3.11.95 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{3/2}}{x^6} \, dx\) [1095]

Optimal. Leaf size=131 \[ -\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} a^5 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]

[Out]

-1/5*(-a^2*c*x^2+c)^(3/2)/x^5-1/2*a*(-a^2*c*x^2+c)^(3/2)/x^4-7/15*a^2*(-a^2*c*x^2+c)^(3/2)/x^3+1/4*a^5*c^(3/2)
*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))-1/4*a^3*c*(-a^2*c*x^2+c)^(1/2)/x^2

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Rubi [A]
time = 0.20, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6286, 1821, 849, 821, 272, 43, 65, 214} \begin {gather*} -\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} a^5 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^6,x]

[Out]

-1/4*(a^3*c*Sqrt[c - a^2*c*x^2])/x^2 - (c - a^2*c*x^2)^(3/2)/(5*x^5) - (a*(c - a^2*c*x^2)^(3/2))/(2*x^4) - (7*
a^2*(c - a^2*c*x^2)^(3/2))/(15*x^3) + (a^5*c^(3/2)*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/4

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])

Rule 6286

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]

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^6} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^6} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} \int \frac {\left (-10 a c-7 a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x^5} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}+\frac {\int \frac {\left (28 a^2 c^2+10 a^3 c^2 x\right ) \sqrt {c-a^2 c x^2}}{x^4} \, dx}{20 c}\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{2} \left (a^3 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}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} \left (a^3 c\right ) \text {Subst}\left (\int \frac {\sqrt {c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}-\frac {1}{8} \left (a^5 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {a^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} \left (a^3 c\right ) \text {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^3 c \sqrt {c-a^2 c x^2}}{4 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{5 x^5}-\frac {a \left (c-a^2 c x^2\right )^{3/2}}{2 x^4}-\frac {7 a^2 \left (c-a^2 c x^2\right )^{3/2}}{15 x^3}+\frac {1}{4} a^5 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 104, normalized size = 0.79 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-12-30 a x-16 a^2 x^2+15 a^3 x^3+28 a^4 x^4\right )}{60 x^5}-\frac {1}{4} a^5 c^{3/2} \log (x)+\frac {1}{4} a^5 c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^6,x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-12 - 30*a*x - 16*a^2*x^2 + 15*a^3*x^3 + 28*a^4*x^4))/(60*x^5) - (a^5*c^(3/2)*Log[x])/
4 + (a^5*c^(3/2)*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(685\) vs. \(2(107)=214\).
time = 0.07, size = 686, normalized size = 5.24

method	result	size
risch	\(-\frac {\left (28 x^{6} a^{6}+15 x^{5} a^{5}-44 a^{4} x^{4}-45 a^{3} x^{3}+4 a^{2} x^{2}+30 a x +12\right ) c^{2}}{60 x^{5} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {a^{5} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{4}\)	\(105\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 c \,x^{5}}+2 a^{4} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )+2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )}{3}\right )+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}-\frac {a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )}{4}\right )+2 a^{5} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a^{5} \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )\right )+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )\)	\(686\)

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^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (107) = 214\).
time = 0.47, size = 221, normalized size = 1.69 \begin {gather*} \frac {{\left (a^{2} c^{\frac {3}{2}} x^{2} - c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1} a^{2}}{3 \, x^{3}} - \frac {a^{6} c^{\frac {5}{2}} \log \left (\frac {\sqrt {-a^{2} c x^{2} + c} - \sqrt {c}}{\sqrt {-a^{2} c x^{2} + c} + \sqrt {c}}\right ) + \frac {2 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{6} c^{3} + \sqrt {-a^{2} c x^{2} + c} a^{6} c^{4}\right )}}{{\left (a^{2} c x^{2} - c\right )}^{2} + 2 \, {\left (a^{2} c x^{2} - c\right )} c + c^{2}}}{8 \, a c} + \frac {{\left (2 \, a^{4} c^{\frac {3}{2}} x^{4} + a^{2} c^{\frac {3}{2}} x^{2} - 3 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, x^{5}} \end {gather*}

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^6,x, algorithm="maxima")

[Out]

