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

Optimal. Leaf size=176 \[ \frac{4096 c^6 \left (1-a^2 x^2\right )^{3/2}}{3465 a (c-a c x)^{3/2}}+\frac{1024 c^5 \left (1-a^2 x^2\right )^{3/2}}{1155 a \sqrt{c-a c x}}+\frac{128 c^4 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a c x}}{231 a}+\frac{32 c^3 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{99 a}+\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a} \]

[Out]

(4096*c^6*(1 - a^2*x^2)^(3/2))/(3465*a*(c - a*c*x)^(3/2)) + (1024*c^5*(1 - a^2*x^2)^(3/2))/(1155*a*Sqrt[c - a*
c*x]) + (128*c^4*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(231*a) + (32*c^3*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))
/(99*a) + (2*c^2*(c - a*c*x)^(5/2)*(1 - a^2*x^2)^(3/2))/(11*a)

________________________________________________________________________________________

Rubi [A]  time = 0.128885, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6127, 657, 649} \[ \frac{4096 c^6 \left (1-a^2 x^2\right )^{3/2}}{3465 a (c-a c x)^{3/2}}+\frac{1024 c^5 \left (1-a^2 x^2\right )^{3/2}}{1155 a \sqrt{c-a c x}}+\frac{128 c^4 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a c x}}{231 a}+\frac{32 c^3 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{99 a}+\frac{2 c^2 \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4096*c^6*(1 - a^2*x^2)^(3/2))/(3465*a*(c - a*c*x)^(3/2)) + (1024*c^5*(1 - a^2*x^2)^(3/2))/(1155*a*Sqrt[c - a*
c*x]) + (128*c^4*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(231*a) + (32*c^3*(c - a*c*x)^(3/2)*(1 - a^2*x^2)^(3/2))
/(99*a) + (2*c^2*(c - a*c*x)^(5/2)*(1 - a^2*x^2)^(3/2))/(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^{\tanh ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=c \int (c-a c x)^{7/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{2 c^2 (c-a c x)^{5/2} \left (1-a^2 x^2\right )^{3/2}}{11 a}+\frac{1}{11} \left (16 c^2\right ) \int (c-a c x)^{5/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{32 c^3 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{99 a}+\frac{2 c^2 (c-a c x)^{5/2} \left (1-a^2 x^2\right )^{3/2}}{11 a}+\frac{1}{33} \left (64 c^3\right ) \int (c-a c x)^{3/2} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{128 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{231 a}+\frac{32 c^3 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{99 a}+\frac{2 c^2 (c-a c x)^{5/2} \left (1-a^2 x^2\right )^{3/2}}{11 a}+\frac{1}{231} \left (512 c^4\right ) \int \sqrt{c-a c x} \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1024 c^5 \left (1-a^2 x^2\right )^{3/2}}{1155 a \sqrt{c-a c x}}+\frac{128 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{231 a}+\frac{32 c^3 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{99 a}+\frac{2 c^2 (c-a c x)^{5/2} \left (1-a^2 x^2\right )^{3/2}}{11 a}+\frac{\left (2048 c^5\right ) \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx}{1155}\\ &=\frac{4096 c^6 \left (1-a^2 x^2\right )^{3/2}}{3465 a (c-a c x)^{3/2}}+\frac{1024 c^5 \left (1-a^2 x^2\right )^{3/2}}{1155 a \sqrt{c-a c x}}+\frac{128 c^4 \sqrt{c-a c x} \left (1-a^2 x^2\right )^{3/2}}{231 a}+\frac{32 c^3 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{99 a}+\frac{2 c^2 (c-a c x)^{5/2} \left (1-a^2 x^2\right )^{3/2}}{11 a}\\ \end{align*}

Mathematica [A]  time = 0.0500722, size = 70, normalized size = 0.4 \[ \frac{2 c^4 (a x+1)^{3/2} \left (315 a^4 x^4-1820 a^3 x^3+4530 a^2 x^2-6396 a x+5419\right ) \sqrt{c-a c x}}{3465 a \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*c^4*(1 + a*x)^(3/2)*Sqrt[c - a*c*x]*(5419 - 6396*a*x + 4530*a^2*x^2 - 1820*a^3*x^3 + 315*a^4*x^4))/(3465*a*
Sqrt[1 - a*x])

