### 3.310 $$\int e^{3 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx$$

Optimal. Leaf size=309 $\frac{92 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{472 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1576 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{9/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{38 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}$

[Out]

(1576*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(315*a^4*Sqrt[1 - 1/(a*x)]) + (472*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x
])/(315*a^3*Sqrt[1 - 1/(a*x)]) + (92*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(105*a^2*Sqrt[1 - 1/(a*x)]) + (38*
Sqrt[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(63*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^4*Sqrt[c - a*c*x])/(9
*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 +
1/(a*x)])])/(a^(9/2)*Sqrt[1 - 1/(a*x)])

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Rubi [A]  time = 0.324192, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.304, Rules used = {6176, 6181, 98, 152, 12, 93, 206} $\frac{92 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{472 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1576 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{9/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{38 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1576*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(315*a^4*Sqrt[1 - 1/(a*x)]) + (472*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x
])/(315*a^3*Sqrt[1 - 1/(a*x)]) + (92*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(105*a^2*Sqrt[1 - 1/(a*x)]) + (38*
Sqrt[1 + 1/(a*x)]*x^3*Sqrt[c - a*c*x])/(63*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1/(a*x)]*x^4*Sqrt[c - a*c*x])/(9
*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 +
1/(a*x)])])/(a^(9/2)*Sqrt[1 - 1/(a*x)])

Rule 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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]

Rule 98

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

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{11/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{\left (2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{19}{2 a}-\frac{17 x}{2 a^2}}{x^{9/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{9 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\frac{69}{2 a^2}+\frac{57 x}{2 a^3}}{x^{7/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{63 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{177}{2 a^3}-\frac{69 x}{a^4}}{x^{5/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{315 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (16 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\frac{591}{4 a^4}+\frac{177 x}{2 a^5}}{x^{3/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{945 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{\left (32 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int -\frac{945}{8 a^5 \sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{945 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^5 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{a^5 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{a^{9/2} \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.108288, size = 130, normalized size = 0.42 $\frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} \left (35 a^4 x^4+95 a^3 x^3+138 a^2 x^2+236 a x+788\right )-630 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{315 a^{9/2} \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(788 + 236*a*x + 138*a^2*x^2 + 95*a^3*x^3 + 35*a^4*x^4) - 630*Sq
rt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(315*a^(9/2)*Sqrt[1 - 1/(a*x)
])

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Maple [A]  time = 0.177, size = 161, normalized size = 0.5 \begin{align*} -{\frac{2\,ax-2}{ \left ( 315\,ax+315 \right ){a}^{4}}\sqrt{-c \left ( ax-1 \right ) } \left ( -35\,{x}^{4}{a}^{4}\sqrt{-c \left ( ax+1 \right ) }-95\,{x}^{3}{a}^{3}\sqrt{-c \left ( ax+1 \right ) }-138\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+630\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -236\,xa\sqrt{-c \left ( ax+1 \right ) }-788\,\sqrt{-c \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(-c*(a*x-1))^(1/2)*(-35*x^4*a^4*(-c*(a*x+1))^(1/2)-95*x^3*a^3*(
-c*(a*x+1))^(1/2)-138*x^2*a^2*(-c*(a*x+1))^(1/2)+630*c^(1/2)*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(
1/2))-236*x*a*(-c*(a*x+1))^(1/2)-788*(-c*(a*x+1))^(1/2))/(-c*(a*x+1))^(1/2)/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c} x^{3}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.69984, size = 749, normalized size = 2.42 \begin{align*} \left [\frac{2 \,{\left (315 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{315 \,{\left (a^{5} x - a^{4}\right )}}, -\frac{2 \,{\left (630 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{315 \,{\left (a^{5} x - a^{4}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/315*(315*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-
c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (35*a^5*x^5 + 130*a^4*x^4 + 233*a^3*x^3 + 374*a^2
*x^2 + 1024*a*x + 788)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*x - a^4), -2/315*(630*sqrt(2)*(a*x - 1
)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (35*a^5*x^5 + 130*a
^4*x^4 + 233*a^3*x^3 + 374*a^2*x^2 + 1024*a*x + 788)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*x - a^4)
]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 1.28071, size = 244, normalized size = 0.79 \begin{align*} -\frac{1260 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 2584 \, \sqrt{2} \sqrt{-c}}{315 \, a^{4} \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (630 \, \sqrt{2} c^{\frac{9}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - 35 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} + 45 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c - 63 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2} + 105 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3} - 630 \, \sqrt{-a c x - c} c^{4}\right )}}{315 \, a^{4} c^{4} \mathrm{sgn}\left (-a c x - c\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(1260*I*sqrt(2)*sqrt(-c)*arctan(-I) - 2584*sqrt(2)*sqrt(-c))/(a^4*sgn(c)) - 2/315*(630*sqrt(2)*c^(9/2)*
arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c)) - 35*(a*c*x + c)^4*sqrt(-a*c*x - c) + 45*(a*c*x + c)^3*sqrt(-a*c*
x - c)*c - 63*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^2 + 105*(-a*c*x - c)^(3/2)*c^3 - 630*sqrt(-a*c*x - c)*c^4)/(a^4
*c^4*sgn(-a*c*x - c))