3.121 \(\int \frac {(a+b \log (c (d+e x)^n))^{5/2}}{(f+g x)^2} \, dx\)

Optimal. Leaf size=88 \[ \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x) (e f-d g)}-\frac {5 b e n \text {Int}\left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{f+g x},x\right )}{2 (e f-d g)} \]

[Out]

(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(5/2)/(-d*g+e*f)/(g*x+f)-5/2*b*e*n*Unintegrable((a+b*ln(c*(e*x+d)^n))^(3/2)/(g*x
+f),x)/(-d*g+e*f)

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x)^2,x]

[Out]

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(5/2))/((e*f - d*g)*(f + g*x)) - (5*b*e*n*Defer[Int][(a + b*Log[c*(d + e
*x)^n])^(3/2)/(f + g*x), x])/(2*(e*f - d*g))

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^2} \, dx &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(e f-d g) (f+g x)}-\frac {(5 b e n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{f+g x} \, dx}{2 (e f-d g)}\\ \end {align*}

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Mathematica [A]  time = 7.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x)^2,x]

[Out]

Integrate[(a + b*Log[c*(d + e*x)^n])^(5/2)/(f + g*x)^2, 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((a+b*log(c*(e*x+d)^n))^(5/2)/(g*x+f)^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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}}{{\left (g x + f\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{\frac {5}{2}}}{\left (g x +f \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(e*x+d)^n)+a)^(5/2)/(g*x+f)^2,x)

[Out]

int((b*ln(c*(e*x+d)^n)+a)^(5/2)/(g*x+f)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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