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 71, normalized size = 0.4 \begin{align*}{\frac{2\, \left ( 315\,{x}^{4}{a}^{4}-1820\,{x}^{3}{a}^{3}+4530\,{a}^{2}{x}^{2}-6396\,ax+5419 \right ) \left ( ax+1 \right ) ^{2}}{3465\, \left ( ax-1 \right ) ^{4}a} \left ( -acx+c \right ) ^{{\frac{9}{2}}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(a*x+1)^2*(315*a^4*x^4-1820*a^3*x^3+4530*a^2*x^2-6396*a*x+5419)*(-a*c*x+c)^(9/2)/a/(a*x-1)^4/(-a^2*x^2+
1)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.02965, size = 203, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (35 \, a^{6} c^{\frac{9}{2}} x^{6} - 175 \, a^{5} c^{\frac{9}{2}} x^{5} + 360 \, a^{4} c^{\frac{9}{2}} x^{4} - 422 \, a^{3} c^{\frac{9}{2}} x^{3} + 459 \, a^{2} c^{\frac{9}{2}} x^{2} - 1451 \, a c^{\frac{9}{2}} x - 2902 \, c^{\frac{9}{2}}\right )}}{385 \, \sqrt{a x + 1} a} + \frac{2 \,{\left (35 \, a^{5} c^{\frac{9}{2}} x^{5} - 185 \, a^{4} c^{\frac{9}{2}} x^{4} + 422 \, a^{3} c^{\frac{9}{2}} x^{3} - 634 \, a^{2} c^{\frac{9}{2}} x^{2} + 1591 \, a c^{\frac{9}{2}} x + 2867 \, c^{\frac{9}{2}}\right )}}{315 \, \sqrt{a x + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-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 + 360*a^4*c^(9/2)*x^4 - 422*a^3*c^(9/2)*x^3 + 459*a^2*c^(9/2)*
x^2 - 1451*a*c^(9/2)*x - 2902*c^(9/2))/(sqrt(a*x + 1)*a) + 2/315*(35*a^5*c^(9/2)*x^5 - 185*a^4*c^(9/2)*x^4 + 4
22*a^3*c^(9/2)*x^3 - 634*a^2*c^(9/2)*x^2 + 1591*a*c^(9/2)*x + 2867*c^(9/2))/(sqrt(a*x + 1)*a)

________________________________________________________________________________________

Fricas [A]  time = 1.70215, size = 212, normalized size = 1.2 \begin{align*} -\frac{2 \,{\left (315 \, a^{5} c^{4} x^{5} - 1505 \, a^{4} c^{4} x^{4} + 2710 \, a^{3} c^{4} x^{3} - 1866 \, a^{2} c^{4} x^{2} - 977 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{3465 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*a^5*c^4*x^5 - 1505*a^4*c^4*x^4 + 2710*a^3*c^4*x^3 - 1866*a^2*c^4*x^2 - 977*a*c^4*x + 5419*c^4)*sq
rt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.48155, size = 117, normalized size = 0.66 \begin{align*} -\frac{2 \,{\left (4096 \, \sqrt{2} c^{\frac{7}{2}} - \frac{315 \,{\left (a c x + c\right )}^{\frac{11}{2}} - 3080 \,{\left (a c x + c\right )}^{\frac{9}{2}} c + 11880 \,{\left (a c x + c\right )}^{\frac{7}{2}} c^{2} - 22176 \,{\left (a c x + c\right )}^{\frac{5}{2}} c^{3} + 18480 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{4}}{c^{2}}\right )} c^{2}}{3465 \, a{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(4096*sqrt(2)*c^(7/2) - (315*(a*c*x + c)^(11/2) - 3080*(a*c*x + c)^(9/2)*c + 11880*(a*c*x + c)^(7/2)*c
^2 - 22176*(a*c*x + c)^(5/2)*c^3 + 18480*(a*c*x + c)^(3/2)*c^4)/c^2)*c^2/(a*abs(c))