3.262 \(\int \sqrt {1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x) \, dx\)

Optimal. Leaf size=61 \[ -\frac {(a+b x)^2}{4 b}+\frac {\sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{2 b}+\frac {\sinh ^{-1}(a+b x)^2}{4 b} \]

[Out]

-1/4*(b*x+a)^2/b+1/4*arcsinh(b*x+a)^2/b+1/2*(b*x+a)*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b

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Rubi [A]  time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5867, 5682, 5675, 30} \[ -\frac {(a+b x)^2}{4 b}+\frac {\sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{2 b}+\frac {\sinh ^{-1}(a+b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x],x]

[Out]

-(a + b*x)^2/(4*b) + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(2*b) + ArcSinh[a + b*x]^2/(4*b)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5675

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

Rule 5682

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

Rule 5867

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> D
ist[1/d, Subst[Int[(C/d^2 + (C*x^2)/d^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A,
B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]

Rubi steps

\begin {align*} \int \sqrt {1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {1+x^2} \sinh ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{2 b}-\frac {\operatorname {Subst}(\int x \, dx,x,a+b x)}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)}{2 b}+\frac {\sinh ^{-1}(a+b x)^2}{4 b}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 61, normalized size = 1.00 \[ \frac {2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)-b x (2 a+b x)+\sinh ^{-1}(a+b x)^2}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x],x]

[Out]

(-(b*x*(2*a + b*x)) + 2*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x] + ArcSinh[a + b*x]^2)/(4*
b)

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fricas [A]  time = 0.60, size = 98, normalized size = 1.61 \[ -\frac {b^{2} x^{2} + 2 \, a b x - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*arcsinh(b*x + a), x)

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maple [A]  time = 0.10, size = 91, normalized size = 1.49 \[ \frac {2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x b -b^{2} x^{2}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) a -2 a b x +\arcsinh \left (b x +a \right )^{2}-a^{2}-1}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2),x)

[Out]

1/4*(2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a)*x*b-b^2*x^2+2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*arcsinh(b*x+a)
*a-2*a*b*x+arcsinh(b*x+a)^2-a^2-1)/b

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maxima [B]  time = 0.71, size = 238, normalized size = 3.90 \[ -\frac {1}{4} \, {\left (x^{2} + \frac {2 \, a x}{b} + \frac {2 \, \operatorname {arsinh}\left (b x + a\right ) \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{2}} - \frac {\operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b - \frac {1}{2} \, {\left (\frac {a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x - \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b}\right )} \operatorname {arsinh}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(x^2 + 2*a*x/b + 2*arcsinh(b*x + a)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 - arc
sinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))^2/b^2)*b - 1/2*(a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a
^2*b^2 + 4*(a^2 + 1)*b^2))/b - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x - (a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4
*a^2*b^2 + 4*(a^2 + 1)*b^2))/b - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/b)*arcsinh(b*x + a)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {asinh}\left (a+b\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2),x)

[Out]

int(asinh(a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(b*x+a)*(b**2*x**2+2*a*b*x+a**2+1)**(1/2),x)

[Out]

Integral(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x), x)

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