3.82 \(\int \frac{(d+e x)^3 (d^2-e^2 x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}+\frac{19 e^{11} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^3} \]

[Out]

(-19*e^9*Sqrt[d^2 - e^2*x^2])/(256*d^2*x^2) + (19*e^7*(d^2 - e^2*x^2)^(3/2))/(384*d^2*x^4) - (19*e^5*(d^2 - e^
2*x^2)^(5/2))/(480*d^2*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(11*x^11) - (3*e*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (3
7*e^2*(d^2 - e^2*x^2)^(7/2))/(99*d*x^9) - (19*e^3*(d^2 - e^2*x^2)^(7/2))/(80*d^2*x^8) - (74*e^4*(d^2 - e^2*x^2
)^(7/2))/(693*d^3*x^7) + (19*e^11*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(256*d^3)

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Rubi [A]  time = 0.329047, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}+\frac{19 e^{11} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]

[Out]

(-19*e^9*Sqrt[d^2 - e^2*x^2])/(256*d^2*x^2) + (19*e^7*(d^2 - e^2*x^2)^(3/2))/(384*d^2*x^4) - (19*e^5*(d^2 - e^
2*x^2)^(5/2))/(480*d^2*x^6) - (d*(d^2 - e^2*x^2)^(7/2))/(11*x^11) - (3*e*(d^2 - e^2*x^2)^(7/2))/(10*x^10) - (3
7*e^2*(d^2 - e^2*x^2)^(7/2))/(99*d*x^9) - (19*e^3*(d^2 - e^2*x^2)^(7/2))/(80*d^2*x^8) - (74*e^4*(d^2 - e^2*x^2
)^(7/2))/(693*d^3*x^7) + (19*e^11*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(256*d^3)

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^{12}} \, dx &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{\int \frac{\left (d^2-e^2 x^2\right )^{5/2} \left (-33 d^4 e-37 d^3 e^2 x-11 d^2 e^3 x^2\right )}{x^{11}} \, dx}{11 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}+\frac{\int \frac{\left (370 d^5 e^2+209 d^4 e^3 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx}{110 d^4}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{\int \frac{\left (-1881 d^6 e^3-740 d^5 e^4 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx}{990 d^6}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}+\frac{\int \frac{\left (5920 d^7 e^4+1881 d^6 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2}}{x^8} \, dx}{7920 d^8}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac{\left (19 e^5\right ) \int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^7} \, dx}{80 d^2}\\ &=-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac{\left (19 e^5\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{5/2}}{x^4} \, dx,x,x^2\right )}{160 d^2}\\ &=-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac{\left (19 e^7\right ) \operatorname{Subst}\left (\int \frac{\left (d^2-e^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )}{192 d^2}\\ &=\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac{\left (19 e^9\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d^2-e^2 x}}{x^2} \, dx,x,x^2\right )}{256 d^2}\\ &=-\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}-\frac{\left (19 e^{11}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{512 d^2}\\ &=-\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac{\left (19 e^9\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^2}\\ &=-\frac{19 e^9 \sqrt{d^2-e^2 x^2}}{256 d^2 x^2}+\frac{19 e^7 \left (d^2-e^2 x^2\right )^{3/2}}{384 d^2 x^4}-\frac{19 e^5 \left (d^2-e^2 x^2\right )^{5/2}}{480 d^2 x^6}-\frac{d \left (d^2-e^2 x^2\right )^{7/2}}{11 x^{11}}-\frac{3 e \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac{37 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 d x^9}-\frac{19 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{80 d^2 x^8}-\frac{74 e^4 \left (d^2-e^2 x^2\right )^{7/2}}{693 d^3 x^7}+\frac{19 e^{11} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{256 d^3}\\ \end{align*}

Mathematica [C]  time = 0.0676529, size = 112, normalized size = 0.44 \[ -\frac{\left (d^2-e^2 x^2\right )^{7/2} \left (99 e^{11} x^{11} \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )+297 e^{11} x^{11} \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};1-\frac{e^2 x^2}{d^2}\right )+259 d^9 e^2 x^2+74 d^7 e^4 x^4+63 d^{11}\right )}{693 d^{10} x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^3*(d^2 - e^2*x^2)^(5/2))/x^12,x]

[Out]

-((d^2 - e^2*x^2)^(7/2)*(63*d^11 + 259*d^9*e^2*x^2 + 74*d^7*e^4*x^4 + 99*e^11*x^11*Hypergeometric2F1[7/2, 5, 9
/2, 1 - (e^2*x^2)/d^2] + 297*e^11*x^11*Hypergeometric2F1[7/2, 6, 9/2, 1 - (e^2*x^2)/d^2]))/(693*d^10*x^11)

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Maple [A]  time = 0.569, size = 303, normalized size = 1.2 \begin{align*} -{\frac{d}{11\,{x}^{11}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{37\,{e}^{2}}{99\,d{x}^{9}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{74\,{e}^{4}}{693\,{d}^{3}{x}^{7}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{3}}{80\,{d}^{2}{x}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{5}}{480\,{d}^{4}{x}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{19\,{e}^{7}}{1920\,{d}^{6}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{9}}{1280\,{d}^{8}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{19\,{e}^{11}}{1280\,{d}^{8}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{19\,{e}^{11}}{768\,{d}^{6}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{19\,{e}^{11}}{256\,{d}^{4}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{19\,{e}^{11}}{256\,{d}^{2}}\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}{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^12,x)

[Out]

