3.3.38 \(\int \frac {h+i x}{(f+g x) (a+b \log (c (d+e x)^n))^2} \, dx\) [238]

Optimal. Leaf size=143 \[ \frac {e^{-\frac {a}{b n}} i (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e g n^2}-\frac {i (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(g h-f i) \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right )}{g} \]

[Out]

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

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Rubi [A]
time = 0.14, 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 {h+i x}{(f+g x) \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[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {h+238 x}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx &=\int \left (\frac {238}{g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {-238 f+g h}{g (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}\right ) \, dx\\ &=\frac {238 \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac {(-238 f+g h) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}\\ &=\frac {238 \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{e g}+\frac {(-238 f+g h) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}\\ &=-\frac {238 (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(-238 f+g h) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac {238 \text {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{b e g n}\\ &=-\frac {238 (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(-238 f+g h) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}+\frac {\left (238 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e g n^2}\\ &=\frac {238 e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e g n^2}-\frac {238 (d+e x)}{b e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {(-238 f+g h) \int \frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx}{g}\\ \end {align*}

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Mathematica [A]
time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {h+i x}{(f+g x) \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[(h + i*x)/((f + g*x)*(a + b*Log[c*(d + e*x)^n])^2),x]

[Out]

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

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Maple [A]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {i x +h}{\left (g x +f \right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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