3.2.58 \(\int \frac {\sqrt {a+b \log (c (d+e x)^n)}}{(f+g x)^{3/2}} \, dx\) [158]

Optimal. Leaf size=78 \[ -\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{g \sqrt {f+g x}}+\frac {b e n \text {Int}\left (\frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{g} \]

[Out]

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

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Rubi [A]
time = 0.17, 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 = {} \begin {gather*} \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^(3/2),x]

[Out]

(-2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(g*Sqrt[f + g*x]) + (b*e*n*Defer[Int][1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a +
b*Log[c*(d + e*x)^n]]), x])/g

Rubi steps

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

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Mathematica [A]
time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^(3/2),x]

[Out]

Integrate[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^(3/2), x]

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Maple [A]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}}{\left (g x +f \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^(1/2)/(g*x+f)^(3/2),x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^(1/2)/(g*x+f)^(3/2),x)

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Maxima [A]
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((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*log((x*e + d)^n*c) + a)/(g*x + f)^(3/2), x)

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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((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**(1/2)/(g*x+f)**(3/2),x)

[Out]

Integral(sqrt(a + b*log(c*(d + e*x)**n))/(f + g*x)**(3/2), x)

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Giac [A]
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((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*log((x*e + d)^n*c) + a)/(g*x + f)^(3/2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{{\left (f+g\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^(1/2)/(f + g*x)^(3/2),x)

[Out]

int((a + b*log(c*(d + e*x)^n))^(1/2)/(f + g*x)^(3/2), x)

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