1/3*(a^2*c^(3/2)*x^2 - c^(3/2))*sqrt(a*x + 1)*sqrt(-a*x + 1)*a^2/x^3 - 1/8*(a^6*c^(5/2)*log((sqrt(-a^2*c*x^2 +
c) - sqrt(c))/(sqrt(-a^2*c*x^2 + c) + sqrt(c))) + 2*((-a^2*c*x^2 + c)^(3/2)*a^6*c^3 + sqrt(-a^2*c*x^2 + c)*a^
6*c^4)/((a^2*c*x^2 - c)^2 + 2*(a^2*c*x^2 - c)*c + c^2))/(a*c) + 1/15*(2*a^4*c^(3/2)*x^4 + a^2*c^(3/2)*x^2 - 3*
c^(3/2))*sqrt(a*x + 1)*sqrt(-a*x + 1)/x^5

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Fricas [A]
time = 0.37, size = 209, normalized size = 1.60 \begin {gather*} \left [\frac {15 \, a^{5} c^{\frac {3}{2}} x^{5} \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 (28 \, a^{4} c x^{4} + 15 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} - 30 \, a c x - 12 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{120 \, x^{5}}, \frac {15 \, a^{5} \sqrt {-c} c x^{5} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (28 \, a^{4} c x^{4} + 15 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} - 30 \, a c x - 12 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{60 \, x^{5}}\right ] \end {gather*}

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^6,x, algorithm="fricas")

[Out]

[1/120*(15*a^5*c^(3/2)*x^5*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + 2*(28*a^4*c*x^4 + 15
*a^3*c*x^3 - 16*a^2*c*x^2 - 30*a*c*x - 12*c)*sqrt(-a^2*c*x^2 + c))/x^5, 1/60*(15*a^5*sqrt(-c)*c*x^5*arctan(sqr
t(-a^2*c*x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (28*a^4*c*x^4 + 15*a^3*c*x^3 - 16*a^2*c*x^2 - 30*a*c*x - 12*c)*s
qrt(-a^2*c*x^2 + c))/x^5]

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Sympy [C] Result contains complex when optimal does not.
time = 11.64, size = 484, normalized size = 3.69 \begin {gather*} a^{2} c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + 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 ) \end {gather*}

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**6,x)

[Out]

a**2*c*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs
(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), Tr
ue)) + 2*a*c*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - a**3*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a*sqrt
(c)/(8*x**3*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-
I*a**4*sqrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*I*a*sqrt(c)/(8*x**3*sqrt(1 -
1/(a**2*x**2))) + I*sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True)) + 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))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (107) = 214\).
time = 0.43, size = 414, normalized size = 3.16 \begin {gather*} -\frac {a^{5} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{2 \, \sqrt {-c}} + \frac {15 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{9} a^{5} c^{2} - 60 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{8} a^{4} \sqrt {-c} c^{2} {\left | a \right |} + 90 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{5} c^{3} + 240 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{4} \sqrt {-c} c^{3} {\left | a \right |} - 40 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{4} \sqrt {-c} c^{4} {\left | a \right |} - 90 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{5} c^{5} + 80 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{4} \sqrt {-c} c^{5} {\left | a \right |} - 15 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{5} c^{6} - 28 \, a^{4} \sqrt {-c} c^{6} {\left | a \right |}}{30 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{5}} \end {gather*}

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^6,x, algorithm="giac")

[Out]

-1/2*a^5*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c) + 1/30*(15*(sqrt(-a^2*c)*x - s
qrt(-a^2*c*x^2 + c))^9*a^5*c^2 - 60*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^8*a^4*sqrt(-c)*c^2*abs(a) + 90*(sq
rt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^7*a^5*c^3 + 240*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^6*a^4*sqrt(-c)*c^
3*abs(a) - 40*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^4*a^4*sqrt(-c)*c^4*abs(a) - 90*(sqrt(-a^2*c)*x - sqrt(-a
^2*c*x^2 + c))^3*a^5*c^5 + 80*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^4*sqrt(-c)*c^5*abs(a) - 15*(sqrt(-a^
2*c)*x - sqrt(-a^2*c*x^2 + c))*a^5*c^6 - 28*a^4*sqrt(-c)*c^6*abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^
2 - c)^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^6\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(x^6*(a^2*x^2 - 1)),x)

[Out]

-int(((c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(x^6*(a^2*x^2 - 1)), x)

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