3.7.23 \(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\) [623]

Optimal. Leaf size=176 \[ -\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {77 c^{9/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a} \]

[Out]

-77/384*c^3*x*(-a^2*c*x^2+c)^(3/2)-77/480*c^2*x*(-a^2*c*x^2+c)^(5/2)-11/80*c*x*(-a^2*c*x^2+c)^(7/2)+11/90*(-a^
2*c*x^2+c)^(9/2)/a+1/10*(a*x+1)*(-a^2*c*x^2+c)^(9/2)/a-77/256*c^(9/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))
/a-77/256*c^4*x*(-a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6276, 685, 655, 201, 223, 209} \begin {gather*} -\frac {77 c^{9/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a}-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2),x]

[Out]

(-77*c^4*x*Sqrt[c - a^2*c*x^2])/256 - (77*c^3*x*(c - a^2*c*x^2)^(3/2))/384 - (77*c^2*x*(c - a^2*c*x^2)^(5/2))/
480 - (11*c*x*(c - a^2*c*x^2)^(7/2))/80 + (11*(c - a^2*c*x^2)^(9/2))/(90*a) + ((1 + a*x)*(c - a^2*c*x^2)^(9/2)
)/(10*a) - (77*c^(9/2)*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(256*a)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[2*c*d*((m + p)/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rule 6276

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps

\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2} \, dx\right )\\ &=\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{10} (11 c) \int (1+a x) \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{10} (11 c) \int \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{80} \left (77 c^2\right ) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{96} \left (77 c^3\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{128} \left (77 c^4\right ) \int \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{256} \left (77 c^5\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{256} \left (77 c^5\right ) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {77 c^{9/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 167, normalized size = 0.95 \begin {gather*} \frac {c^4 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (2560-10615 a x-2185 a^2 x^2+16390 a^3 x^3+9210 a^4 x^4-15048 a^5 x^5-10552 a^6 x^6+7216 a^7 x^7+5584 a^8 x^8-1408 a^9 x^9-1152 a^{10} x^{10}\right )+6930 \sqrt {1-a x} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{11520 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2),x]

[Out]

(c^4*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(2560 - 10615*a*x - 2185*a^2*x^2 + 16390*a^3*x^3 + 9210*a^4*x^4 - 1504
8*a^5*x^5 - 10552*a^6*x^6 + 7216*a^7*x^7 + 5584*a^8*x^8 - 1408*a^9*x^9 - 1152*a^10*x^10) + 6930*Sqrt[1 - a*x]*
ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(11520*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(453\) vs. \(2(144)=288\).
time = 0.21, size = 454, normalized size = 2.58

method	result	size
risch	\(-\frac {\left (1152 a^{9} x^{9}+2560 a^{8} x^{8}-3024 a^{7} x^{7}-10240 a^{6} x^{6}+312 a^{5} x^{5}+15360 a^{4} x^{4}+6150 a^{3} x^{3}-10240 a^{2} x^{2}-8055 a x +2560\right ) \left (a^{2} x^{2}-1\right ) c^{5}}{11520 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {77 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{5}}{256 \sqrt {a^{2} c}}\)	\(138\)
default	\(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{10}+\frac {9 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}+\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}+\frac {\frac {2 \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {9}{2}}}{9}-2 a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{16 a^{2} c}+\frac {7 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}\right )}{a}\)	\(454\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/10*x*(-a^2*c*x^2+c)^(9/2)+9/10*c*(1/8*x*(-a^2*c*x^2+c)^(7/2)+7/8*c*(1/6*x*(-a^2*c*x^2+c)^(5/2)+5/6*c*(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/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+
c)^(1/2))))))+2/a*(1/9*(-a^2*c*(x-1/a)^2-2*(x-1/a)*a*c)^(9/2)-a*c*(-1/16*(-2*a^2*c*(x-1/a)-2*a*c)/a^2/c*(-a^2*
c*(x-1/a)^2-2*(x-1/a)*a*c)^(7/2)+7/8*c*(-1/12*(-2*a^2*c*(x-1/a)-2*a*c)/a^2/c*(-a^2*c*(x-1/a)^2-2*(x-1/a)*a*c)^
(5/2)+5/6*c*(-1/8*(-2*a^2*c*(x-1/a)-2*a*c)/a^2/c*(-a^2*c*(x-1/a)^2-2*(x-1/a)*a*c)^(3/2)+3/4*c*(-1/4*(-2*a^2*c*
(x-1/a)-2*a*c)/a^2/c*(-a^2*c*(x-1/a)^2-2*(x-1/a)*a*c)^(1/2)+1/2*c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c
*(x-1/a)^2-2*(x-1/a)*a*c)^(1/2)))))))

