3.1.71 \(\int e^{\frac {3}{2} \tanh ^{-1}(a x)} x^3 \, dx\) [71]

Optimal. Leaf size=290 \[ -\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}+\frac {123 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}-\frac {123 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4} \]

[Out]

-41/64*(-a*x+1)^(1/4)*(a*x+1)^(3/4)/a^4-1/4*x^2*(-a*x+1)^(1/4)*(a*x+1)^(7/4)/a^2-1/32*(-a*x+1)^(1/4)*(a*x+1)^(
7/4)*(4*a*x+11)/a^4-123/128*arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a^4*2^(1/2)-123/128*arctan(1+(-a*x
+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a^4*2^(1/2)+123/256*ln(1-(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/
(a*x+1)^(1/2))/a^4*2^(1/2)-123/256*ln(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^4
*2^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6261, 102, 152, 52, 65, 246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {123 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{64 \sqrt {2} a^4}-\frac {123 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{64 \sqrt {2} a^4}-\frac {\sqrt [4]{1-a x} (a x+1)^{7/4} (4 a x+11)}{32 a^4}-\frac {41 \sqrt [4]{1-a x} (a x+1)^{3/4}}{64 a^4}+\frac {123 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{128 \sqrt {2} a^4}-\frac {x^2 \sqrt [4]{1-a x} (a x+1)^{7/4}}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*ArcTanh[a*x])/2)*x^3,x]

[Out]

(-41*(1 - a*x)^(1/4)*(1 + a*x)^(3/4))/(64*a^4) - (x^2*(1 - a*x)^(1/4)*(1 + a*x)^(7/4))/(4*a^2) - ((1 - a*x)^(1
/4)*(1 + a*x)^(7/4)*(11 + 4*a*x))/(32*a^4) + (123*ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*S
qrt[2]*a^4) - (123*ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(64*Sqrt[2]*a^4) + (123*Log[1 + Sqrt
[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4) - (123*Log[1 + Sqrt[1
- a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)])/(128*Sqrt[2]*a^4)

Rule 52

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 210

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

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 6261

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; Fre
eQ[{a, m, n}, x] &&  !IntegerQ[(n - 1)/2]

Rubi steps

\begin {align*} \int e^{\frac {3}{2} \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1+a x)^{3/4}}{(1-a x)^{3/4}} \, dx\\ &=-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\int \frac {x \left (-2-\frac {3 a x}{2}\right ) (1+a x)^{3/4}}{(1-a x)^{3/4}} \, dx}{4 a^2}\\ &=-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}+\frac {41 \int \frac {(1+a x)^{3/4}}{(1-a x)^{3/4}} \, dx}{64 a^3}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}+\frac {123 \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{128 a^3}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}-\frac {123 \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{32 a^4}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}-\frac {123 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{32 a^4}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}-\frac {123 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 a^4}-\frac {123 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 a^4}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}-\frac {123 \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 a^4}-\frac {123 \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 a^4}+\frac {123 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}+\frac {123 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}+\frac {123 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {123 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}\\ &=-\frac {41 \sqrt [4]{1-a x} (1+a x)^{3/4}}{64 a^4}-\frac {x^2 \sqrt [4]{1-a x} (1+a x)^{7/4}}{4 a^2}-\frac {\sqrt [4]{1-a x} (1+a x)^{7/4} (11+4 a x)}{32 a^4}+\frac {123 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}-\frac {123 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{128 \sqrt {2} a^4}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.12, size = 103, normalized size = 0.36 \begin {gather*} \frac {-\frac {8 e^{\frac {3}{2} \tanh ^{-1}(a x)} \left (41+183 e^{2 \tanh ^{-1}(a x)}+147 e^{4 \tanh ^{-1}(a x)}+133 e^{6 \tanh ^{-1}(a x)}\right )}{\left (1+e^{2 \tanh ^{-1}(a x)}\right )^4}-123 \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {\tanh ^{-1}(a x)-2 \log \left (e^{\frac {1}{2} \tanh ^{-1}(a x)}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{256 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((3*ArcTanh[a*x])/2)*x^3,x]

[Out]

((-8*E^((3*ArcTanh[a*x])/2)*(41 + 183*E^(2*ArcTanh[a*x]) + 147*E^(4*ArcTanh[a*x]) + 133*E^(6*ArcTanh[a*x])))/(
1 + E^(2*ArcTanh[a*x]))^4 - 123*RootSum[1 + #1^4 & , (ArcTanh[a*x] - 2*Log[E^(ArcTanh[a*x]/2) - #1])/#1 & ])/(
256*a^4)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (223) = 446\).
time = 0.39, size = 557, normalized size = 1.92 \begin {gather*} -\frac {492 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{4} \sqrt {\frac {\sqrt {2} {\left (a^{13} x - a^{12}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} + {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - \sqrt {2} a^{4} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - 1\right ) + 492 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{4} \sqrt {-\frac {\sqrt {2} {\left (a^{13} x - a^{12}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} - \sqrt {2} a^{4} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {1}{4}} + 1\right ) + 123 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{13} x - a^{12}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} + {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 123 \, \sqrt {2} a^{4} \frac {1}{a^{16}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{13} x - a^{12}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{16}}^{\frac {3}{4}} - {\left (a^{9} x - a^{8}\right )} \sqrt {\frac {1}{a^{16}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) + 4 \, {\left (16 \, a^{3} x^{3} + 24 \, a^{2} x^{2} + 30 \, a x + 63\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{256 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256*(492*sqrt(2)*a^4*(a^(-16))^(1/4)*arctan(sqrt(2)*a^4*sqrt((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 +
1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4)
- sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(1/4) - 1) + 492*sqrt(2)*a^4*(a^(-16))^(1/4)*arcta
n(sqrt(2)*a^4*sqrt(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x - a^
8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-16))^(1/4) - sqrt(2)*a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x
- 1))*(a^(-16))^(1/4) + 1) + 123*sqrt(2)*a^4*(a^(-16))^(1/4)*log((sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2
+ 1)/(a*x - 1))*(a^(-16))^(3/4) + (a^9*x - a^8)*sqrt(a^(-16)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - 123*sqrt(2)*a
^4*(a^(-16))^(1/4)*log(-(sqrt(2)*(a^13*x - a^12)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-16))^(3/4) - (a^9*x
- a^8)*sqrt(a^(-16)) + sqrt(-a^2*x^2 + 1))/(a*x - 1)) + 4*(16*a^3*x^3 + 24*a^2*x^2 + 30*a*x + 63)*sqrt(-a^2*x^
2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^4

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*((a*x + 1)/sqrt(-a**2*x**2 + 1))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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