3.3.38 \(\int e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [238]

Optimal. Leaf size=38 \[ -\frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c} \]

[Out]

2/3*(-a*c*x+c)^(3/2)/a/c-4*(-a*c*x+c)^(1/2)/a

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Rubi [A]
time = 0.03, antiderivative size = 38, 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)^{3/2}}{3 a c}-\frac {4 \sqrt {c-a c x}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[c - a*c*x])/a + (2*(c - a*c*x)^(3/2))/(3*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)} \sqrt {c-a c x} \, dx &=\int \frac {(1+a x) \sqrt {c-a c x}}{1-a x} \, dx\\ &=c \int \frac {1+a x}{\sqrt {c-a c x}} \, dx\\ &=c \int \left (\frac {2}{\sqrt {c-a c x}}-\frac {\sqrt {c-a c x}}{c}\right ) \, dx\\ &=-\frac {4 \sqrt {c-a c x}}{a}+\frac {2 (c-a c x)^{3/2}}{3 a c}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.61 \begin {gather*} -\frac {2 (5+a x) \sqrt {c-a c x}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(5 + a*x)*Sqrt[c - a*c*x])/(3*a)

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Maple [A]
time = 1.28, size = 33, normalized size = 0.87

method	result	size
gosper	\(-\frac {2 \sqrt {-c x a +c}\, \left (a x +5\right )}{3 a}\)	\(20\)
trager	\(-\frac {2 \sqrt {-c x a +c}\, \left (a x +5\right )}{3 a}\)	\(20\)
risch	\(\frac {2 c \left (a x +5\right ) \left (a x -1\right )}{3 a \sqrt {-c \left (a x -1\right )}}\)	\(27\)
derivativedivides	\(\frac {\frac {2 \left (-c x a +c \right )^{\frac {3}{2}}}{3}-4 c \sqrt {-c x a +c}}{a c}\)	\(33\)
default	\(\frac {\frac {2 \left (-c x a +c \right )^{\frac {3}{2}}}{3}-4 c \sqrt {-c x a +c}}{a c}\)	\(33\)

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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/c/a*(1/3*(-a*c*x+c)^(3/2)-2*c*(-a*c*x+c)^(1/2))

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Maxima [A]
time = 0.26, size = 30, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} - 6 \, \sqrt {-a c x + c} c\right )}}{3 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/3*((-a*c*x + c)^(3/2) - 6*sqrt(-a*c*x + c)*c)/(a*c)

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Fricas [A]
time = 0.35, size = 19, normalized size = 0.50 \begin {gather*} -\frac {2 \, \sqrt {-a c x + c} {\left (a x + 5\right )}}{3 \, 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)^(1/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(-a*c*x + c)*(a*x + 5)/a

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Sympy [A]
time = 2.52, size = 31, normalized size = 0.82 \begin {gather*} - \frac {2 \cdot \left (2 c \sqrt {- a c x + c} - \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\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)**(1/2),x)

[Out]

-2*(2*c*sqrt(-a*c*x + c) - (-a*c*x + c)**(3/2)/3)/(a*c)

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Giac [A]
time = 0.41, size = 44, normalized size = 1.16 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {-a c x + c} - \frac {{\left (-a c x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-a c x + c} c}{c}\right )}}{3 \, 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)^(1/2),x, algorithm="giac")

[Out]

-2/3*(3*sqrt(-a*c*x + c) - ((-a*c*x + c)^(3/2) - 3*sqrt(-a*c*x + c)*c)/c)/a

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Mupad [B]
time = 0.03, size = 32, normalized size = 0.84 \begin {gather*} \frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {c-a\,c\,x}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(c - a*c*x)^(3/2))/(3*a*c) - (4*(c - a*c*x)^(1/2))/a

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