3.3.42 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\) [242]

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

[Out]

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

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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 {4}{7 a (c-a c x)^{7/2}}-\frac {2}{5 a c (c-a c x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - a*c*x)^(7/2),x]

[Out]

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

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 \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\int \frac {1+a x}{(1-a x) (c-a c x)^{7/2}} \, dx\\ &=c \int \frac {1+a x}{(c-a c x)^{9/2}} \, dx\\ &=c \int \left (\frac {2}{(c-a c x)^{9/2}}-\frac {1}{c (c-a c x)^{7/2}}\right ) \, dx\\ &=\frac {4}{7 a (c-a c x)^{7/2}}-\frac {2}{5 a c (c-a c x)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 34, normalized size = 0.85 \begin {gather*} \frac {2 (3+7 a x) \sqrt {c-a c x}}{35 a c^4 (-1+a x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - a*c*x)^(7/2),x]

[Out]

(2*(3 + 7*a*x)*Sqrt[c - a*c*x])/(35*a*c^4*(-1 + a*x)^4)

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Maple [A]
time = 1.18, size = 33, normalized size = 0.82

method	result	size
gosper	\(\frac {\frac {2 a x}{5}+\frac {6}{35}}{a \left (-c x a +c \right )^{\frac {7}{2}}}\)	\(21\)
trager	\(\frac {2 \left (7 a x +3\right ) \sqrt {-c x a +c}}{35 c^{4} \left (a x -1\right )^{4} a}\)	\(31\)
derivativedivides	\(\frac {\frac {4 c}{7 \left (-c x a +c \right )^{\frac {7}{2}}}-\frac {2}{5 \left (-c x a +c \right )^{\frac {5}{2}}}}{a c}\)	\(33\)
default	\(\frac {\frac {4 c}{7 \left (-c x a +c \right )^{\frac {7}{2}}}-\frac {2}{5 \left (-c x a +c \right )^{\frac {5}{2}}}}{a c}\)	\(33\)

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)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 26, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (7 \, a c x + 3 \, c\right )}}{35 \, {\left (-a c x + c\right )}^{\frac {7}{2}} 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)^(7/2),x, algorithm="maxima")

[Out]

2/35*(7*a*c*x + 3*c)/((-a*c*x + c)^(7/2)*a*c)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
time = 0.35, size = 66, normalized size = 1.65 \begin {gather*} \frac {2 \, \sqrt {-a c x + c} {\left (7 \, a x + 3\right )}}{35 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \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)^(7/2),x, algorithm="fricas")

[Out]

2/35*sqrt(-a*c*x + c)*(7*a*x + 3)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)

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Sympy [A]
time = 34.61, size = 31, normalized size = 0.78 \begin {gather*} \frac {4}{7 a \left (- a c x + c\right )^{\frac {7}{2}}} - \frac {2}{5 a c \left (- a c x + c\right )^{\frac {5}{2}}} \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)**(7/2),x)

[Out]

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

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Giac [A]
time = 0.41, size = 36, normalized size = 0.90 \begin {gather*} -\frac {2 \, {\left (7 \, a c x + 3 \, c\right )}}{35 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} 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)^(7/2),x, algorithm="giac")

[Out]

-2/35*(7*a*c*x + 3*c)/((a*c*x - c)^3*sqrt(-a*c*x + c)*a*c)

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Mupad [B]
time = 0.78, size = 20, normalized size = 0.50 \begin {gather*} \frac {14\,a\,x+6}{35\,a\,{\left (c-a\,c\,x\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(14*a*x + 6)/(35*a*(c - a*c*x)^(7/2))

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