-1/11*d*(-e^2*x^2+d^2)^(7/2)/x^11-37/99*e^2*(-e^2*x^2+d^2)^(7/2)/d/x^9-74/693*e^4*(-e^2*x^2+d^2)^(7/2)/d^3/x^7
-19/80*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^8-19/480*e^5/d^4/x^6*(-e^2*x^2+d^2)^(7/2)+19/1920*e^7/d^6/x^4*(-e^2*x^2+
d^2)^(7/2)-19/1280*e^9/d^8/x^2*(-e^2*x^2+d^2)^(7/2)-19/1280*e^11/d^8*(-e^2*x^2+d^2)^(5/2)-19/768*e^11/d^6*(-e^
2*x^2+d^2)^(3/2)-19/256*e^11/d^4*(-e^2*x^2+d^2)^(1/2)+19/256*e^11/d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^
2*x^2+d^2)^(1/2))/x)-3/10*e*(-e^2*x^2+d^2)^(7/2)/x^10

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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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.16834, size = 417, normalized size = 1.64 \begin{align*} -\frac{65835 \, e^{11} x^{11} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (94720 \, e^{10} x^{10} + 65835 \, d e^{9} x^{9} + 47360 \, d^{2} e^{8} x^{8} - 251790 \, d^{3} e^{7} x^{7} - 629760 \, d^{4} e^{6} x^{6} - 201432 \, d^{5} e^{5} x^{5} + 657920 \, d^{6} e^{4} x^{4} + 587664 \, d^{7} e^{3} x^{3} - 89600 \, d^{8} e^{2} x^{2} - 266112 \, d^{9} e x - 80640 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{887040 \, d^{3} x^{11}} \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^12,x, algorithm="fricas")

[Out]

-1/887040*(65835*e^11*x^11*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (94720*e^10*x^10 + 65835*d*e^9*x^9 + 47360*d^2
*e^8*x^8 - 251790*d^3*e^7*x^7 - 629760*d^4*e^6*x^6 - 201432*d^5*e^5*x^5 + 657920*d^6*e^4*x^4 + 587664*d^7*e^3*
x^3 - 89600*d^8*e^2*x^2 - 266112*d^9*e*x - 80640*d^10)*sqrt(-e^2*x^2 + d^2))/(d^3*x^11)

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Sympy [C]  time = 54.7222, size = 2422, normalized size = 9.54 \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**12,x)

[Out]

d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(11*x**10) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(99*d**2*x**8) + 8*
e**5*sqrt(d**2/(e**2*x**2) - 1)/(693*d**4*x**6) + 16*e**7*sqrt(d**2/(e**2*x**2) - 1)/(1155*d**6*x**4) + 64*e**
9*sqrt(d**2/(e**2*x**2) - 1)/(3465*d**8*x**2) + 128*e**11*sqrt(d**2/(e**2*x**2) - 1)/(3465*d**10), Abs(d**2)/(
Abs(e**2)*Abs(x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(11*x**10) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(
99*d**2*x**8) + 8*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(693*d**4*x**6) + 16*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(
1155*d**6*x**4) + 64*I*e**9*sqrt(-d**2/(e**2*x**2) + 1)/(3465*d**8*x**2) + 128*I*e**11*sqrt(-d**2/(e**2*x**2)
+ 1)/(3465*d**10), True)) + 3*d**6*e*Piecewise((-d**2/(10*e*x**11*sqrt(d**2/(e**2*x**2) - 1)) + 9*e/(80*x**9*s
qrt(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/(768*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)
) + d**5*e**2*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) + 1
6*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)/(1
05*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)) - 5*d**4*e**3*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)/(Ab
s(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)) - 5*d**3
*e**4*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)) + d**2*e**5*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)/(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)) + 3*d*e**6*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)) + e**7*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))

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Giac [B]  time = 1.2886, size = 1007, normalized size = 3.96 \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^12,x, algorithm="giac")

[Out]

1/14192640*x^11*(4158*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^22/x + 8470*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*e^20/x^2 -
3465*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*e^18/x^3 - 40590*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*e^16/x^4 - 57750*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)^5*e^14/x^5 + 6930*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*e^12/x^6 + 138600*(d*e + sqrt(-x
^2*e^2 + d^2)*e)^7*e^10/x^7 + 244860*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*e^8/x^8 + 152460*(d*e + sqrt(-x^2*e^2 +
d^2)*e)^9*e^6/x^9 - 568260*(d*e + sqrt(-x^2*e^2 + d^2)*e)^10*e^4/x^10 + 630*e^24)*e^9/((d*e + sqrt(-x^2*e^2 +
d^2)*e)^11*d^3) + 19/256*e^11*log(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d^3 + 1/14192640*(
568260*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^30*e^152/x - 152460*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^30*e^150/x^2 -
244860*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^30*e^148/x^3 - 138600*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^30*e^146/x^
4 - 6930*(d*e + sqrt(-x^2*e^2 + d^2)*e)^5*d^30*e^144/x^5 + 57750*(d*e + sqrt(-x^2*e^2 + d^2)*e)^6*d^30*e^142/x
^6 + 40590*(d*e + sqrt(-x^2*e^2 + d^2)*e)^7*d^30*e^140/x^7 + 3465*(d*e + sqrt(-x^2*e^2 + d^2)*e)^8*d^30*e^138/
x^8 - 8470*(d*e + sqrt(-x^2*e^2 + d^2)*e)^9*d^30*e^136/x^9 - 4158*(d*e + sqrt(-x^2*e^2 + d^2)*e)^10*d^30*e^134
/x^10 - 630*(d*e + sqrt(-x^2*e^2 + d^2)*e)^11*d^30*e^132/x^11)*e^(-143)/d^33