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Maxima [A]
time = 0.51, size = 192, normalized size = 1.09 \begin {gather*} \frac {1}{10} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} x - \frac {11}{80} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} c x - \frac {77}{480} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c^{2} x - \frac {77}{384} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{3} x - \frac {35}{64} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{4} x + \frac {63}{256} \, \sqrt {-a^{2} c x^{2} + c} c^{4} x - \frac {35 \, c^{6} \arcsin \left (a x - 2\right )}{64 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {63 \, c^{\frac {9}{2}} \arcsin \left (a x\right )}{256 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{9 \, a} + \frac {35 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{4}}{32 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

1/10*(-a^2*c*x^2 + c)^(9/2)*x - 11/80*(-a^2*c*x^2 + c)^(7/2)*c*x - 77/480*(-a^2*c*x^2 + c)^(5/2)*c^2*x - 77/38
4*(-a^2*c*x^2 + c)^(3/2)*c^3*x - 35/64*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^4*x + 63/256*sqrt(-a^2*c*x^2 + c)*c^4
*x - 35/64*c^6*arcsin(a*x - 2)/(a*(-c)^(3/2)) + 63/256*c^(9/2)*arcsin(a*x)/a + 2/9*(-a^2*c*x^2 + c)^(9/2)/a +
35/32*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^4/a

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Fricas [A]
time = 0.41, size = 329, normalized size = 1.87 \begin {gather*} \left [\frac {3465 \, \sqrt {-c} c^{4} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt {-a^{2} c x^{2} + c}}{23040 \, a}, \frac {3465 \, c^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt {-a^{2} c x^{2} + c}}{11520 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

[1/23040*(3465*sqrt(-c)*c^4*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(1152*a^9*c^4*x^9 +
2560*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 - 10240*a^6*c^4*x^6 + 312*a^5*c^4*x^5 + 15360*a^4*c^4*x^4 + 6150*a^3*c^4*
x^3 - 10240*a^2*c^4*x^2 - 8055*a*c^4*x + 2560*c^4)*sqrt(-a^2*c*x^2 + c))/a, 1/11520*(3465*c^(9/2)*arctan(sqrt(
-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (1152*a^9*c^4*x^9 + 2560*a^8*c^4*x^8 - 3024*a^7*c^4*x^7 - 10240
*a^6*c^4*x^6 + 312*a^5*c^4*x^5 + 15360*a^4*c^4*x^4 + 6150*a^3*c^4*x^3 - 10240*a^2*c^4*x^2 - 8055*a*c^4*x + 256
0*c^4)*sqrt(-a^2*c*x^2 + c))/a]

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Sympy [C] Result contains complex when optimal does not.
time = 150.00, size = 1340, normalized size = 7.61 \begin {gather*} a^{8} c^{4} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{11}}{10 \sqrt {a^{2} x^{2} - 1}} - \frac {9 i \sqrt {c} x^{9}}{80 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{7}}{480 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} x^{5}}{1920 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} x^{3}}{768 a^{6} \sqrt {a^{2} x^{2} - 1}} + \frac {7 i \sqrt {c} x}{256 a^{8} \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{256 a^{9}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{11}}{10 \sqrt {- a^{2} x^{2} + 1}} + \frac {9 \sqrt {c} x^{9}}{80 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{7}}{480 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} x^{5}}{1920 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} x^{3}}{768 a^{6} \sqrt {- a^{2} x^{2} + 1}} - \frac {7 \sqrt {c} x}{256 a^{8} \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} \operatorname {asin}{\left (a x \right )}}{256 a^{9}} & \text {otherwise} \end {cases}\right ) + 2 a^{7} c^{4} \left (\begin {cases} \frac {x^{8} \sqrt {- a^{2} c x^{2} + c}}{9} - \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{63 a^{2}} - \frac {2 x^{4} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 x^{2} \sqrt {- a^{2} c x^{2} + c}}{315 a^{6}} - \frac {16 \sqrt {- a^{2} c x^{2} + c}}{315 a^{8}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{8}}{8} & \text {otherwise} \end {cases}\right ) - 2 a^{6} c^{4} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{9}}{8 \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} x^{7}}{48 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{5}}{192 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{3}}{384 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i \sqrt {c} x}{128 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{128 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{9}}{8 \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} x^{7}}{48 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{5}}{192 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{3}}{384 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 \sqrt {c} x}{128 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} \operatorname {asin}{\left (a x \right )}}{128 a^{7}} & \text {otherwise} \end {cases}\right ) - 6 a^{5} c^{4} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 6 a^{3} c^{4} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) - 2 a c^{4} \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 ) - c^{4} \left (\begin {cases} \frac {i \sqrt {c} x \sqrt {a^{2} x^{2} - 1}}{2} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a**2*c*x**2+c)**(9/2),x)

[Out]

