3.4.100 \(\int \frac {(a+b \log (c (d+e x)^n))^3 (f+g \log (h (i+j x)^m))}{x^2} \, dx\) [400]

Optimal. Leaf size=37 \[ \text {Int}\left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, 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 {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (400+j x)^m\right )\right )}{x^2} \, dx &=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (400+j x)^m\right )\right )}{x^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{3} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m))/x^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))^3*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="maxima")

[Out]

-3*a^2*b*f*n*(log(x*e + d)/d - log(x)/d)*e - 3*a^2*b*f*log((x*e + d)^n*c)/x - a^3*f/x + integrate(((g*log(h) +
f)*b^3*log((x*e + d)^n)^3 + (g*log(h) + f)*b^3*log(c)^3 + 3*(g*log(h) + f)*a*b^2*log(c)^2 + 3*a^2*b*g*log(c)*
log(h) + a^3*g*log(h) + 3*((g*log(h) + f)*b^3*log(c) + (g*log(h) + f)*a*b^2)*log((x*e + d)^n)^2 + (b^3*g*log((
x*e + d)^n)^3 + b^3*g*log(c)^3 + 3*a*b^2*g*log(c)^2 + 3*a^2*b*g*log(c) + a^3*g + 3*(b^3*g*log(c) + a*b^2*g)*lo
g((x*e + d)^n)^2 + 3*(b^3*g*log(c)^2 + 2*a*b^2*g*log(c) + a^2*b*g)*log((x*e + d)^n))*log((j*x + I)^m) + 3*((g*
log(h) + f)*b^3*log(c)^2 + 2*(g*log(h) + f)*a*b^2*log(c) + a^2*b*g*log(h))*log((x*e + d)^n))/x^2, x)

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Fricas [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))^3*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x^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))^3*(f + g*log(h*(i + j*x)^m)))/x^2,x)

[Out]

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

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