3.243 \(\int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{9/2} \, dx\)

Optimal. Leaf size=141 \[ \frac{256 c^7 \left (1-a^2 x^2\right )^{5/2}}{1155 a (c-a c x)^{5/2}}+\frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt{c-a c x}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2} \sqrt{c-a c x}}{11 a} \]

[Out]

(256*c^7*(1 - a^2*x^2)^(5/2))/(1155*a*(c - a*c*x)^(5/2)) + (64*c^6*(1 - a^2*x^2)^(5/2))/(231*a*(c - a*c*x)^(3/
2)) + (8*c^5*(1 - a^2*x^2)^(5/2))/(33*a*Sqrt[c - a*c*x]) + (2*c^4*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(5/2))/(11*a)

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Rubi [A]  time = 0.108989, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6127, 657, 649} \[ \frac{256 c^7 \left (1-a^2 x^2\right )^{5/2}}{1155 a (c-a c x)^{5/2}}+\frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt{c-a c x}}+\frac{2 c^4 \left (1-a^2 x^2\right )^{5/2} \sqrt{c-a c x}}{11 a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a*c*x)^(9/2),x]

[Out]

(256*c^7*(1 - a^2*x^2)^(5/2))/(1155*a*(c - a*c*x)^(5/2)) + (64*c^6*(1 - a^2*x^2)^(5/2))/(231*a*(c - a*c*x)^(3/
2)) + (8*c^5*(1 - a^2*x^2)^(5/2))/(33*a*Sqrt[c - a*c*x]) + (2*c^4*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(5/2))/(11*a)

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]

Rule 657

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*Simplify[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, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

Rule 649

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*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

\begin{align*} \int e^{3 \tanh ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=c^3 \int (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{2 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}+\frac{1}{11} \left (12 c^4\right ) \int \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2} \, dx\\ &=\frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt{c-a c x}}+\frac{2 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}+\frac{1}{33} \left (32 c^5\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt{c-a c x}}+\frac{2 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}+\frac{1}{231} \left (128 c^6\right ) \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac{256 c^7 \left (1-a^2 x^2\right )^{5/2}}{1155 a (c-a c x)^{5/2}}+\frac{64 c^6 \left (1-a^2 x^2\right )^{5/2}}{231 a (c-a c x)^{3/2}}+\frac{8 c^5 \left (1-a^2 x^2\right )^{5/2}}{33 a \sqrt{c-a c x}}+\frac{2 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{5/2}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0450786, size = 62, normalized size = 0.44 \[ -\frac{2 c^4 (a x+1)^{5/2} \left (105 a^3 x^3-455 a^2 x^2+755 a x-533\right ) \sqrt{c-a c x}}{1155 a \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a*c*x)^(9/2),x]

[Out]

(-2*c^4*(1 + a*x)^(5/2)*Sqrt[c - a*c*x]*(-533 + 755*a*x - 455*a^2*x^2 + 105*a^3*x^3))/(1155*a*Sqrt[1 - a*x])

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Maple [A]  time = 0.039, size = 63, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( 105\,{x}^{3}{a}^{3}-455\,{a}^{2}{x}^{2}+755\,ax-533 \right ) \left ( ax+1 \right ) ^{4}}{1155\, \left ( ax-1 \right ) ^{3}a} \left ( -acx+c \right ) ^{{\frac{9}{2}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

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)^(9/2),x)

[Out]

2/1155*(a*x+1)^4*(105*a^3*x^3-455*a^2*x^2+755*a*x-533)*(-a*c*x+c)^(9/2)/a/(a*x-1)^3/(-a^2*x^2+1)^(3/2)

