3.513 \(\int \frac{1}{x \sqrt{a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt{a}}\right )}{\sqrt{a} m n p q} \]

[Out]

(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]/Sqrt[a]])/(Sqrt[a]*m*n*p*q)

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Rubi [A]  time = 1.10166, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt{a}}\right )}{\sqrt{a} m n p q} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]/Sqrt[a]])/(Sqrt[a]*m*n*p*q)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a + b \left (c \left (d \left (e \left (f x\right )^{m}\right )^{n}\right )^{p}\right )^{q}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*(c*(d*(e*(f*x)**m)**n)**p)**q)**(1/2),x)

[Out]

Integral(1/(x*sqrt(a + b*(c*(d*(e*(f*x)**m)**n)**p)**q)), x)

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Mathematica [A]  time = 2.22159, size = 51, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt{a}}\right )}{\sqrt{a} m n p q} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]),x]

[Out]

(-2*ArcTanh[Sqrt[a + b*(c*(d*(e*(f*x)^m)^n)^p)^q]/Sqrt[a]])/(Sqrt[a]*m*n*p*q)

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Maple [A]  time = 0.032, size = 46, normalized size = 0.9 \[ -2\,{\frac{1}{mnpq\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( c \left ( d \left ( e \left ( fx \right ) ^{m} \right ) ^{n} \right ) ^{p} \right ) ^{q}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*(c*(d*(e*(f*x)^m)^n)^p)^q)^(1/2),x)

[Out]

-2*arctanh((a+b*(c*(d*(e*(f*x)^m)^n)^p)^q)^(1/2)/a^(1/2))/m/n/p/q/a^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(((((f*x)^m*e)^n*d)^p*c)^q*b + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288374, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left ({\left (\sqrt{a} b e^{\left (m n p q \log \left (f x\right ) + n p q \log \left (e\right ) + p q \log \left (d\right ) + q \log \left (c\right )\right )} - 2 \, \sqrt{b e^{\left (m n p q \log \left (f x\right ) + n p q \log \left (e\right ) + p q \log \left (d\right ) + q \log \left (c\right )\right )} + a} a + 2 \, a^{\frac{3}{2}}\right )} e^{\left (-m n p q \log \left (f x\right ) - n p q \log \left (e\right ) - p q \log \left (d\right ) - q \log \left (c\right )\right )}\right )}{\sqrt{a} m n p q}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b e^{\left (m n p q \log \left (f x\right ) + n p q \log \left (e\right ) + p q \log \left (d\right ) + q \log \left (c\right )\right )} + a} \sqrt{-a}}\right )}{\sqrt{-a} m n p q}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(((((f*x)^m*e)^n*d)^p*c)^q*b + a)*x),x, algorithm="fricas")

[Out]

[log((sqrt(a)*b*e^(m*n*p*q*log(f*x) + n*p*q*log(e) + p*q*log(d) + q*log(c)) - 2*
sqrt(b*e^(m*n*p*q*log(f*x) + n*p*q*log(e) + p*q*log(d) + q*log(c)) + a)*a + 2*a^
(3/2))*e^(-m*n*p*q*log(f*x) - n*p*q*log(e) - p*q*log(d) - q*log(c)))/(sqrt(a)*m*
n*p*q), 2*arctan(a/(sqrt(b*e^(m*n*p*q*log(f*x) + n*p*q*log(e) + p*q*log(d) + q*l
og(c)) + a)*sqrt(-a)))/(sqrt(-a)*m*n*p*q)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a + b \left (c \left (d \left (e \left (f x\right )^{m}\right )^{n}\right )^{p}\right )^{q}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b*(c*(d*(e*(f*x)**m)**n)**p)**q)**(1/2),x)

[Out]

Integral(1/(x*sqrt(a + b*(c*(d*(e*(f*x)**m)**n)**p)**q)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (\left (\left (\left (f x\right )^{m} e\right )^{n} d\right )^{p} c\right )^{q} b + a} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(((((f*x)^m*e)^n*d)^p*c)^q*b + a)*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(((((f*x)^m*e)^n*d)^p*c)^q*b + a)*x), x)