3.4.36 \(\int e^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\) [336]

Optimal. Leaf size=142 \[ \frac {152 c \sqrt {1-\frac {1}{a^2 x^2}} x}{105 a^2 \sqrt {c-a c x}}+\frac {38 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}}{105 a^2}+\frac {6 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}}{35 a^2 c}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x^2 (c-a c x)^{3/2}}{7 a c} \]

[Out]

6/35*x*(-a*c*x+c)^(3/2)*(1-1/a^2/x^2)^(1/2)/a^2/c-2/7*x^2*(-a*c*x+c)^(3/2)*(1-1/a^2/x^2)^(1/2)/a/c+152/105*c*x
*(1-1/a^2/x^2)^(1/2)/a^2/(-a*c*x+c)^(1/2)+38/105*x*(1-1/a^2/x^2)^(1/2)*(-a*c*x+c)^(1/2)/a^2

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Rubi [A]
time = 0.16, antiderivative size = 185, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6311, 6316, 79, 47, 37} \begin {gather*} -\frac {208 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {104 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {26 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-208*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(105*a^3*Sqrt[1 - 1/(a*x)]) + (104*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x
])/(105*a^2*Sqrt[1 - 1/(a*x)]) - (26*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(35*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt
[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(7*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 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^{-\coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{5/2} \, 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 {1-\frac {x}{a}}{x^{9/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {\left (13 \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 )}{7 a \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {26 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {\left (52 \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 )}{35 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {104 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {26 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {\left (104 \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 )}{105 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {208 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {104 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {26 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-104+52 a x-39 a^2 x^2+15 a^3 x^3\right )}{105 a^3 \sqrt {1-\frac {1}{a x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-104 + 52*a*x - 39*a^2*x^2 + 15*a^3*x^3))/(105*a^3*Sqrt[1 - 1/(a*x)])

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Maple [A]
time = 0.14, size = 65, normalized size = 0.46

method	result	size
risch	\(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (15 a^{3} x^{3}-39 a^{2} x^{2}+52 a x -104\right ) \left (a x +1\right )}{105 \sqrt {-c \left (a x -1\right )}\, a^{3}}\)	\(59\)
gosper	\(\frac {2 \left (a x +1\right ) \left (15 a^{3} x^{3}-39 a^{2} x^{2}+52 a x -104\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{105 a^{3} \left (a x -1\right )}\)	\(64\)
default	\(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (15 a^{3} x^{3}-39 a^{2} x^{2}+52 a x -104\right )}{105 \left (a x -1\right ) a^{3}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*(15*a^3*x^3-39*a^2*x^2+52*a*x-104)/(a*x-1)/a^3

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Maxima [A]
time = 0.28, size = 83, normalized size = 0.58 \begin {gather*} \frac {2 \, {\left (15 \, a^{4} \sqrt {-c} x^{4} - 24 \, a^{3} \sqrt {-c} x^{3} + 13 \, a^{2} \sqrt {-c} x^{2} - 52 \, a \sqrt {-c} x - 104 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{105 \, {\left (a^{4} x - a^{3}\right )} \sqrt {a x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^4*sqrt(-c)*x^4 - 24*a^3*sqrt(-c)*x^3 + 13*a^2*sqrt(-c)*x^2 - 52*a*sqrt(-c)*x - 104*sqrt(-c))*(a*x
- 1)/((a^4*x - a^3)*sqrt(a*x + 1))

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Fricas [A]
time = 0.35, size = 69, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left (15 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 52 \, a x - 104\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{4} x - a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*a^4*x^4 - 24*a^3*x^3 + 13*a^2*x^2 - 52*a*x - 104)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^4*x
- a^3)

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Sympy [A]
time = 106.24, size = 230, normalized size = 1.62 \begin {gather*} \frac {304 c \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{105 a^{3} \sqrt {- a c x + c}} - \frac {76 \sqrt {- a c x + c} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{105 a^{3}} - \frac {62 \left (- a c x + c\right )^{\frac {3}{2}} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{105 a^{3} c} + \frac {24 \left (- a c x + c\right )^{\frac {5}{2}} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{35 a^{3} c^{2}} - \frac {2 \left (- a c x + c\right )^{\frac {7}{2}} \sqrt {- \frac {a c x}{- a c x - c} + \frac {c}{- a c x - c}}}{7 a^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

304*c*sqrt(-a*c*x/(-a*c*x - c) + c/(-a*c*x - c))/(105*a**3*sqrt(-a*c*x + c)) - 76*sqrt(-a*c*x + c)*sqrt(-a*c*x
/(-a*c*x - c) + c/(-a*c*x - c))/(105*a**3) - 62*(-a*c*x + c)**(3/2)*sqrt(-a*c*x/(-a*c*x - c) + c/(-a*c*x - c))
/(105*a**3*c) + 24*(-a*c*x + c)**(5/2)*sqrt(-a*c*x/(-a*c*x - c) + c/(-a*c*x - c))/(35*a**3*c**2) - 2*(-a*c*x +
c)**(7/2)*sqrt(-a*c*x/(-a*c*x - c) + c/(-a*c*x - c))/(7*a**3*c**3)

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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^2*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),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 = 1.29, size = 88, normalized size = 0.62 \begin {gather*} \frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (15\,a^3\,x^3-9\,a^2\,x^2+4\,a\,x-48\right )}{105\,a^3}-\frac {304\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{105\,a^3\,\left (a\,x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(4*a*x - 9*a^2*x^2 + 15*a^3*x^3 - 48))/(105*a^3) - (304*(c -
a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2))/(105*a^3*(a*x - 1))

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