Optimal. Leaf size=107 \[ \frac {b e^2 e^{\frac {2 A}{B}} \text {Ei}\left (-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B}\right )}{B (b c-a d)^2 g^3}-\frac {d e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{B (b c-a d)^2 g^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2550, 2395,
2346, 2209} \begin {gather*} \frac {b e^2 e^{\frac {2 A}{B}} \text {Ei}\left (-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B}\right )}{B g^3 (b c-a d)^2}-\frac {d e e^{A/B} \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{B g^3 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2209
Rule 2346
Rule 2395
Rule 2550
Rubi steps
\begin {align*} \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \, dx &=\int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 89, normalized size = 0.83 \begin {gather*} \frac {e e^{A/B} \left (b e e^{A/B} \text {Ei}\left (-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{B}\right )-d \text {Ei}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )\right )}{B (b c-a d)^2 g^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 9.96, size = 117, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {e \left (\frac {d \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B}-\frac {b e \,{\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (1, 2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 A}{B}\right )}{B}\right )}{\left (a d -c b \right )^{2} g^{3}}\) | \(117\) |
default | \(\frac {e \left (\frac {d \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B}-\frac {b e \,{\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (1, 2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 A}{B}\right )}{B}\right )}{\left (a d -c b \right )^{2} g^{3}}\) | \(117\) |
risch | \(\frac {e d \,{\mathrm e}^{\frac {A}{B}} \expIntegral \left (1, \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{g^{3} \left (a d -c b \right )^{2} B}-\frac {e^{2} b \,{\mathrm e}^{\frac {2 A}{B}} \expIntegral \left (1, 2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 A}{B}\right )}{g^{3} \left (a d -c b \right )^{2} B}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 122, normalized size = 1.14 \begin {gather*} -\frac {d e^{\left (\frac {A}{B} + 1\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {A}{B} - 1\right )}}{b x + a}\right ) - b e^{\left (\frac {2 \, A}{B} + 2\right )} \operatorname {log\_integral}\left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e^{\left (-\frac {2 \, A}{B} - 2\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{{\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} g^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{A a^{3} + 3 A a^{2} b x + 3 A a b^{2} x^{2} + A b^{3} x^{3} + B a^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )} + 3 B a^{2} b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )} + 3 B a b^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )} + B b^{3} x^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx}{g^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
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]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________