3.130 \(\int x (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=146 \[ \frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac{b c^3 d x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{2 b c d x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{b d x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}} \]

[Out]

-(b*d*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^3*Sqrt[d + c^2*d*x^2])/(15*Sqrt[1 + c^2*x^2]
) - (b*c^3*d*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5
*c^2*d)

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Rubi [A]  time = 0.0740795, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5717, 194} \[ \frac{\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac{b c^3 d x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{2 b c d x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{b d x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]

[Out]

-(b*d*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2]) - (2*b*c*d*x^3*Sqrt[d + c^2*d*x^2])/(15*Sqrt[1 + c^2*x^2]
) - (b*c^3*d*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) + ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5
*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]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac{\left (b d \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 c \sqrt{1+c^2 x^2}}\\ &=\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}-\frac{\left (b d \sqrt{d+c^2 d x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt{1+c^2 x^2}}\\ &=-\frac{b d x \sqrt{d+c^2 d x^2}}{5 c \sqrt{1+c^2 x^2}}-\frac{2 b c d x^3 \sqrt{d+c^2 d x^2}}{15 \sqrt{1+c^2 x^2}}-\frac{b c^3 d x^5 \sqrt{d+c^2 d x^2}}{25 \sqrt{1+c^2 x^2}}+\frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.12883, size = 102, normalized size = 0.7 \[ \frac{d \sqrt{c^2 d x^2+d} \left (15 a \left (c^2 x^2+1\right )^3-b c x \left (3 c^4 x^4+10 c^2 x^2+15\right ) \sqrt{c^2 x^2+1}+15 b \left (c^2 x^2+1\right )^3 \sinh ^{-1}(c x)\right )}{75 c^2 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(d*Sqrt[d + c^2*d*x^2]*(15*a*(1 + c^2*x^2)^3 - b*c*x*Sqrt[1 + c^2*x^2]*(15 + 10*c^2*x^2 + 3*c^4*x^4) + 15*b*(1
+ c^2*x^2)^3*ArcSinh[c*x]))/(75*c^2*(1 + c^2*x^2))

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Maple [B]  time = 0.149, size = 559, normalized size = 3.8 \begin{align*}{\frac{a}{5\,{c}^{2}d} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+b \left ({\frac{ \left ( -1+5\,{\it Arcsinh} \left ( cx \right ) \right ) d}{800\,{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ( 16\,{c}^{6}{x}^{6}+16\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}+28\,{c}^{4}{x}^{4}+20\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+13\,{c}^{2}{x}^{2}+5\,cx\sqrt{{c}^{2}{x}^{2}+1}+1 \right ) }+{\frac{ \left ( -1+3\,{\it Arcsinh} \left ( cx \right ) \right ) d}{96\,{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{ \left ( -1+{\it Arcsinh} \left ( cx \right ) \right ) d}{16\,{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{ \left ( 1+{\it Arcsinh} \left ( cx \right ) \right ) d}{16\,{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{ \left ( 1+3\,{\it Arcsinh} \left ( cx \right ) \right ) d}{96\,{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{ \left ( 1+5\,{\it Arcsinh} \left ( cx \right ) \right ) d}{800\,{c}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) } \left ( 16\,{c}^{6}{x}^{6}-16\,{c}^{5}{x}^{5}\sqrt{{c}^{2}{x}^{2}+1}+28\,{c}^{4}{x}^{4}-20\,{c}^{3}{x}^{3}\sqrt{{c}^{2}{x}^{2}+1}+13\,{c}^{2}{x}^{2}-5\,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*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x)

[Out]

1/5*a/c^2/d*(c^2*d*x^2+d)^(5/2)+b*(1/800*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4
*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+5*arcsinh(c*x))*d/c^2/(c^2*x^2+1)+
1/96*(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*a
rcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/16*(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))*
d/c^2/(c^2*x^2+1)+1/16*(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))*d/c^2/(c^2*x^2
+1)+1/96*(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))*d/c^2/(c^2*x^2+1)+1/800*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*
x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*(1+5*arcsinh(c*x))*d/c^2/(c^2*x^2+1))

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Maxima [A]  time = 1.205, size = 115, normalized size = 0.79 \begin{align*} \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}} b \operatorname{arsinh}\left (c x\right )}{5 \, c^{2} d} + \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}} a}{5 \, c^{2} d} - \frac{{\left (3 \, c^{4} d^{\frac{5}{2}} x^{5} + 10 \, c^{2} d^{\frac{5}{2}} x^{3} + 15 \, d^{\frac{5}{2}} x\right )} b}{75 \, c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

1/5*(c^2*d*x^2 + d)^(5/2)*b*arcsinh(c*x)/(c^2*d) + 1/5*(c^2*d*x^2 + d)^(5/2)*a/(c^2*d) - 1/75*(3*c^4*d^(5/2)*x
^5 + 10*c^2*d^(5/2)*x^3 + 15*d^(5/2)*x)*b/(c*d)

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Fricas [A]  time = 2.4402, size = 373, normalized size = 2.55 \begin{align*} \frac{15 \,{\left (b c^{6} d x^{6} + 3 \, b c^{4} d x^{4} + 3 \, b c^{2} d x^{2} + b d\right )} \sqrt{c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (15 \, a c^{6} d x^{6} + 45 \, a c^{4} d x^{4} + 45 \, a c^{2} d x^{2} + 15 \, a d -{\left (3 \, b c^{5} d x^{5} + 10 \, b c^{3} d x^{3} + 15 \, b c d x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{c^{2} d x^{2} + d}}{75 \,{\left (c^{4} x^{2} + c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/75*(15*(b*c^6*d*x^6 + 3*b*c^4*d*x^4 + 3*b*c^2*d*x^2 + b*d)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))
+ (15*a*c^6*d*x^6 + 45*a*c^4*d*x^4 + 45*a*c^2*d*x^2 + 15*a*d - (3*b*c^5*d*x^5 + 10*b*c^3*d*x^3 + 15*b*c*d*x)*s
qrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^4*x^2 + c^2)

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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(x*(c**2*d*x**2+d)**(3/2)*(a+b*asinh(c*x)),x)

[Out]

Timed out

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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*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError