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

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {(f+g x)^{3/2}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x + f\right )}^{\frac {3}{2}}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{\frac {3}{2}}}{b \ln \left (c \left (e x +d \right )^{n}\right )+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (g^{2} x^{2} + 2 \, f g x + f^{2}\right )} \sqrt {g x + f}}{5 \, {\left (b g \log \left ({\left (e x + d\right )}^{n}\right ) + b g \log \relax (c) + a g\right )}} + \int \frac {2 \, {\left (b e g^{2} n x^{2} + 2 \, b e f g n x + b e f^{2} n\right )} \sqrt {g x + f}}{5 \, {\left (b^{2} d g \log \relax (c)^{2} + 2 \, a b d g \log \relax (c) + a^{2} d g + {\left (b^{2} e g x + b^{2} d g\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + {\left (b^{2} e g \log \relax (c)^{2} + 2 \, a b e g \log \relax (c) + a^{2} e g\right )} x + 2 \, {\left (b^{2} d g \log \relax (c) + a b d g + {\left (b^{2} e g \log \relax (c) + a b e g\right )} x\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(g^2*x^2 + 2*f*g*x + f^2)*sqrt(g*x + f)/(b*g*log((e*x + d)^n) + b*g*log(c) + a*g) + integrate(2/5*(b*e*g^2
*n*x^2 + 2*b*e*f*g*n*x + b*e*f^2*n)*sqrt(g*x + f)/(b^2*d*g*log(c)^2 + 2*a*b*d*g*log(c) + a^2*d*g + (b^2*e*g*x
+ b^2*d*g)*log((e*x + d)^n)^2 + (b^2*e*g*log(c)^2 + 2*a*b*e*g*log(c) + a^2*e*g)*x + 2*(b^2*d*g*log(c) + a*b*d*
g + (b^2*e*g*log(c) + a*b*e*g)*x)*log((e*x + d)^n)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (f+g\,x\right )}^{3/2}}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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