3.228 \(\int e^{\tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx\)

Optimal. Leaf size=106 \[ \frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a (c-a c x)^{3/2}}+\frac {16 c^3 \left (1-a^2 x^2\right )^{3/2}}{35 a \sqrt {c-a c x}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a} \]

[Out]

64/105*c^4*(-a^2*x^2+1)^(3/2)/a/(-a*c*x+c)^(3/2)+16/35*c^3*(-a^2*x^2+1)^(3/2)/a/(-a*c*x+c)^(1/2)+2/7*c^2*(-a^2
*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/a

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Rubi [A]  time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 657, 649} \[ \frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a (c-a c x)^{3/2}}+\frac {16 c^3 \left (1-a^2 x^2\right )^{3/2}}{35 a \sqrt {c-a c x}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - a*c*x)^(5/2),x]

[Out]

(64*c^4*(1 - a^2*x^2)^(3/2))/(105*a*(c - a*c*x)^(3/2)) + (16*c^3*(1 - a^2*x^2)^(3/2))/(35*a*Sqrt[c - a*c*x]) +
(2*c^2*Sqrt[c - a*c*x]*(1 - a^2*x^2)^(3/2))/(7*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)^{5/2} \, dx &=c \int (c-a c x)^{3/2} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {2 c^2 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{7 a}+\frac {1}{7} \left (8 c^2\right ) \int \sqrt {c-a c x} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {16 c^3 \left (1-a^2 x^2\right )^{3/2}}{35 a \sqrt {c-a c x}}+\frac {2 c^2 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{7 a}+\frac {1}{35} \left (32 c^3\right ) \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {64 c^4 \left (1-a^2 x^2\right )^{3/2}}{105 a (c-a c x)^{3/2}}+\frac {16 c^3 \left (1-a^2 x^2\right )^{3/2}}{35 a \sqrt {c-a c x}}+\frac {2 c^2 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{7 a}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 54, normalized size = 0.51 \[ \frac {2 c^2 (a x+1)^{3/2} \left (15 a^2 x^2-54 a x+71\right ) \sqrt {c-a c x}}{105 a \sqrt {1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a*x]*(c - a*c*x)^(5/2),x]

[Out]

(2*c^2*(1 + a*x)^(3/2)*Sqrt[c - a*c*x]*(71 - 54*a*x + 15*a^2*x^2))/(105*a*Sqrt[1 - a*x])

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fricas [A]  time = 0.63, size = 69, normalized size = 0.65 \[ -\frac {2 \, {\left (15 \, a^{3} c^{2} x^{3} - 39 \, a^{2} c^{2} x^{2} + 17 \, a c^{2} x + 71 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{105 \, {\left (a^{2} x - a\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(15*a^3*c^2*x^3 - 39*a^2*c^2*x^2 + 17*a*c^2*x + 71*c^2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)

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giac [A]  time = 0.20, size = 61, normalized size = 0.58 \[ -\frac {2 \, {\left (64 \, \sqrt {2} c^{\frac {3}{2}} - \frac {15 \, {\left (a c x + c\right )}^{\frac {7}{2}} - 84 \, {\left (a c x + c\right )}^{\frac {5}{2}} c + 140 \, {\left (a c x + c\right )}^{\frac {3}{2}} c^{2}}{c^{2}}\right )} c^{2}}{105 \, a {\left | c \right |}} \]

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)^(5/2),x, algorithm="giac")

[Out]

-2/105*(64*sqrt(2)*c^(3/2) - (15*(a*c*x + c)^(7/2) - 84*(a*c*x + c)^(5/2)*c + 140*(a*c*x + c)^(3/2)*c^2)/c^2)*
c^2/(a*abs(c))

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maple [A]  time = 0.03, size = 55, normalized size = 0.52 \[ \frac {2 \left (a x +1\right )^{2} \left (15 a^{2} x^{2}-54 a x +71\right ) \left (-a c x +c \right )^{\frac {5}{2}}}{105 a \left (a x -1\right )^{2} \sqrt {-a^{2} x^{2}+1}} \]

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)^(5/2),x)

[Out]

2/105*(a*x+1)^2*(15*a^2*x^2-54*a*x+71)*(-a*c*x+c)^(5/2)/a/(a*x-1)^2/(-a^2*x^2+1)^(1/2)

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maxima [A]  time = 0.35, size = 106, normalized size = 1.00 \[ \frac {2 \, {\left (3 \, a^{4} c^{\frac {5}{2}} x^{4} - 9 \, a^{3} c^{\frac {5}{2}} x^{3} + 11 \, a^{2} c^{\frac {5}{2}} x^{2} - 23 \, a c^{\frac {5}{2}} x - 46 \, c^{\frac {5}{2}}\right )}}{21 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (3 \, a^{3} c^{\frac {5}{2}} x^{3} - 11 \, a^{2} c^{\frac {5}{2}} x^{2} + 29 \, a c^{\frac {5}{2}} x + 43 \, c^{\frac {5}{2}}\right )}}{15 \, \sqrt {a x + 1} a} \]

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)^(5/2),x, algorithm="maxima")

[Out]

2/21*(3*a^4*c^(5/2)*x^4 - 9*a^3*c^(5/2)*x^3 + 11*a^2*c^(5/2)*x^2 - 23*a*c^(5/2)*x - 46*c^(5/2))/(sqrt(a*x + 1)
*a) + 2/15*(3*a^3*c^(5/2)*x^3 - 11*a^2*c^(5/2)*x^2 + 29*a*c^(5/2)*x + 43*c^(5/2))/(sqrt(a*x + 1)*a)

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mupad [B]  time = 0.95, size = 68, normalized size = 0.64 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {176\,c^2\,x}{105}+\frac {142\,c^2}{105\,a}-\frac {44\,a\,c^2\,x^2}{105}-\frac {16\,a^2\,c^2\,x^3}{35}+\frac {2\,a^3\,c^2\,x^4}{7}\right )}{\sqrt {1-a^2\,x^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c - a*c*x)^(1/2)*((176*c^2*x)/105 + (142*c^2)/(105*a) - (44*a*c^2*x^2)/105 - (16*a^2*c^2*x^3)/35 + (2*a^3*c^
2*x^4)/7))/(1 - a^2*x^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

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)**(5/2),x)

[Out]

Integral((-c*(a*x - 1))**(5/2)*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)

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