3.11.97 \(\int \frac {e^{2 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{3/2}}{x^8} \, dx\) [1097]

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

[Out]

-1/7*(-a^2*c*x^2+c)^(3/2)/x^7-1/3*a*(-a^2*c*x^2+c)^(3/2)/x^6-11/35*a^2*(-a^2*c*x^2+c)^(3/2)/x^5-1/4*a^3*(-a^2*
c*x^2+c)^(3/2)/x^4-22/105*a^4*(-a^2*c*x^2+c)^(3/2)/x^3+1/8*a^7*c^(3/2)*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))-1
/8*a^5*c*(-a^2*c*x^2+c)^(1/2)/x^2

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Rubi [A]
time = 0.25, antiderivative size = 181, normalized size of antiderivative = 1.00, 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 = {6286, 1821, 849, 821, 272, 43, 65, 214} \begin {gather*} -\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 {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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(a^5*c*Sqrt[c - a^2*c*x^2])/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 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^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 ) \text {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 ) \text {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 ) \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^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*}

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Mathematica [A]
time = 0.13, size = 120, normalized size = 0.66 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-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\right )}{840 x^7}-\frac {1}{8} a^7 c^{3/2} \log (x)+\frac {1}{8} a^7 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^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] Leaf count of result is larger than twice the leaf count of optimal. \(859\) vs. \(2(149)=298\).
time = 0.07, size = 860, normalized size = 4.75

method	result	size
risch	\(-\frac {\left (176 a^{8} x^{8}+105 a^{7} x^{7}-88 x^{6} a^{6}-35 x^{5} a^{5}-232 a^{4} x^{4}-350 a^{3} x^{3}+24 a^{2} x^{2}+280 a x +120\right ) c^{2}}{840 x^{7} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {a^{7} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\)	\(121\)
default	\(-\frac {16 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{35 c \,x^{5}}+2 a^{6} \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 \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,x^{6}}+\frac {a^{2} \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 )}{6}\right )+2 a^{4} \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^{3} \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 )-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{7 c \,x^{7}}+2 a^{7} \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^{7} \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^{5} \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 )\)	\(860\)

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,method=_RETURNVERBOSE)

[Out]

-16/35*a^2/c/x^5*(-a^2*c*x^2+c)^(5/2)+2*a^6*(-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*(-1/6/c
/x^6*(-a^2*c*x^2+c)^(5/2)+1/6*a^2*(-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^4*(-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^3
*(-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)))))-1/7/c/x^7*(-a^2*c*x^2+c)^(5/2)+2
*a^7*(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^7*(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^5*(
-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 [A]
time = 0.49, size = 287, normalized size = 1.59 \begin {gather*} -\frac {3 \, a^{8} 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 (3 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{8} c^{3} - 8 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{8} c^{4} - 3 \, \sqrt {-a^{2} c x^{2} + c} a^{8} c^{5}\right )}}{{\left (a^{2} c x^{2} - c\right )}^{3} + 3 \, {\left (a^{2} c x^{2} - c\right )}^{2} c + 3 \, {\left (a^{2} c x^{2} - c\right )} c^{2} + c^{3}}}{48 \, 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} a^{2}}{15 \, x^{5}} + \frac {{\left (8 \, a^{6} c^{\frac {3}{2}} x^{6} + 4 \, a^{4} c^{\frac {3}{2}} x^{4} + 3 \, a^{2} c^{\frac {3}{2}} x^{2} - 15 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{105 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-1/48*(3*a^8*c^(5/2)*log((sqrt(-a^2*c*x^2 + c) - sqrt(c))/(sqrt(-a^2*c*x^2 + c) + sqrt(c))) - 2*(3*(-a^2*c*x^2
+ c)^(5/2)*a^8*c^3 - 8*(-a^2*c*x^2 + c)^(3/2)*a^8*c^4 - 3*sqrt(-a^2*c*x^2 + c)*a^8*c^5)/((a^2*c*x^2 - c)^3 +
3*(a^2*c*x^2 - c)^2*c + 3*(a^2*c*x^2 - c)*c^2 + c^3))/(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)*a^2/x^5 + 1/105*(8*a^6*c^(3/2)*x^6 + 4*a^4*c^(3/2)*x^4 + 3*a^2*c^(3/2)*x^2
- 15*c^(3/2))*sqrt(a*x + 1)*sqrt(-a*x + 1)/x^7

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Fricas [A]
time = 0.35, size = 245, normalized size = 1.35 \begin {gather*} \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 {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^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] Result contains complex when optimal does not.
time = 17.30, size = 660, normalized size = 3.65 \begin {gather*} 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 {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**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] Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (149) = 298\).
time = 0.44, size = 529, normalized size = 2.92 \begin {gather*} -\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 {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^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

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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^8\,\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^8*(a^2*x^2 - 1)),x)

[Out]

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

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