3.5.83 \(\int \frac {\sinh ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx\) [483]

Optimal. Leaf size=42 \[ \frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {c+a^2 c x^2}} \]

[Out]

2/7*arcsinh(a*x)^(7/2)*(a^2*x^2+1)^(1/2)/a/(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783} \begin {gather*} \frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {a^2 c x^2+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^(5/2)/Sqrt[c + a^2*c*x^2],x]

[Out]

(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[c + a^2*c*x^2])

Rule 5783

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

Rubi steps

\begin {align*} \int \frac {\sinh ^{-1}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {\sinh ^{-1}(a x)^{5/2}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {c+a^2 c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {c+a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a*x]^(5/2)/Sqrt[c + a^2*c*x^2],x]

[Out]

(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Maple [A]
time = 1.22, size = 36, normalized size = 0.86

method	result	size
default	\(\frac {2 \arcsinh \left (a x \right )^{\frac {7}{2}} \sqrt {a^{2} x^{2}+1}}{7 a \sqrt {c \left (a^{2} x^{2}+1\right )}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/7*arcsinh(a*x)^(7/2)/a/(c*(a^2*x^2+1))^(1/2)*(a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^(5/2)/sqrt(a^2*c*x^2 + c), x)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**(5/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^(5/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(5/2)/sqrt(a^2*c*x^2 + c), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {asinh}\left (a\,x\right )}^{5/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a*x)^(5/2)/(c + a^2*c*x^2)^(1/2),x)

[Out]

int(asinh(a*x)^(5/2)/(c + a^2*c*x^2)^(1/2), x)

________________________________________________________________________________________