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]

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

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Rubi [A]  time = 0.13, antiderivative size = 176, normalized size of antiderivative = 1.00, 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 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]

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 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 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*}

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Mathematica [A]  time = 0.05, size = 70, normalized size = 0.40 \[ \frac {2 c^4 (a x+1)^{3/2} \left (315 a^4 x^4-1820 a^3 x^3+4530 a^2 x^2-6396 a x+5419\right ) \sqrt {c-a c x}}{3465 a \sqrt {1-a x}} \]

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])

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fricas [A]  time = 0.48, size = 91, normalized size = 0.52 \[ -\frac {2 \, {\left (315 \, a^{5} c^{4} x^{5} - 1505 \, a^{4} c^{4} x^{4} + 2710 \, a^{3} c^{4} x^{3} - 1866 \, a^{2} c^{4} x^{2} - 977 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{3465 \, {\left (a^{2} x - a\right )}} \]

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)

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giac [A]  time = 0.27, size = 87, normalized size = 0.49 \[ -\frac {2 \, {\left (4096 \, \sqrt {2} c^{\frac {7}{2}} - \frac {315 \, {\left (a c x + c\right )}^{\frac {11}{2}} - 3080 \, {\left (a c x + c\right )}^{\frac {9}{2}} c + 11880 \, {\left (a c x + c\right )}^{\frac {7}{2}} c^{2} - 22176 \, {\left (a c x + c\right )}^{\frac {5}{2}} c^{3} + 18480 \, {\left (a c x + c\right )}^{\frac {3}{2}} c^{4}}{c^{2}}\right )} c^{2}}{3465 \, a {\left | c \right |}} \]

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))

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maple [A]  time = 0.03, size = 71, normalized size = 0.40 \[ \frac {2 \left (a x +1\right )^{2} \left (315 x^{4} a^{4}-1820 x^{3} a^{3}+4530 a^{2} x^{2}-6396 a x +5419\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{3465 a \left (a x -1\right )^{4} \sqrt {-a^{2} x^{2}+1}} \]

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)

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maxima [A]  time = 0.36, size = 150, normalized size = 0.85 \[ \frac {2 \, {\left (35 \, a^{6} c^{\frac {9}{2}} x^{6} - 175 \, a^{5} c^{\frac {9}{2}} x^{5} + 360 \, a^{4} c^{\frac {9}{2}} x^{4} - 422 \, a^{3} c^{\frac {9}{2}} x^{3} + 459 \, a^{2} c^{\frac {9}{2}} x^{2} - 1451 \, a c^{\frac {9}{2}} x - 2902 \, c^{\frac {9}{2}}\right )}}{385 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (35 \, a^{5} c^{\frac {9}{2}} x^{5} - 185 \, a^{4} c^{\frac {9}{2}} x^{4} + 422 \, a^{3} c^{\frac {9}{2}} x^{3} - 634 \, a^{2} c^{\frac {9}{2}} x^{2} + 1591 \, a c^{\frac {9}{2}} x + 2867 \, c^{\frac {9}{2}}\right )}}{315 \, \sqrt {a x + 1} a} \]

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)

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mupad [B]  time = 1.12, size = 90, normalized size = 0.51 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {8884\,c^4\,x}{3465}+\frac {10838\,c^4}{3465\,a}-\frac {5686\,a\,c^4\,x^2}{3465}+\frac {1688\,a^2\,c^4\,x^3}{3465}+\frac {482\,a^3\,c^4\,x^4}{693}-\frac {68\,a^4\,c^4\,x^5}{99}+\frac {2\,a^5\,c^4\,x^6}{11}\right )}{\sqrt {1-a^2\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c - a*c*x)^(1/2)*((8884*c^4*x)/3465 + (10838*c^4)/(3465*a) - (5686*a*c^4*x^2)/3465 + (1688*a^2*c^4*x^3)/3465
+ (482*a^3*c^4*x^4)/693 - (68*a^4*c^4*x^5)/99 + (2*a^5*c^4*x^6)/11))/(1 - a^2*x^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

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]

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

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