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

Optimal. Leaf size=105 \[ \frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac{b c x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}-\frac{b x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}} \]

[Out]

-(b*x*Sqrt[d + c^2*d*x^2])/(3*c*Sqrt[1 + c^2*x^2]) - (b*c*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) + ((d
+ c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*d)

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Rubi [A]  time = 0.0674952, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {5717} \[ \frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac{b c x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}-\frac{b x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]

[Out]

-(b*x*Sqrt[d + c^2*d*x^2])/(3*c*Sqrt[1 + c^2*x^2]) - (b*c*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[1 + c^2*x^2]) + ((d
+ c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2*d)

Rule 5717

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

Rubi steps

\begin{align*} \int x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac{\left (b \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c \sqrt{1+c^2 x^2}}\\ &=-\frac{b x \sqrt{d+c^2 d x^2}}{3 c \sqrt{1+c^2 x^2}}-\frac{b c x^3 \sqrt{d+c^2 d x^2}}{9 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0915695, size = 92, normalized size = 0.88 \[ \frac{\sqrt{c^2 d x^2+d} \left (3 a \left (c^2 x^2+1\right )^2-b c x \left (c^2 x^2+3\right ) \sqrt{c^2 x^2+1}+3 b \left (c^2 x^2+1\right )^2 \sinh ^{-1}(c x)\right )}{9 c^2 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]

[Out]

(Sqrt[d + c^2*d*x^2]*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2) + 3*b*(1 + c^2*x^2)^2*ArcSin
h[c*x]))/(9*c^2*(1 + c^2*x^2))

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Maple [B]  time = 0.122, size = 321, normalized size = 3.1 \begin{align*}{\frac{a}{3\,{c}^{2}d} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+b \left ({\frac{-1+3\,{\it Arcsinh} \left ( cx \right ) }{72\,{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ( 4\,{c}^{4}{x}^{4}+4\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+5\,{c}^{2}{x}^{2}+3\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{-1+{\it Arcsinh} \left ( cx \right ) }{8\,{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}+cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{1+{\it Arcsinh} \left ( cx \right ) }{8\,{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ({c}^{2}{x}^{2}-cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{1+3\,{\it Arcsinh} \left ( cx \right ) }{72\,{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ( 4\,{c}^{4}{x}^{4}-4\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+5\,{c}^{2}{x}^{2}-3\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a/c^2/d*(c^2*d*x^2+d)^(3/2)+b*(1/72*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2
+3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsinh(c*x))/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^
2+1)^(1/2)+1)*(-1+arcsinh(c*x))/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(1
+arcsinh(c*x))/c^2/(c^2*x^2+1)+1/72*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c
*x*(c^2*x^2+1)^(1/2)+1)*(1+3*arcsinh(c*x))/c^2/(c^2*x^2+1))

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Maxima [A]  time = 1.13595, size = 99, normalized size = 0.94 \begin{align*} \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} b \operatorname{arsinh}\left (c x\right )}{3 \, c^{2} d} - \frac{{\left (c^{2} d^{\frac{3}{2}} x^{3} + 3 \, d^{\frac{3}{2}} x\right )} b}{9 \, c d} + \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a}{3 \, c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(c^2*d*x^2 + d)^(3/2)*b*arcsinh(c*x)/(c^2*d) - 1/9*(c^2*d^(3/2)*x^3 + 3*d^(3/2)*x)*b/(c*d) + 1/3*(c^2*d*x^
2 + d)^(3/2)*a/(c^2*d)

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Fricas [A]  time = 2.58918, size = 273, normalized size = 2.6 \begin{align*} \frac{3 \,{\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} -{\left (b c^{3} x^{3} + 3 \, b c x\right )} \sqrt{c^{2} x^{2} + 1} + 3 \, a\right )} \sqrt{c^{2} d x^{2} + d}}{9 \,{\left (c^{4} x^{2} + c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(3*(b*c^4*x^4 + 2*b*c^2*x^2 + b)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (3*a*c^4*x^4 + 6*a*c^2
*x^2 - (b*c^3*x^3 + 3*b*c*x)*sqrt(c^2*x^2 + 1) + 3*a)*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError