3.160 \(\int \frac {1}{\sqrt {f+g x} \sqrt {a+b \log (c (d+e x)^n)}} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {1}{\sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right ) \]

[Out]

Unintegrable(1/(g*x+f)^(1/2)/(a+b*ln(c*(e*x+d)^n))^(1/2),x)

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Defer[Int][1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx &=\int \frac {1}{\sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Integrate[1/(Sqrt[f + g*x]*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^(1/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {g x + f} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^(1/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(g*x + f)*sqrt(b*log((e*x + d)^n*c) + a)), x)

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maple [A]  time = 0.57, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {g x +f}\, \sqrt {b \ln \left (c \left (e x +d \right )^{n}\right )+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x+f)^(1/2)/(b*ln(c*(e*x+d)^n)+a)^(1/2),x)

[Out]

int(1/(g*x+f)^(1/2)/(b*ln(c*(e*x+d)^n)+a)^(1/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {g x + f} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^(1/2)/(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(g*x + f)*sqrt(b*log((e*x + d)^n*c) + a)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {f+g\,x}\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((f + g*x)^(1/2)*(a + b*log(c*(d + e*x)^n))^(1/2)),x)

[Out]

int(1/((f + g*x)^(1/2)*(a + b*log(c*(d + e*x)^n))^(1/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}} \sqrt {f + g x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)**(1/2)/(a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*log(c*(d + e*x)**n))*sqrt(f + g*x)), x)

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