3.3.41 \(\int \frac {1}{(f+g x) (h+i x)^2 (a+b \log (c (d+e x)^n))^2} \, dx\) [241]

Optimal. Leaf size=123 \[ \frac {g^2 \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{(g h-f i)^2}-\frac {i \text {Int}\left (\frac {1}{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g h-f i}-\frac {g i \text {Int}\left (\frac {1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{(g h-f i)^2} \]

[Out]

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

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Rubi [A]
time = 0.15, 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 {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(h+241 x)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac {241}{(241 f-g h) (h+241 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {241 g}{(241 f-g h)^2 (h+241 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {g^2}{(241 f-g h)^2 (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=-\frac {(241 g) \int \frac {1}{(h+241 x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{(241 f-g h)^2}+\frac {g^2 \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{(241 f-g h)^2}+\frac {241 \int \frac {1}{(h+241 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{241 f-g h}\\ \end {align*}

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Mathematica [A]
time = 14.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((f + g*x)*(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

Integrate[1/((f + g*x)*(h + i*x)^2*(a + b*Log[c*(d + e*x)^n])^2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (g x +f \right ) \left (i x +h \right )^{2} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

(x*e + d)/((b^2*g*n*log(c) + a*b*g*n)*x^3*e + ((-2*I*g*h*n + f*n)*b^2*log(c) + (-2*I*g*h*n + f*n)*a*b)*x^2*e -
((g*h^2*n + 2*I*f*h*n)*b^2*log(c) + (g*h^2*n + 2*I*f*h*n)*a*b)*x*e - (b^2*f*h^2*n*log(c) + a*b*f*h^2*n)*e + (
b^2*g*n*x^3*e - b^2*f*h^2*n*e + (-2*I*g*h*n + f*n)*b^2*x^2*e - (g*h^2*n + 2*I*f*h*n)*b^2*x*e)*log((x*e + d)^n)
) + integrate((2*g*x^2*e + I*f*h*e + (-I*g*h + 2*f)*d + (3*d*g + f*e)*x)/((b^2*g^2*n*log(c) + a*b*g^2*n)*x^5*e
+ ((-3*I*g^2*h*n + 2*f*g*n)*b^2*log(c) + (-3*I*g^2*h*n + 2*f*g*n)*a*b)*x^4*e - ((3*g^2*h^2*n + 6*I*f*g*h*n -
f^2*n)*b^2*log(c) + (3*g^2*h^2*n + 6*I*f*g*h*n - f^2*n)*a*b)*x^3*e + ((I*g^2*h^3*n - 6*f*g*h^2*n - 3*I*f^2*h*n
)*b^2*log(c) + (I*g^2*h^3*n - 6*f*g*h^2*n - 3*I*f^2*h*n)*a*b)*x^2*e + ((2*I*f*g*h^3*n - 3*f^2*h^2*n)*b^2*log(c
) + (2*I*f*g*h^3*n - 3*f^2*h^2*n)*a*b)*x*e + (I*b^2*f^2*h^3*n*log(c) + I*a*b*f^2*h^3*n)*e + (b^2*g^2*n*x^5*e +
I*b^2*f^2*h^3*n*e + (-3*I*g^2*h*n + 2*f*g*n)*b^2*x^4*e - (3*g^2*h^2*n + 6*I*f*g*h*n - f^2*n)*b^2*x^3*e + (I*g
^2*h^3*n - 6*f*g*h^2*n - 3*I*f^2*h*n)*b^2*x^2*e + (2*I*f*g*h^3*n - 3*f^2*h^2*n)*b^2*x*e)*log((x*e + d)^n)), 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(1/(g*x+f)/(i*x+h)^2/(a+b*log(c*(e*x+d)^n))^2,x, algorithm="fricas")

[Out]

(x*e + ((b^2*g*n^2*x^3 - b^2*f*h^2*n^2 + (-2*I*b^2*g*h + b^2*f)*n^2*x^2 - (b^2*g*h^2 + 2*I*b^2*f*h)*n^2*x)*e*l
og(x*e + d) + (b^2*g*n*x^3 - b^2*f*h^2*n + (-2*I*b^2*g*h + b^2*f)*n*x^2 - (b^2*g*h^2 + 2*I*b^2*f*h)*n*x)*e*log
(c) + (a*b*g*n*x^3 - a*b*f*h^2*n + (-2*I*a*b*g*h + a*b*f)*n*x^2 - (a*b*g*h^2 + 2*I*a*b*f*h)*n*x)*e)*integral((
-I*d*g*h + 3*d*g*x + 2*d*f + (2*g*x^2 + I*f*h + f*x)*e)/((b^2*g^2*n^2*x^5 + I*b^2*f^2*h^3*n^2 + (-3*I*b^2*g^2*
h + 2*b^2*f*g)*n^2*x^4 - (3*b^2*g^2*h^2 + 6*I*b^2*f*g*h - b^2*f^2)*n^2*x^3 + (I*b^2*g^2*h^3 - 6*b^2*f*g*h^2 -
3*I*b^2*f^2*h)*n^2*x^2 + (2*I*b^2*f*g*h^3 - 3*b^2*f^2*h^2)*n^2*x)*e*log(x*e + d) + (b^2*g^2*n*x^5 + I*b^2*f^2*
h^3*n + (-3*I*b^2*g^2*h + 2*b^2*f*g)*n*x^4 - (3*b^2*g^2*h^2 + 6*I*b^2*f*g*h - b^2*f^2)*n*x^3 + (I*b^2*g^2*h^3
- 6*b^2*f*g*h^2 - 3*I*b^2*f^2*h)*n*x^2 + (2*I*b^2*f*g*h^3 - 3*b^2*f^2*h^2)*n*x)*e*log(c) + (a*b*g^2*n*x^5 + I*
a*b*f^2*h^3*n + (-3*I*a*b*g^2*h + 2*a*b*f*g)*n*x^4 - (3*a*b*g^2*h^2 + 6*I*a*b*f*g*h - a*b*f^2)*n*x^3 + (I*a*b*
g^2*h^3 - 6*a*b*f*g*h^2 - 3*I*a*b*f^2*h)*n*x^2 + (2*I*a*b*f*g*h^3 - 3*a*b*f^2*h^2)*n*x)*e), x) + d)/((b^2*g*n^
2*x^3 - b^2*f*h^2*n^2 + (-2*I*b^2*g*h + b^2*f)*n^2*x^2 - (b^2*g*h^2 + 2*I*b^2*f*h)*n^2*x)*e*log(x*e + d) + (b^
2*g*n*x^3 - b^2*f*h^2*n + (-2*I*b^2*g*h + b^2*f)*n*x^2 - (b^2*g*h^2 + 2*I*b^2*f*h)*n*x)*e*log(c) + (a*b*g*n*x^
3 - a*b*f*h^2*n + (-2*I*a*b*g*h + a*b*f)*n*x^2 - (a*b*g*h^2 + 2*I*a*b*f*h)*n*x)*e)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right ) \left (h + i x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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