3.127 \(\int \frac{\sqrt{d+c^2 d x^2} (a+b \sinh ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=106 \[ -\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}-\frac{b c \sqrt{c^2 d x^2+d}}{6 x^2 \sqrt{c^2 x^2+1}}+\frac{b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}} \]

[Out]

-(b*c*Sqrt[d + c^2*d*x^2])/(6*x^2*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*d*x^3)
+ (b*c^3*Sqrt[d + c^2*d*x^2]*Log[x])/(3*Sqrt[1 + c^2*x^2])

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Rubi [A]  time = 0.0945453, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {5723, 14} \[ -\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}-\frac{b c \sqrt{c^2 d x^2+d}}{6 x^2 \sqrt{c^2 x^2+1}}+\frac{b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

-(b*c*Sqrt[d + c^2*d*x^2])/(6*x^2*Sqrt[1 + c^2*x^2]) - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*d*x^3)
+ (b*c^3*Sqrt[d + c^2*d*x^2]*Log[x])/(3*Sqrt[1 + c^2*x^2])

Rule 5723

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n)/(d*f*(m + 1)), x] - Dist[(b*c*n*d^IntPart[p]*(d + e
*x^2)^FracPart[p])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*Arc
Sinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^3}+\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int \frac{1+c^2 x^2}{x^3} \, dx}{3 \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 d x^3}+\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \int \left (\frac{1}{x^3}+\frac{c^2}{x}\right ) \, dx}{3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c \sqrt{d+c^2 d x^2}}{6 x^2 \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 d x^3}+\frac{b c^3 \sqrt{d+c^2 d x^2} \log (x)}{3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.186435, size = 131, normalized size = 1.24 \[ \frac{b c^3 \log (x) \sqrt{d \left (c^2 x^2+1\right )}}{3 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left (2 a \left (c^2 x^2+1\right )^2+b c x \left (3 c^2 x^2+1\right ) \sqrt{c^2 x^2+1}+2 b \left (c^2 x^2+1\right )^2 \sinh ^{-1}(c x)\right )}{6 x^3 \left (c^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]

[Out]

-(Sqrt[d + c^2*d*x^2]*(2*a*(1 + c^2*x^2)^2 + b*c*x*Sqrt[1 + c^2*x^2]*(1 + 3*c^2*x^2) + 2*b*(1 + c^2*x^2)^2*Arc
Sinh[c*x]))/(6*x^3*(1 + c^2*x^2)) + (b*c^3*Sqrt[d*(1 + c^2*x^2)]*Log[x])/(3*Sqrt[1 + c^2*x^2])

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Maple [B]  time = 0.171, size = 946, normalized size = 8.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x)

[Out]

-1/3*a/d/x^3*(c^2*d*x^2+d)^(3/2)-2/3*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-b*(d*(c^2*x^2+
1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)*c^8+b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^
2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^7-1/6*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^2*x^2+1
)*c^8+1/6*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*c^6-3*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x
^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6+b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*a
rcsinh(c*x)*c^5-1/3*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-1/2*b*(d*(c^2*x^2+1))^
(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+1/6*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x*
c^4-10/3*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4+1/3*b*(d*(c^2*x^2+1))^
(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-1/6*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*
x^2+1)*x/(c^2*x^2+1)*c^4-1/2*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*c^3-5/3*b*(d*(c
^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x/(c^2*x^2+1)*arcsinh(c*x)*c^2-1/6*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4
+3*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*c-1/3*b*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcs
inh(c*x)+1/3*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.04533, size = 470, normalized size = 4.43 \begin{align*} -\frac{2 \,{\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 (b c^{5} x^{5} + b c^{3} x^{3}\right )} \sqrt{d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} + d x^{4} + \sqrt{c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} + d}{c^{2} x^{4} + x^{2}}\right ) +{\left (2 \, a c^{4} x^{4} + 4 \, a c^{2} x^{2} -{\left (b c x^{3} - b c x\right )} \sqrt{c^{2} x^{2} + 1} + 2 \, a\right )} \sqrt{c^{2} d x^{2} + d}}{6 \,{\left (c^{2} x^{5} + x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/6*(2*(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)) - (b*c^5*x^5 + b*c^3*x^
3)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 + d*x^4 + sqrt(c^2*d*x^2 + d)*sqrt(c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) + d)/(
c^2*x^4 + x^2)) + (2*a*c^4*x^4 + 4*a*c^2*x^2 - (b*c*x^3 - b*c*x)*sqrt(c^2*x^2 + 1) + 2*a)*sqrt(c^2*d*x^2 + d))
/(c^2*x^5 + x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))*(c**2*d*x**2+d)**(1/2)/x**4,x)

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/x^4, x)