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

Optimal. Leaf size=129 \[ \frac {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8} \]

[Out]

1/11*(-a^2*x^2+1)^(5/2)/a/c^5/(-a*x+1)^8+1/33*(-a^2*x^2+1)^(5/2)/a/c^5/(-a*x+1)^7+2/231*(-a^2*x^2+1)^(5/2)/a/c
^5/(-a*x+1)^6+2/1155*(-a^2*x^2+1)^(5/2)/a/c^5/(-a*x+1)^5

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 659, 651} \[ \frac {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - a^2*x^2)^(5/2)/(11*a*c^5*(1 - a*x)^8) + (1 - a^2*x^2)^(5/2)/(33*a*c^5*(1 - a*x)^7) + (2*(1 - a^2*x^2)^(5/
2))/(231*a*c^5*(1 - a*x)^6) + (2*(1 - a^2*x^2)^(5/2))/(1155*a*c^5*(1 - a*x)^5)

Rule 651

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

Rule 659

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

Rule 6127

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

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^8} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {1}{11} \left (3 c^2\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^7} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {1}{33} (2 c) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^6} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {2}{231} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 51, normalized size = 0.40 \[ -\frac {(a x+1)^{5/2} \left (2 a^3 x^3-16 a^2 x^2+61 a x-152\right )}{1155 a c^5 (1-a x)^{11/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/1155*((1 + a*x)^(5/2)*(-152 + 61*a*x - 16*a^2*x^2 + 2*a^3*x^3))/(a*c^5*(1 - a*x)^(11/2))

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 171, normalized size = 1.33 \[ \frac {152 \, a^{6} x^{6} - 912 \, a^{5} x^{5} + 2280 \, a^{4} x^{4} - 3040 \, a^{3} x^{3} + 2280 \, a^{2} x^{2} - 912 \, a x - {\left (2 \, a^{5} x^{5} - 12 \, a^{4} x^{4} + 31 \, a^{3} x^{3} - 46 \, a^{2} x^{2} - 243 \, a x - 152\right )} \sqrt {-a^{2} x^{2} + 1} + 152}{1155 \, {\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, a^{2} c^{5} x + a c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(152*a^6*x^6 - 912*a^5*x^5 + 2280*a^4*x^4 - 3040*a^3*x^3 + 2280*a^2*x^2 - 912*a*x - (2*a^5*x^5 - 12*a^4
*x^4 + 31*a^3*x^3 - 46*a^2*x^2 - 243*a*x - 152)*sqrt(-a^2*x^2 + 1) + 152)/(a^7*c^5*x^6 - 6*a^6*c^5*x^5 + 15*a^
5*c^5*x^4 - 20*a^4*c^5*x^3 + 15*a^3*c^5*x^2 - 6*a^2*c^5*x + a*c^5)

________________________________________________________________________________________

giac [C]  time = 0.57, size = 509, normalized size = 3.95 \[ \frac {\frac {48 i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}{c^{3}} + \frac {\frac {5 \, {\left (63 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{5} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 385 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 990 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1386 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1155 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 693 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}} + \frac {22 \, {\left (35 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 180 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 378 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 420 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}} + \frac {99 \, {\left (5 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 21 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 35 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)}}{27720 \, c^{2} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27720*(48*I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)/c^3 + (5*(63*(2*c/(a*c*x - c) + 1)^5*sqrt(-2*c/(a*c*x - c) - 1)
- 385*(2*c/(a*c*x - c) + 1)^4*sqrt(-2*c/(a*c*x - c) - 1) + 990*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c)
- 1) - 1386*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) - 1155*(-2*c/(a*c*x - c) - 1)^(3/2) - 693*sqrt(
-2*c/(a*c*x - c) - 1))/c^3 + 22*(35*(2*c/(a*c*x - c) + 1)^4*sqrt(-2*c/(a*c*x - c) - 1) - 180*(2*c/(a*c*x - c)
+ 1)^3*sqrt(-2*c/(a*c*x - c) - 1) + 378*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) + 420*(-2*c/(a*c*x
- c) - 1)^(3/2) + 315*sqrt(-2*c/(a*c*x - c) - 1))/c^3 + 99*(5*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c) -
1) - 21*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) - 35*(-2*c/(a*c*x - c) - 1)^(3/2) - 35*sqrt(-2*c/(a
*c*x - c) - 1))/c^3)/(sgn(1/(a*c*x - c))*sgn(a)*sgn(c)))/(c^2*abs(a))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 57, normalized size = 0.44 \[ -\frac {\left (2 x^{3} a^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x +1\right )^{4}}{1155 \left (a x -1\right )^{4} c^{5} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1155*(2*a^3*x^3-16*a^2*x^2+61*a*x-152)*(a*x+1)^4/(a*x-1)^4/c^5/(-a^2*x^2+1)^(3/2)/a

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 462, normalized size = 3.58 \[ -\frac {8}{11 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{6} c^{5} x^{5} - 5 \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} + 10 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 10 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 5 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {28}{33 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} + 6 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {58}{231 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac {1}{1155 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {1}{1155 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac {2 \, x}{1155 \, \sqrt {-a^{2} x^{2} + 1} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/11/(sqrt(-a^2*x^2 + 1)*a^6*c^5*x^5 - 5*sqrt(-a^2*x^2 + 1)*a^5*c^5*x^4 + 10*sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 -
10*sqrt(-a^2*x^2 + 1)*a^3*c^5*x^2 + 5*sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2*x^2 + 1)*a*c^5) - 28/33/(sqrt(
-a^2*x^2 + 1)*a^5*c^5*x^4 - 4*sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 + 6*sqrt(-a^2*x^2 + 1)*a^3*c^5*x^2 - 4*sqrt(-a^2*
x^2 + 1)*a^2*c^5*x + sqrt(-a^2*x^2 + 1)*a*c^5) - 58/231/(sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 - 3*sqrt(-a^2*x^2 + 1)
*a^3*c^5*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2*x^2 + 1)*a*c^5) + 1/1155/(sqrt(-a^2*x^2 + 1)*a^3*c^5
*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^5*x + sqrt(-a^2*x^2 + 1)*a*c^5) - 1/1155/(sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqr
t(-a^2*x^2 + 1)*a*c^5) + 2/1155*x/(sqrt(-a^2*x^2 + 1)*c^5)

________________________________________________________________________________________

mupad [B]  time = 0.87, size = 604, normalized size = 4.68 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^6}{693\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^6}{231\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {20\,a^6}{99\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}+\frac {4\,a^7}{11\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^6\,\sqrt {-a^2}}+\frac {80\,a^9}{693\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,{\left (-a^2\right )}^{3/2}}+\frac {32\,a^{11}}{693\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{5/2}}\right )}{a^6\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^5}{315\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^5}{105\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {4\,a^5}{9\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}-\frac {16\,a^2\,{\left (-a^2\right )}^{3/2}}{63\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4}+\frac {32\,a^6}{315\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^5\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {3\,a^4}{35\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}-\frac {2\,a^4}{35\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}+\frac {a^5}{7\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,\sqrt {-a^2}}+\frac {2\,a^7}{35\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{3/2}}\right )}{a^4\,\sqrt {-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1 - a^2*x^2)^(1/2)*((32*a^6)/(693*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - (16*a^6)/(231*c^5*(x*(-a^2)^(1/2)
- (-a^2)^(1/2)/a)^3) + (20*a^6)/(99*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^5) + (4*a^7)/(11*c^5*(x*(-a^2)^(1/2
) - (-a^2)^(1/2)/a)^6*(-a^2)^(1/2)) + (80*a^9)/(693*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^4*(-a^2)^(3/2)) + (3
2*a^11)/(693*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^2*(-a^2)^(5/2))))/(a^6*(-a^2)^(1/2)) - ((1 - a^2*x^2)^(1/2)
*((32*a^5)/(315*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - (16*a^5)/(105*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^3
) + (4*a^5)/(9*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^5) - (16*a^2*(-a^2)^(3/2))/(63*c^5*(x*(-a^2)^(1/2) - (-a^
2)^(1/2)/a)^4) + (32*a^6)/(315*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^2*(-a^2)^(1/2))))/(a^5*(-a^2)^(1/2)) - ((
1 - a^2*x^2)^(1/2)*((3*a^4)/(35*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^3) - (2*a^4)/(35*c^5*(x*(-a^2)^(1/2) - (
-a^2)^(1/2)/a)) + a^5/(7*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^4*(-a^2)^(1/2)) + (2*a^7)/(35*c^5*(x*(-a^2)^(1/
2) - (-a^2)^(1/2)/a)^2*(-a^2)^(3/2))))/(a^4*(-a^2)^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {3 a x}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(Integral(3*a*x/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*
x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2
+ 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**2/(-a**7*x**7*sqrt(-a**2*x*
*2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 +
1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-
a**2*x**2 + 1)), x) + Integral(a**3*x**3/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) -
9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a*
*2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**7*x**7
*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt
(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2
+ 1) - sqrt(-a**2*x**2 + 1)), x))/c**5

________________________________________________________________________________________