3.4.50 \(\int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx\) [350]

Optimal. Leaf size=281 \[ \frac {1312 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {656 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}} \]

[Out]

-82/9*x^2*(-a*c*x+c)^(1/2)/a^2/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-8/9*x^3*(-a*c*x+c)^(1/2)/a/(1-1/a/x)^(1/2)/(1+1
/a/x)^(1/2)+2/9*x^4*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+1312/45*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/
a^4/(1-1/a/x)^(1/2)-656/45*x*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^3/(1-1/a/x)^(1/2)+164/15*x^2*(1+1/a/x)^(1/2)*(
-a*c*x+c)^(1/2)/a^2/(1-1/a/x)^(1/2)

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Rubi [A]
time = 0.18, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6311, 6316, 91, 79, 47, 37} \begin {gather*} \frac {1312 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {656 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {164 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]),x]

[Out]

(1312*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(45*a^4*Sqrt[1 - 1/(a*x)]) - (656*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x]
)/(45*a^3*Sqrt[1 - 1/(a*x)]) - (82*x^2*Sqrt[c - a*c*x])/(9*a^2*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) + (164*Sqr
t[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)]) - (8*x^3*Sqrt[c - a*c*x])/(9*a*Sqrt[1 - 1/(a*x)
]*Sqrt[1 + 1/(a*x)]) + (2*x^4*Sqrt[c - a*c*x])/(9*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 6311

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

Rule 6316

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

Rubi steps

\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{7/2} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{11/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {-\frac {14}{a}+\frac {9 x}{2 a^2}}{x^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{9 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (41 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{9 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (82 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (328 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{15 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {656 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (656 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{45 a^4 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {1312 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {656 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 73, normalized size = 0.26 \begin {gather*} \frac {2 \sqrt {c-a c x} \left (656+328 a x-82 a^2 x^2+41 a^3 x^3-20 a^4 x^4+5 a^5 x^5\right )}{45 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]),x]

[Out]

(2*Sqrt[c - a*c*x]*(656 + 328*a*x - 82*a^2*x^2 + 41*a^3*x^3 - 20*a^4*x^4 + 5*a^5*x^5))/(45*a^5*Sqrt[1 - 1/(a^2
*x^2)]*x)

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Maple [A]
time = 0.10, size = 81, normalized size = 0.29

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 a^{4} \left (a x -1\right )^{2}}\)	\(80\)
default	\(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (5 a^{5} x^{5}-20 a^{4} x^{4}+41 a^{3} x^{3}-82 a^{2} x^{2}+328 a x +656\right )}{45 \left (a x -1\right )^{2} a^{4}}\)	\(81\)
risch	\(-\frac {2 \left (5 a^{4} x^{4}-25 a^{3} x^{3}+66 a^{2} x^{2}-148 a x +476\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{45 a^{4} \sqrt {-c \left (a x -1\right )}}-\frac {8 c \sqrt {\frac {a x -1}{a x +1}}}{a^{4} \sqrt {-c \left (a x -1\right )}}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/45*((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2*(a*x+1)*(-c*(a*x-1))^(1/2)*(5*a^5*x^5-20*a^4*x^4+41*a^3*x^3-82*a^2*x^2+
328*a*x+656)/a^4

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Maxima [A]
time = 0.30, size = 117, normalized size = 0.42 \begin {gather*} \frac {2 \, {\left (5 \, a^{6} \sqrt {-c} x^{6} - 15 \, a^{5} \sqrt {-c} x^{5} + 21 \, a^{4} \sqrt {-c} x^{4} - 41 \, a^{3} \sqrt {-c} x^{3} + 246 \, a^{2} \sqrt {-c} x^{2} + 984 \, a \sqrt {-c} x + 656 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{45 \, {\left (a^{6} x^{2} - 2 \, a^{5} x + a^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

2/45*(5*a^6*sqrt(-c)*x^6 - 15*a^5*sqrt(-c)*x^5 + 21*a^4*sqrt(-c)*x^4 - 41*a^3*sqrt(-c)*x^3 + 246*a^2*sqrt(-c)*
x^2 + 984*a*sqrt(-c)*x + 656*sqrt(-c))*(a*x - 1)^2/((a^6*x^2 - 2*a^5*x + a^4)*(a*x + 1)^(3/2))

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Fricas [A]
time = 0.34, size = 77, normalized size = 0.27 \begin {gather*} \frac {2 \, {\left (5 \, a^{5} x^{5} - 20 \, a^{4} x^{4} + 41 \, a^{3} x^{3} - 82 \, a^{2} x^{2} + 328 \, a x + 656\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{45 \, {\left (a^{5} x - a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

2/45*(5*a^5*x^5 - 20*a^4*x^4 + 41*a^3*x^3 - 82*a^2*x^2 + 328*a*x + 656)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x +
1))/(a^5*x - a^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 1.41, size = 74, normalized size = 0.26 \begin {gather*} \frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^5\,x^5-20\,a^4\,x^4+41\,a^3\,x^3-82\,a^2\,x^2+328\,a\,x+656\right )}{45\,a^4\,\left (a\,x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(328*a*x - 82*a^2*x^2 + 41*a^3*x^3 - 20*a^4*x^4 + 5*a^5*x^5 +
656))/(45*a^4*(a*x - 1))

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