3.1104 \(\int \frac{e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{5/2}}{x^3} \, dx\)

Optimal. Leaf size=151 \[ -\frac{1}{2} a^2 c^2 (6 a x+1) \sqrt{c-a^2 c x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\frac{1}{2} a^2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{a c (a x+12) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2} \]

[Out]

-(a^2*c^2*(1 + 6*a*x)*Sqrt[c - a^2*c*x^2])/2 - (a*c*(12 + a*x)*(c - a^2*c*x^2)^(3/2))/(6*x) - (c - a^2*c*x^2)^
(5/2)/(2*x^2) - 3*a^2*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + (a^2*c^(5/2)*ArcTanh[Sqrt[c - a^2*c*
x^2]/Sqrt[c]])/2

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Rubi [A]  time = 0.331636, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.37, Rules used = {6151, 1807, 813, 815, 844, 217, 203, 266, 63, 208} \[ -\frac{1}{2} a^2 c^2 (6 a x+1) \sqrt{c-a^2 c x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\frac{1}{2} a^2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )-\frac{a c (a x+12) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2))/x^3,x]

[Out]

-(a^2*c^2*(1 + 6*a*x)*Sqrt[c - a^2*c*x^2])/2 - (a*c*(12 + a*x)*(c - a^2*c*x^2)^(3/2))/(6*x) - (c - a^2*c*x^2)^
(5/2)/(2*x^2) - 3*a^2*c^(5/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + (a^2*c^(5/2)*ArcTanh[Sqrt[c - a^2*c*
x^2]/Sqrt[c]])/2

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 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 815

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*(a + c*x^2)^p)/(c*e^2*(m + 2*p + 1)*(m
+ 2*p + 2)), x] + Dist[(2*p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 844

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 217

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 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 )^{5/2}}{x^3} \, dx &=c \int \frac{(1+a x)^2 \left (c-a^2 c x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac{1}{2} \int \frac{\left (-4 a c+a^2 c x\right ) \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac{a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}+\frac{1}{4} \int \frac{\left (-2 a^2 c^2-24 a^3 c^2 x\right ) \sqrt{c-a^2 c x^2}}{x} \, dx\\ &=-\frac{1}{2} a^2 c^2 (1+6 a x) \sqrt{c-a^2 c x^2}-\frac{a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac{\int \frac{4 a^4 c^4+24 a^5 c^4 x}{x \sqrt{c-a^2 c x^2}} \, dx}{8 a^2 c}\\ &=-\frac{1}{2} a^2 c^2 (1+6 a x) \sqrt{c-a^2 c x^2}-\frac{a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac{1}{2} \left (a^2 c^3\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx-\left (3 a^3 c^3\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{1}{2} a^2 c^2 (1+6 a x) \sqrt{c-a^2 c x^2}-\frac{a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac{1}{4} \left (a^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )-\left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{1}{2} a^2 c^2 (1+6 a x) \sqrt{c-a^2 c x^2}-\frac{a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\frac{1}{2} c^2 \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{1}{2} a^2 c^2 (1+6 a x) \sqrt{c-a^2 c x^2}-\frac{a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac{\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\frac{1}{2} a^2 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.260316, size = 151, normalized size = 1. \[ -\frac{c^2 \left (2 a^4 x^4+6 a^3 x^3-2 a^2 x^2+12 a x+3\right ) \sqrt{c-a^2 c x^2}}{6 x^2}+\frac{1}{2} a^2 c^{5/2} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+3 a^2 c^{5/2} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )-\frac{1}{2} a^2 c^{5/2} \log (x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2))/x^3,x]

[Out]

-(c^2*Sqrt[c - a^2*c*x^2]*(3 + 12*a*x - 2*a^2*x^2 + 6*a^3*x^3 + 2*a^4*x^4))/(6*x^2) + 3*a^2*c^(5/2)*ArcTan[(a*
x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] - (a^2*c^(5/2)*Log[x])/2 + (a^2*c^(5/2)*Log[c + Sqrt[c]*Sqrt[
c - a^2*c*x^2]])/2

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Maple [B]  time = 0.048, size = 399, normalized size = 2.6 \begin{align*} -2\,{\frac{a \left ( -{a}^{2}c{x}^{2}+c \right ) ^{7/2}}{cx}}-2\,{a}^{3}x \left ( -{a}^{2}c{x}^{2}+c \right ) ^{5/2}-{\frac{5\,{a}^{3}cx}{2} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{a}^{3}{c}^{2}x}{4}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{15\,{a}^{3}{c}^{3}}{4}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{{a}^{2}}{10} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}c}{6} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{2}{c}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }-{\frac{{a}^{2}{c}^{2}}{2}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{2\,{a}^{2}}{5} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{3}cx}{2} \left ( -c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}{c}^{2}x}{4}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+{\frac{3\,{a}^{3}{c}^{3}}{4}\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{1}{2\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{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)^(5/2)/x^3,x)

[Out]

-2*a/c/x*(-a^2*c*x^2+c)^(7/2)-2*a^3*x*(-a^2*c*x^2+c)^(5/2)-5/2*a^3*c*x*(-a^2*c*x^2+c)^(3/2)-15/4*a^3*c^2*x*(-a
^2*c*x^2+c)^(1/2)-15/4*a^3*c^3/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))-1/10*a^2*(-a^2*c*x^2
+c)^(5/2)-1/6*a^2*c*(-a^2*c*x^2+c)^(3/2)+1/2*a^2*c^(5/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-1/2*a^2*(-
a^2*c*x^2+c)^(1/2)*c^2-2/5*a^2*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(5/2)+1/2*a^3*c*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a
))^(3/2)*x+3/4*a^3*c^2*(-c*a^2*(x-1/a)^2-2*a*c*(x-1/a))^(1/2)*x+3/4*a^3*c^3/(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))-1/2/c/x^2*(-a^2*c*x^2+c)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \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)^(5/2)/x^3,x, algorithm="maxima")

