3.1097 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=181 \[ \frac{1}{8} a^7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{a^5 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7} \]

[Out]

-(a^5*c*Sqrt[c - a^2*c*x^2])/(8*x^2) - (c - a^2*c*x^2)^(3/2)/(7*x^7) - (a*(c - a^2*c*x^2)^(3/2))/(3*x^6) - (11
*a^2*(c - a^2*c*x^2)^(3/2))/(35*x^5) - (a^3*(c - a^2*c*x^2)^(3/2))/(4*x^4) - (22*a^4*(c - a^2*c*x^2)^(3/2))/(1
05*x^3) + (a^7*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/8

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Rubi [A]  time = 0.371403, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6151, 1807, 835, 807, 266, 47, 63, 208} \[ \frac{1}{8} a^7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{a^5 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^8,x]

[Out]

-(a^5*c*Sqrt[c - a^2*c*x^2])/(8*x^2) - (c - a^2*c*x^2)^(3/2)/(7*x^7) - (a*(c - a^2*c*x^2)^(3/2))/(3*x^6) - (11
*a^2*(c - a^2*c*x^2)^(3/2))/(35*x^5) - (a^3*(c - a^2*c*x^2)^(3/2))/(4*x^4) - (22*a^4*(c - a^2*c*x^2)^(3/2))/(1
05*x^3) + (a^7*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/8

Rule 6151

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^(n/2), Int[x^m*(c
+ d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] ||
GtQ[c, 0]) && IGtQ[n/2, 0]

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{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx &=c \int \frac{(1+a x)^2 \sqrt{c-a^2 c x^2}}{x^8} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{1}{7} \int \frac{\left (-14 a c-11 a^2 c x\right ) \sqrt{c-a^2 c x^2}}{x^7} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}+\frac{\int \frac{\left (66 a^2 c^2+42 a^3 c^2 x\right ) \sqrt{c-a^2 c x^2}}{x^6} \, dx}{42 c}\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{\int \frac{\left (-210 a^3 c^3-132 a^4 c^3 x\right ) \sqrt{c-a^2 c x^2}}{x^5} \, dx}{210 c^2}\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}+\frac{\int \frac{\left (528 a^4 c^4+210 a^5 c^4 x\right ) \sqrt{c-a^2 c x^2}}{x^4} \, dx}{840 c^3}\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}+\frac{1}{4} \left (a^5 c\right ) \int \frac{\sqrt{c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}+\frac{1}{8} \left (a^5 c\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{a^5 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}-\frac{1}{16} \left (a^7 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{a^5 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}+\frac{1}{8} \left (a^5 c\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )\\ &=-\frac{a^5 c \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac{a \left (c-a^2 c x^2\right )^{3/2}}{3 x^6}-\frac{11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac{a^3 \left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac{22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}+\frac{1}{8} a^7 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.188234, size = 120, normalized size = 0.66 \[ \frac{1}{8} a^7 c^{3/2} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-\frac{1}{8} a^7 c^{3/2} \log (x)+\frac{c \left (176 a^6 x^6+105 a^5 x^5+88 a^4 x^4+70 a^3 x^3-144 a^2 x^2-280 a x-120\right ) \sqrt{c-a^2 c x^2}}{840 x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^8,x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-120 - 280*a*x - 144*a^2*x^2 + 70*a^3*x^3 + 88*a^4*x^4 + 105*a^5*x^5 + 176*a^6*x^6))/(
840*x^7) - (a^7*c^(3/2)*Log[x])/8 + (a^7*c^(3/2)*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/8

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Maple [B]  time = 0.093, size = 436, normalized size = 2.4 \begin{align*} -{\frac{{a}^{7}}{24} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{1}{7\,c{x}^{7}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{a}{3\,c{x}^{6}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{8}x}{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{7}}{8}{c}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }-{\frac{{a}^{7}c}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}+{a}^{8}c\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }x+{{a}^{8}{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{2\,{a}^{6}}{3\,cx} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{a}^{8}cx\sqrt{-{a}^{2}c{x}^{2}+c}-{{a}^{8}{c}^{2}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{7\,{a}^{3}}{12\,c{x}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{17\,{a}^{5}}{24\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{4}}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{a}^{2}}{35\,c{x}^{5}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{2\,{a}^{7}}{3} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x)

[Out]

-1/24*a^7*(-a^2*c*x^2+c)^(3/2)-1/7/c/x^7*(-a^2*c*x^2+c)^(5/2)-1/3*a/c/x^6*(-a^2*c*x^2+c)^(5/2)-2/3*a^8*x*(-a^2
*c*x^2+c)^(3/2)+1/8*a^7*c^(3/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-1/8*a^7*(-a^2*c*x^2+c)^(1/2)*c+a^8*
c*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)*x+a^8*c^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-c*a^2*(x-1/a)^2-2*a*
c*(x-1/a))^(1/2))-2/3*a^6/c/x*(-a^2*c*x^2+c)^(5/2)-a^8*c*x*(-a^2*c*x^2+c)^(1/2)-a^8*c^2/(a^2*c)^(1/2)*arctan((
a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))-7/12*a^3/c/x^4*(-a^2*c*x^2+c)^(5/2)-17/24*a^5/c/x^2*(-a^2*c*x^2+c)^(5/2)-
2/3*a^4/c/x^3*(-a^2*c*x^2+c)^(5/2)-16/35*a^2/c/x^5*(-a^2*c*x^2+c)^(5/2)-2/3*a^7*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a
))^(3/2)

