3.3.36 \(\int e^{2 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx\) [236]

Optimal. Leaf size=40 \[ -\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c} \]

[Out]

-4/5*(-a*c*x+c)^(5/2)/a+2/7*(-a*c*x+c)^(7/2)/a/c

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Rubi [A]
time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6265, 21, 45} \begin {gather*} \frac {2 (c-a c x)^{7/2}}{7 a c}-\frac {4 (c-a c x)^{5/2}}{5 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(c - a*c*x)^(5/2))/(5*a) + (2*(c - a*c*x)^(7/2))/(7*a*c)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6265

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

Rubi steps

\begin {align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\int \frac {(1+a x) (c-a c x)^{5/2}}{1-a x} \, dx\\ &=c \int (1+a x) (c-a c x)^{3/2} \, dx\\ &=c \int \left (2 (c-a c x)^{3/2}-\frac {(c-a c x)^{5/2}}{c}\right ) \, dx\\ &=-\frac {4 (c-a c x)^{5/2}}{5 a}+\frac {2 (c-a c x)^{7/2}}{7 a c}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.85 \begin {gather*} -\frac {2 c^2 (-1+a x)^2 (9+5 a x) \sqrt {c-a c x}}{35 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c^2*(-1 + a*x)^2*(9 + 5*a*x)*Sqrt[c - a*c*x])/(35*a)

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Maple [A]
time = 1.74, size = 33, normalized size = 0.82

method	result	size
gosper	\(-\frac {2 \left (-c x a +c \right )^{\frac {5}{2}} \left (5 a x +9\right )}{35 a}\)	\(21\)
derivativedivides	\(\frac {\frac {2 \left (-c x a +c \right )^{\frac {7}{2}}}{7}-\frac {4 c \left (-c x a +c \right )^{\frac {5}{2}}}{5}}{a c}\)	\(33\)
default	\(\frac {\frac {2 \left (-c x a +c \right )^{\frac {7}{2}}}{7}-\frac {4 c \left (-c x a +c \right )^{\frac {5}{2}}}{5}}{a c}\)	\(33\)
trager	\(-\frac {2 c^{2} \left (5 a^{3} x^{3}-a^{2} x^{2}-13 a x +9\right ) \sqrt {-c x a +c}}{35 a}\)	\(40\)
risch	\(\frac {2 c^{3} \left (5 a^{3} x^{3}-a^{2} x^{2}-13 a x +9\right ) \left (a x -1\right )}{35 a \sqrt {-c \left (a x -1\right )}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/c/a*(1/7*(-a*c*x+c)^(7/2)-2/5*c*(-a*c*x+c)^(5/2))

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Maxima [A]
time = 0.25, size = 32, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (5 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 14 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c\right )}}{35 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(-a*c*x + c)^(7/2) - 14*(-a*c*x + c)^(5/2)*c)/(a*c)

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Fricas [A]
time = 0.36, size = 49, normalized size = 1.22 \begin {gather*} -\frac {2 \, {\left (5 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 13 \, a c^{2} x + 9 \, c^{2}\right )} \sqrt {-a c x + c}}{35 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(5*a^3*c^2*x^3 - a^2*c^2*x^2 - 13*a*c^2*x + 9*c^2)*sqrt(-a*c*x + c)/a

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Sympy [A]
time = 8.82, size = 76, normalized size = 1.90 \begin {gather*} c^{2} \left (\begin {cases} \sqrt {c} x & \text {for}\: a = 0 \\0 & \text {for}\: c = 0 \\- \frac {2 \left (- a c x + c\right )^{\frac {3}{2}}}{3 a c} & \text {otherwise} \end {cases}\right ) + \frac {2 \left (\frac {c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*Piecewise((sqrt(c)*x, Eq(a, 0)), (0, Eq(c, 0)), (-2*(-a*c*x + c)**(3/2)/(3*a*c), True)) + 2*(c**2*(-a*c*x
+ c)**(3/2)/3 - 2*c*(-a*c*x + c)**(5/2)/5 + (-a*c*x + c)**(7/2)/7)/(a*c)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (32) = 64\).
time = 0.42, size = 141, normalized size = 3.52 \begin {gather*} \frac {2 \, {\left (21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} - 70 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c - 35 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c\right )} c - \frac {3 \, {\left (5 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} + 21 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} c - 35 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-a c x + c} c^{3}\right )}}{c}\right )}}{105 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(21*(a*c*x - c)^2*sqrt(-a*c*x + c) - 70*(-a*c*x + c)^(3/2)*c - 35*((-a*c*x + c)^(3/2) - 3*sqrt(-a*c*x +
c)*c)*c - 3*(5*(a*c*x - c)^3*sqrt(-a*c*x + c) + 21*(a*c*x - c)^2*sqrt(-a*c*x + c)*c - 35*(-a*c*x + c)^(3/2)*c^
2 + 35*sqrt(-a*c*x + c)*c^3)/c)/a

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Mupad [B]
time = 0.77, size = 32, normalized size = 0.80 \begin {gather*} \frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a\,c}-\frac {4\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(c - a*c*x)^(7/2))/(7*a*c) - (4*(c - a*c*x)^(5/2))/(5*a)

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