a**8*c**4*Piecewise((I*a**2*sqrt(c)*x**11/(10*sqrt(a**2*x**2 - 1)) - 9*I*sqrt(c)*x**9/(80*sqrt(a**2*x**2 - 1))
- I*sqrt(c)*x**7/(480*a**2*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**5/(1920*a**4*sqrt(a**2*x**2 - 1)) - 7*I*sqrt
(c)*x**3/(768*a**6*sqrt(a**2*x**2 - 1)) + 7*I*sqrt(c)*x/(256*a**8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*acosh(a*x
)/(256*a**9), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**11/(10*sqrt(-a**2*x**2 + 1)) + 9*sqrt(c)*x**9/(80*sqrt(-a
**2*x**2 + 1)) + sqrt(c)*x**7/(480*a**2*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**5/(1920*a**4*sqrt(-a**2*x**2 + 1)
) + 7*sqrt(c)*x**3/(768*a**6*sqrt(-a**2*x**2 + 1)) - 7*sqrt(c)*x/(256*a**8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*a
sin(a*x)/(256*a**9), True)) + 2*a**7*c**4*Piecewise((x**8*sqrt(-a**2*c*x**2 + c)/9 - x**6*sqrt(-a**2*c*x**2 +
c)/(63*a**2) - 2*x**4*sqrt(-a**2*c*x**2 + c)/(105*a**4) - 8*x**2*sqrt(-a**2*c*x**2 + c)/(315*a**6) - 16*sqrt(-
a**2*c*x**2 + c)/(315*a**8), Ne(a, 0)), (sqrt(c)*x**8/8, True)) - 2*a**6*c**4*Piecewise((I*a**2*sqrt(c)*x**9/(
8*sqrt(a**2*x**2 - 1)) - 7*I*sqrt(c)*x**7/(48*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*x**5/(192*a**2*sqrt(a**2*x**2 -
1)) - 5*I*sqrt(c)*x**3/(384*a**4*sqrt(a**2*x**2 - 1)) + 5*I*sqrt(c)*x/(128*a**6*sqrt(a**2*x**2 - 1)) - 5*I*sq
rt(c)*acosh(a*x)/(128*a**7), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**9/(8*sqrt(-a**2*x**2 + 1)) + 7*sqrt(c)*x**
7/(48*sqrt(-a**2*x**2 + 1)) + sqrt(c)*x**5/(192*a**2*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*x**3/(384*a**4*sqrt(-a*
*2*x**2 + 1)) - 5*sqrt(c)*x/(128*a**6*sqrt(-a**2*x**2 + 1)) + 5*sqrt(c)*asin(a*x)/(128*a**7), True)) - 6*a**5*
c**4*Piecewise((x**6*sqrt(-a**2*c*x**2 + c)/7 - x**4*sqrt(-a**2*c*x**2 + c)/(35*a**2) - 4*x**2*sqrt(-a**2*c*x*
*2 + c)/(105*a**4) - 8*sqrt(-a**2*c*x**2 + c)/(105*a**6), Ne(a, 0)), (sqrt(c)*x**6/6, True)) + 6*a**3*c**4*Pie
cewise((x**4*sqrt(-a**2*c*x**2 + c)/5 - x**2*sqrt(-a**2*c*x**2 + c)/(15*a**2) - 2*sqrt(-a**2*c*x**2 + c)/(15*a
**4), Ne(a, 0)), (sqrt(c)*x**4/4, True)) + 2*a**2*c**4*Piecewise((I*a**2*sqrt(c)*x**5/(4*sqrt(a**2*x**2 - 1))
- 3*I*sqrt(c)*x**3/(8*sqrt(a**2*x**2 - 1)) + I*sqrt(c)*x/(8*a**2*sqrt(a**2*x**2 - 1)) - I*sqrt(c)*acosh(a*x)/(
8*a**3), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**5/(4*sqrt(-a**2*x**2 + 1)) + 3*sqrt(c)*x**3/(8*sqrt(-a**2*x**2
+ 1)) - sqrt(c)*x/(8*a**2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(8*a**3), True)) - 2*a*c**4*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), True)) - c**4*Piecewise((I*
sqrt(c)*x*sqrt(a**2*x**2 - 1)/2 - I*sqrt(c)*acosh(a*x)/(2*a), Abs(a**2*x**2) > 1), (-a**2*sqrt(c)*x**3/(2*sqrt
(-a**2*x**2 + 1)) + sqrt(c)*x/(2*sqrt(-a**2*x**2 + 1)) + sqrt(c)*asin(a*x)/(2*a), True))

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Giac [A]
time = 0.43, size = 164, normalized size = 0.93 \begin {gather*} \frac {77 \, c^{5} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{256 \, \sqrt {-c} {\left | a \right |}} + \frac {1}{11520} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {2560 \, c^{4}}{a} - {\left (8055 \, c^{4} + 2 \, {\left (5120 \, a c^{4} - {\left (3075 \, a^{2} c^{4} + 4 \, {\left (1920 \, a^{3} c^{4} + {\left (39 \, a^{4} c^{4} - 2 \, {\left (640 \, a^{5} c^{4} + {\left (189 \, a^{6} c^{4} - 8 \, {\left (9 \, a^{8} c^{4} x + 20 \, a^{7} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x-1)*(a*x+1)*(-a^2*c*x^2+c)^(9/2),x, algorithm="giac")

[Out]

77/256*c^5*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs(a)) + 1/11520*sqrt(-a^2*c*x^2 + c)*(
2560*c^4/a - (8055*c^4 + 2*(5120*a*c^4 - (3075*a^2*c^4 + 4*(1920*a^3*c^4 + (39*a^4*c^4 - 2*(640*a^5*c^4 + (189
*a^6*c^4 - 8*(9*a^8*c^4*x + 20*a^7*c^4)*x)*x)*x)*x)*x)*x)*x)*x)

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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 )}^{9/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a^2*c*x^2)^(9/2)*(a*x + 1))/(a*x - 1),x)

[Out]

int(((c - a^2*c*x^2)^(9/2)*(a*x + 1))/(a*x - 1), x)

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