^10)*e^(-120)/d^20