3.432 \(\int \frac {(1+c^2 x^2)^{5/2}}{x^2 (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=103 \[ \frac {4 c \text {Int}\left (\frac {\left (c^2 x^2+1\right )^2}{x \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b}-\frac {2 \text {Int}\left (\frac {\left (c^2 x^2+1\right )^2}{x^3 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c}-\frac {\left (c^2 x^2+1\right )^3}{b c x^2 \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

-(c^2*x^2+1)^3/b/c/x^2/(a+b*arcsinh(c*x))-2*Unintegrable((c^2*x^2+1)^2/x^3/(a+b*arcsinh(c*x)),x)/b/c+4*c*Unint
egrable((c^2*x^2+1)^2/x/(a+b*arcsinh(c*x)),x)/b

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Rubi [A]  time = 0.32, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])^2),x]

[Out]

-((1 + c^2*x^2)^3/(b*c*x^2*(a + b*ArcSinh[c*x]))) - (2*Defer[Int][(1 + c^2*x^2)^2/(x^3*(a + b*ArcSinh[c*x])),
x])/(b*c) + (4*c*Defer[Int][(1 + c^2*x^2)^2/(x*(a + b*ArcSinh[c*x])), x])/b

Rubi steps

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

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Mathematica [A]  time = 3.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])^2), x]

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{4} + 2 \, c^{2} x^{2} + 1\right )} \sqrt {c^{2} x^{2} + 1}}{b^{2} x^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {arsinh}\left (c x\right ) + a^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^4*x^4 + 2*c^2*x^2 + 1)*sqrt(c^2*x^2 + 1)/(b^2*x^2*arcsinh(c*x)^2 + 2*a*b*x^2*arcsinh(c*x) + a^2*x^
2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c^2*x^2 + 1)^(5/2)/((b*arcsinh(c*x) + a)^2*x^2), x)

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maple [A]  time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c^{6} x^{6} + 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + 1\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{7} x^{7} + 3 \, c^{5} x^{5} + 3 \, c^{3} x^{3} + c x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{4} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x^{3} + a b c x^{2} + {\left (b^{2} c^{3} x^{4} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x^{3} + b^{2} c x^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (4 \, c^{7} x^{7} + 5 \, c^{5} x^{5} - 2 \, c^{3} x^{3} - 3 \, c x\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 2 \, {\left (4 \, c^{8} x^{8} + 8 \, c^{6} x^{6} + 3 \, c^{4} x^{4} - 2 \, c^{2} x^{2} - 1\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (4 \, c^{9} x^{9} + 11 \, c^{7} x^{7} + 9 \, c^{5} x^{5} + c^{3} x^{3} - c x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{7} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{5} + 2 \, a b c^{3} x^{5} + a b c x^{3} + {\left (b^{2} c^{5} x^{7} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{5} + 2 \, b^{2} c^{3} x^{5} + b^{2} c x^{3} + 2 \, {\left (b^{2} c^{4} x^{6} + b^{2} c^{2} x^{4}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{6} + a b c^{2} x^{4}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^6*x^6 + 3*c^4*x^4 + 3*c^2*x^2 + 1)*(c^2*x^2 + 1) + (c^7*x^7 + 3*c^5*x^5 + 3*c^3*x^3 + c*x)*sqrt(c^2*x^2 +
1))/(a*b*c^3*x^4 + sqrt(c^2*x^2 + 1)*a*b*c^2*x^3 + a*b*c*x^2 + (b^2*c^3*x^4 + sqrt(c^2*x^2 + 1)*b^2*c^2*x^3 +
b^2*c*x^2)*log(c*x + sqrt(c^2*x^2 + 1))) + integrate(((4*c^7*x^7 + 5*c^5*x^5 - 2*c^3*x^3 - 3*c*x)*(c^2*x^2 +
1)^(3/2) + 2*(4*c^8*x^8 + 8*c^6*x^6 + 3*c^4*x^4 - 2*c^2*x^2 - 1)*(c^2*x^2 + 1) + (4*c^9*x^9 + 11*c^7*x^7 + 9*c
^5*x^5 + c^3*x^3 - c*x)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^7 + (c^2*x^2 + 1)*a*b*c^3*x^5 + 2*a*b*c^3*x^5 + a*b*c*x^
3 + (b^2*c^5*x^7 + (c^2*x^2 + 1)*b^2*c^3*x^5 + 2*b^2*c^3*x^5 + b^2*c*x^3 + 2*(b^2*c^4*x^6 + b^2*c^2*x^4)*sqrt(
c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^6 + a*b*c^2*x^4)*sqrt(c^2*x^2 + 1)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c^2\,x^2+1\right )}^{5/2}}{x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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