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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((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.78042, size = 585, normalized size = 3.23 \begin{align*} \left [\frac{105 \, a^{7} c^{\frac{3}{2}} x^{7} \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (176 \, a^{6} c x^{6} + 105 \, a^{5} c x^{5} + 88 \, a^{4} c x^{4} + 70 \, a^{3} c x^{3} - 144 \, a^{2} c x^{2} - 280 \, a c x - 120 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{1680 \, x^{7}}, \frac{105 \, a^{7} \sqrt{-c} c x^{7} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (176 \, a^{6} c x^{6} + 105 \, a^{5} c x^{5} + 88 \, a^{4} c x^{4} + 70 \, a^{3} c x^{3} - 144 \, a^{2} c x^{2} - 280 \, a c x - 120 \, c\right )} \sqrt{-a^{2} c x^{2} + c}}{840 \, x^{7}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm="fricas")

[Out]

[1/1680*(105*a^7*c^(3/2)*x^7*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(176*a^6*c*x^6 +
105*a^5*c*x^5 + 88*a^4*c*x^4 + 70*a^3*c*x^3 - 144*a^2*c*x^2 - 280*a*c*x - 120*c)*sqrt(-a^2*c*x^2 + c))/x^7, 1
/840*(105*a^7*sqrt(-c)*c*x^7*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (176*a^6*c*x^6 + 105*a^5*
c*x^5 + 88*a^4*c*x^4 + 70*a^3*c*x^3 - 144*a^2*c*x^2 - 280*a*c*x - 120*c)*sqrt(-a^2*c*x^2 + c))/x^7]

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Sympy [C]  time = 20.0616, size = 660, normalized size = 3.65 \begin{align*} a^{2} c \left (\begin{cases} \frac{2 i a^{4} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x} + \frac{i a^{2} \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{15 x^{3}} - \frac{i \sqrt{c} \sqrt{a^{2} x^{2} - 1}}{5 x^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{2 a^{4} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x} + \frac{a^{2} \sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{c} \sqrt{- a^{2} x^{2} + 1}}{5 x^{5}} & \text{otherwise} \end{cases}\right ) + 2 a c \left (\begin{cases} \frac{a^{6} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{16} - \frac{a^{5} \sqrt{c}}{16 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{a^{3} \sqrt{c}}{48 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} + \frac{5 a \sqrt{c}}{24 x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{6 a x^{7} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{6} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{16} + \frac{i a^{5} \sqrt{c}}{16 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{i a^{3} \sqrt{c}}{48 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} - \frac{5 i a \sqrt{c}}{24 x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i \sqrt{c}}{6 a x^{7} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \frac{8 a^{7} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 a^{5} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{a^{3} \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{a \sqrt{c} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{8 i a^{7} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105} + \frac{4 i a^{5} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{105 x^{2}} + \frac{i a^{3} \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{35 x^{4}} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{7 x^{6}} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**8,x)

[Out]

a**2*c*Piecewise((2*I*a**4*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x**3) -
I*sqrt(c)*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + a*
*2*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(c)*sqrt(-a**2*x**2 + 1)/(5*x**5), True)) + 2*a*c*Piecewise((a
**6*sqrt(c)*acosh(1/(a*x))/16 - a**5*sqrt(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3*sqrt(c)/(48*x**3*sqrt(-1 +
1/(a**2*x**2))) + 5*a*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))
), 1/Abs(a**2*x**2) > 1), (-I*a**6*sqrt(c)*asin(1/(a*x))/16 + I*a**5*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) -
I*a**3*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a*sqrt(c)/(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I*sqrt(c)
/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True)) + c*Piecewise((8*a**7*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/105 + 4*a**
5*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(105*x**2) + a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(35*x**4) - a*sqrt(c)*sq
rt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (8*I*a**7*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/105 + 4*I*a*
*5*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(105*x**2) + I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(c)
*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))

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Giac [B]  time = 1.19264, size = 714, normalized size = 3.94 \begin{align*} -\frac{a^{7} c^{2} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{4 \, \sqrt{-c}} + \frac{105 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{13} a^{7} c^{2} - 700 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{11} a^{7} c^{3} + 1680 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{10} a^{6} \sqrt{-c} c^{3}{\left | a \right |} - 3395 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{9} a^{7} c^{4} - 7280 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{8} a^{6} \sqrt{-c} c^{4}{\left | a \right |} - 1120 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{6} a^{6} \sqrt{-c} c^{5}{\left | a \right |} + 3395 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{5} a^{7} c^{6} - 2016 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{4} a^{6} \sqrt{-c} c^{6}{\left | a \right |} + 700 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{7} c^{7} + 1232 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{6} \sqrt{-c} c^{7}{\left | a \right |} - 105 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{7} c^{8} - 176 \, a^{6} \sqrt{-c} c^{8}{\left | a \right |}}{420 \,{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm="giac")

[Out]

-1/4*a^7*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) + 1/420*(105*(sqrt(-a^2*c)*x -
sqrt(-a^2*c*x^2 + c))^13*a^7*c^2 - 700*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^11*a^7*c^3 + 1680*(sqrt(-a^2*c
)*x - sqrt(-a^2*c*x^2 + c))^10*a^6*sqrt(-c)*c^3*abs(a) - 3395*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^9*a^7*c^
4 - 7280*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^8*a^6*sqrt(-c)*c^4*abs(a) - 1120*(sqrt(-a^2*c)*x - sqrt(-a^2*
c*x^2 + c))^6*a^6*sqrt(-c)*c^5*abs(a) + 3395*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^7*c^6 - 2016*(sqrt(-a
^2*c)*x - sqrt(-a^2*c*x^2 + c))^4*a^6*sqrt(-c)*c^6*abs(a) + 700*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^3*a^7*
c^7 + 1232*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^6*sqrt(-c)*c^7*abs(a) - 105*(sqrt(-a^2*c)*x - sqrt(-a^2
*c*x^2 + c))*a^7*c^8 - 176*a^6*sqrt(-c)*c^8*abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^7