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Maxima [B]  time = 1.15545, size = 343, normalized size = 2.43 \begin{align*} -\frac{2 \,{\left (35 \, a^{6} c^{\frac{9}{2}} x^{6} - 175 \, a^{5} c^{\frac{9}{2}} x^{5} + 415 \, a^{4} c^{\frac{9}{2}} x^{4} - 741 \, a^{3} c^{\frac{9}{2}} x^{3} + 1482 \, a^{2} c^{\frac{9}{2}} x^{2} - 5928 \, a c^{\frac{9}{2}} x - 11856 \, c^{\frac{9}{2}}\right )}}{385 \, \sqrt{a x + 1} a} - \frac{2 \,{\left (7 \, a^{5} c^{\frac{9}{2}} x^{5} - 37 \, a^{4} c^{\frac{9}{2}} x^{4} + 97 \, a^{3} c^{\frac{9}{2}} x^{3} - 215 \, a^{2} c^{\frac{9}{2}} x^{2} + 860 \, a c^{\frac{9}{2}} x + 1720 \, c^{\frac{9}{2}}\right )}}{21 \, \sqrt{a x + 1} a} - \frac{6 \,{\left (5 \, a^{4} c^{\frac{9}{2}} x^{4} - 29 \, a^{3} c^{\frac{9}{2}} x^{3} + 93 \, a^{2} c^{\frac{9}{2}} x^{2} - 407 \, a c^{\frac{9}{2}} x - 814 \, c^{\frac{9}{2}}\right )}}{35 \, \sqrt{a x + 1} a} - \frac{2 \,{\left (a^{3} c^{\frac{9}{2}} x^{3} - 7 \, a^{2} c^{\frac{9}{2}} x^{2} + 43 \, a c^{\frac{9}{2}} x + 91 \, c^{\frac{9}{2}}\right )}}{5 \, \sqrt{a x + 1} a} \end{align*}

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)^(9/2),x, algorithm="maxima")

[Out]

-2/385*(35*a^6*c^(9/2)*x^6 - 175*a^5*c^(9/2)*x^5 + 415*a^4*c^(9/2)*x^4 - 741*a^3*c^(9/2)*x^3 + 1482*a^2*c^(9/2
)*x^2 - 5928*a*c^(9/2)*x - 11856*c^(9/2))/(sqrt(a*x + 1)*a) - 2/21*(7*a^5*c^(9/2)*x^5 - 37*a^4*c^(9/2)*x^4 + 9
7*a^3*c^(9/2)*x^3 - 215*a^2*c^(9/2)*x^2 + 860*a*c^(9/2)*x + 1720*c^(9/2))/(sqrt(a*x + 1)*a) - 6/35*(5*a^4*c^(9
/2)*x^4 - 29*a^3*c^(9/2)*x^3 + 93*a^2*c^(9/2)*x^2 - 407*a*c^(9/2)*x - 814*c^(9/2))/(sqrt(a*x + 1)*a) - 2/5*(a^
3*c^(9/2)*x^3 - 7*a^2*c^(9/2)*x^2 + 43*a*c^(9/2)*x + 91*c^(9/2))/(sqrt(a*x + 1)*a)

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Fricas [A]  time = 2.21952, size = 204, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (105 \, a^{5} c^{4} x^{5} - 245 \, a^{4} c^{4} x^{4} - 50 \, a^{3} c^{4} x^{3} + 522 \, a^{2} c^{4} x^{2} - 311 \, a c^{4} x - 533 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{1155 \,{\left (a^{2} x - a\right )}} \end{align*}

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)^(9/2),x, algorithm="fricas")

[Out]

2/1155*(105*a^5*c^4*x^5 - 245*a^4*c^4*x^4 - 50*a^3*c^4*x^3 + 522*a^2*c^4*x^2 - 311*a*c^4*x - 533*c^4)*sqrt(-a^
2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.30233, size = 99, normalized size = 0.7 \begin{align*} -\frac{2 \,{\left (512 \, \sqrt{2} c^{\frac{7}{2}} + \frac{105 \,{\left (a c x + c\right )}^{\frac{11}{2}} - 770 \,{\left (a c x + c\right )}^{\frac{9}{2}} c + 1980 \,{\left (a c x + c\right )}^{\frac{7}{2}} c^{2} - 1848 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{3}}{c^{2}}\right )} c^{2}}{1155 \, a{\left | c \right |}} \end{align*}

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)^(9/2),x, algorithm="giac")

[Out]

-2/1155*(512*sqrt(2)*c^(7/2) + (105*(a*c*x + c)^(11/2) - 770*(a*c*x + c)^(9/2)*c + 1980*(a*c*x + c)^(7/2)*c^2
- 1848*(a*c*x + c)^(5/2)*c^3)/c^2)*c^2/(a*abs(c))