[Out]

-integrate((-a^2*c*x^2 + c)^(5/2)*(a*x + 1)^2/((a^2*x^2 - 1)*x^3), x)

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Fricas [A]  time = 2.96939, size = 717, normalized size = 4.75 \begin{align*} \left [\frac{36 \, a^{2} c^{\frac{5}{2}} x^{2} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) + 3 \, a^{2} c^{\frac{5}{2}} x^{2} \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 (2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{12 \, x^{2}}, \frac{3 \, a^{2} \sqrt{-c} c^{2} x^{2} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + 9 \, a^{2} \sqrt{-c} c^{2} x^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) -{\left (2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt{-a^{2} c x^{2} + c}}{6 \, x^{2}}\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)^(5/2)/x^3,x, algorithm="fricas")

[Out]

[1/12*(36*a^2*c^(5/2)*x^2*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + 3*a^2*c^(5/2)*x^2*log(-(a
^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*(2*a^4*c^2*x^4 + 6*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 12*
a*c^2*x + 3*c^2)*sqrt(-a^2*c*x^2 + c))/x^2, 1/6*(3*a^2*sqrt(-c)*c^2*x^2*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/(
a^2*c*x^2 - c)) + 9*a^2*sqrt(-c)*c^2*x^2*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - (2*a^4*c
^2*x^4 + 6*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 12*a*c^2*x + 3*c^2)*sqrt(-a^2*c*x^2 + c))/x^2]

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Sympy [C]  time = 10.2947, size = 401, normalized size = 2.66 \begin{align*} - a^{4} c^{2} \left (\begin{cases} 0 & \text{for}\: c = 0 \\\frac{\sqrt{c} x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} c x^{2} + c\right )^{\frac{3}{2}}}{3 a^{2} c} & \text{otherwise} \end{cases}\right ) - 2 a^{3} c^{2} \left (\begin{cases} \frac{i a^{2} \sqrt{c} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \sqrt{c} \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{\sqrt{c} x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\sqrt{c} \operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) + 2 a c^{2} \left (\begin{cases} - \frac{i a^{2} \sqrt{c} x}{\sqrt{a^{2} x^{2} - 1}} + i a \sqrt{c} \operatorname{acosh}{\left (a x \right )} + \frac{i \sqrt{c}}{x \sqrt{a^{2} x^{2} - 1}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{a^{2} \sqrt{c} x}{\sqrt{- a^{2} x^{2} + 1}} - a \sqrt{c} \operatorname{asin}{\left (a x \right )} - \frac{\sqrt{c}}{x \sqrt{- a^{2} x^{2} + 1}} & \text{otherwise} \end{cases}\right ) + c^{2} \left (\begin{cases} \frac{a^{2} \sqrt{c} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} + \frac{a \sqrt{c}}{2 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{\sqrt{c}}{2 a x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac{i a^{2} \sqrt{c} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a \sqrt{c} \sqrt{1 - \frac{1}{a^{2} x^{2}}}}{2 x} & \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)**(5/2)/x**3,x)

[Out]

-a**4*c**2*Piecewise((0, Eq(c, 0)), (sqrt(c)*x**2/2, Eq(a**2, 0)), (-(-a**2*c*x**2 + c)**(3/2)/(3*a**2*c), Tru
e)) - 2*a**3*c**2*Piecewise((I*a**2*sqrt(c)*x**3/(2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x/(2*sqrt(a**2*x**2 - 1))
- I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (sqrt(c)*x*sqrt(-a**2*x**2 + 1)/2 + sqrt(c)*asin(a*x)/(2*a
), True)) + 2*a*c**2*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*
sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c
)/(x*sqrt(-a**2*x**2 + 1)), True)) + c**2*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 +
1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/
(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True))

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Giac [B]  time = 1.18431, size = 408, normalized size = 2.7 \begin{align*} -\frac{a^{2} c^{3} \arctan \left (-\frac{\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{3 \, a^{3} \sqrt{-c} c^{2} \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \frac{1}{3} \, \sqrt{-a^{2} c x^{2} + c}{\left (a^{2} c^{2} -{\left (a^{4} c^{2} x + 3 \, a^{3} c^{2}\right )} x\right )} - \frac{{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{3} a^{2} c^{3}{\left | a \right |} - 4 \,{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt{-c} c^{3} +{\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )} a^{2} c^{4}{\left | a \right |} + 4 \, a^{3} \sqrt{-c} c^{4}}{{\left ({\left (\sqrt{-a^{2} c} x - \sqrt{-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2}{\left | a \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)^(5/2)/x^3,x, algorithm="giac")

[Out]

-a^2*c^3*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) - 3*a^3*sqrt(-c)*c^2*log(abs(-sqrt
(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/abs(a) + 1/3*sqrt(-a^2*c*x^2 + c)*(a^2*c^2 - (a^4*c^2*x + 3*a^3*c^2)*x) -
((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^3*a^2*c^3*abs(a) - 4*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^3*sq
rt(-c)*c^3 + (sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^2*c^4*abs(a) + 4*a^3*sqrt(-c)*c^4)/(((sqrt(-a^2*c)*x -
sqrt(-a^2*c*x^2 + c))^2 - c)^2*abs(a))