3.9.67 \(\int \frac {(1+a x)^3}{x (1-a^2 x^2)^{3/2}} \, dx\) [867]

Optimal. Leaf size=45 \[ \frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

-arcsin(a*x)-arctanh((-a^2*x^2+1)^(1/2))+4*(a*x+1)/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1819, 858, 222, 272, 65, 214} \begin {gather*} \frac {4 (a x+1)}{\sqrt {1-a^2 x^2}}-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\text {ArcSin}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + a*x)^3/(x*(1 - a^2*x^2)^(3/2)),x]

[Out]

(4*(1 + a*x))/Sqrt[1 - a^2*x^2] - ArcSin[a*x] - ArcTanh[Sqrt[1 - a^2*x^2]]

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 214

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

Rule 222

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

Rule 272

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

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(1+a x)^3}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\int \frac {-1+a x}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-a \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)-\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2}\\ &=\frac {4 (1+a x)}{\sqrt {1-a^2 x^2}}-\sin ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(45)=90\).
time = 0.24, size = 94, normalized size = 2.09 \begin {gather*} -\frac {4 \sqrt {1-a^2 x^2}}{-1+a x}+2 \tanh ^{-1}\left (\sqrt {-a^2} x-\sqrt {1-a^2 x^2}\right )+\frac {a \log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{\sqrt {-a^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + a*x)^3/(x*(1 - a^2*x^2)^(3/2)),x]

[Out]

(-4*Sqrt[1 - a^2*x^2])/(-1 + a*x) + 2*ArcTanh[Sqrt[-a^2]*x - Sqrt[1 - a^2*x^2]] + (a*Log[-(Sqrt[-a^2]*x) + Sqr
t[1 - a^2*x^2]])/Sqrt[-a^2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(41)=82\).
time = 0.59, size = 99, normalized size = 2.20

method	result	size
default	\(a^{3} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+\frac {4}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 a x}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\)	\(99\)
meijerg	\(\frac {-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}}{\sqrt {\pi }}-\frac {a \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {3 a x}{\sqrt {-a^{2} x^{2}+1}}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*(x/a^2/(-a^2*x^2+1)^(1/2)-1/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))+4/(-a^2*x^2+1)^(1/2)
+3*a*x/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2))

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Maxima [A]
time = 0.49, size = 65, normalized size = 1.44 \begin {gather*} \frac {4 \, a x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4}{\sqrt {-a^{2} x^{2} + 1}} - \arcsin \left (a x\right ) - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a*x/sqrt(-a^2*x^2 + 1) + 4/sqrt(-a^2*x^2 + 1) - arcsin(a*x) - log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]
time = 0.40, size = 82, normalized size = 1.82 \begin {gather*} \frac {4 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (a x - 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} - 4}{a x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a*x + 2*(a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (a*x - 1)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 4*sq
rt(-a^2*x^2 + 1) - 4)/(a*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (41) = 82\).
time = 3.83, size = 87, normalized size = 1.93 \begin {gather*} -\frac {a \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {8 \, a}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arcsin(a*x)*sgn(a)/abs(a) - a*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 8*a/((
(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a))

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Mupad [B]
time = 3.49, size = 82, normalized size = 1.82 \begin {gather*} \frac {4\,a\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a*(1 - a^2*x^2)^(1/2))/((x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2)) - (a*asinh(x*(-a^2)^(1/2)))/(-a^2)^
(1/2) - atanh((1 - a^2*x^2)^(1/2))

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