3.247 \(\int \frac {c+d x}{(a+b x)^3 \log (e (\frac {a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=75 \[ \frac {(c+d x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^2 (b c-a d)} \]

[Out]

(e*((b*x+a)/(d*x+c))^n)^(2/n)*(d*x+c)^2*Ei(-2*ln(e*((b*x+a)/(d*x+c))^n)/n)/(-a*d+b*c)/n/(b*x+a)^2

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Rubi [A]  time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2510} \[ \frac {(c+d x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a
*d)*n*(a + b*x)^2)

Rule 2510

Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.))/Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.)
+ (d_.)*(x_))^(q_.))^(r_.)], x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1)*ExpIntegralEi[((m + 1)*Lo
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(p*r)])/(p*r*(b*c - a*d)*(e*(f*(a + b*x)^p*(c + d*x)^q)^r)^((m + 1)/(p*r))
), x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && EqQ[m + n + 2, 0
] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 75, normalized size = 1.00 \[ \frac {(c+d x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*Log[e*((a + b*x)/(c + d*x))^n])/n])/((b*c - a
*d)*n*(a + b*x)^2)

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fricas [A]  time = 1.00, size = 66, normalized size = 0.88 \[ \frac {e^{\frac {2}{n}} \operatorname {log\_integral}\left (\frac {d^{2} x^{2} + 2 \, c d x + c^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{\frac {2}{n}}}\right )}{{\left (b c - a d\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(2/n)*log_integral((d^2*x^2 + 2*c*d*x + c^2)/((b^2*x^2 + 2*a*b*x + a^2)*e^(2/n)))/((b*c - a*d)*n)

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

[Out]

Timed out

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {d x +c}{\left (b x +a \right )^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c+d\,x}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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