Optimal. Leaf size=159 \[ \frac {d e^{-\frac {A}{B}} \text {Ei}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 (b c-a d)^2 e g^3}-\frac {2 b e^{-\frac {2 A}{B}} \text {Ei}\left (\frac {2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B}\right )}{B^2 (b c-a d)^2 e^2 g^3}+\frac {c+d x}{B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \]
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Rubi [A]
time = 0.16, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2552, 2357,
2367, 2336, 2209, 2346} \begin {gather*} -\frac {2 b e^{-\frac {2 A}{B}} \text {Ei}\left (\frac {2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B}\right )}{B^2 e^2 g^3 (b c-a d)^2}+\frac {d e^{-\frac {A}{B}} \text {Ei}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 e g^3 (b c-a d)^2}+\frac {c+d x}{B g^3 (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2336
Rule 2346
Rule 2357
Rule 2367
Rule 2552
Rubi steps
\begin {align*} \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx &=\int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 135, normalized size = 0.85 \begin {gather*} \frac {\frac {d e^{-\frac {A}{B}} \text {Ei}\left (\frac {A}{B}+\log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{e}-\frac {2 b e^{-\frac {2 A}{B}} \text {Ei}\left (\frac {2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{B}\right )}{e^2}+\frac {B (b c-a d) (c+d x)}{(a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}}{B^2 (b c-a d)^2 g^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.96, size = 268, normalized size = 1.69
method | result | size |
derivativedivides | \(-\frac {\frac {b \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-2 \,{\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )\right )}{B^{2}}-\frac {e d \left (-\frac {\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )\right )}{B^{2}}}{e^{2} \left (a d -c b \right )^{2} g^{3}}\) | \(268\) |
default | \(-\frac {\frac {b \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-2 \,{\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )\right )}{B^{2}}-\frac {e d \left (-\frac {\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )\right )}{B^{2}}}{e^{2} \left (a d -c b \right )^{2} g^{3}}\) | \(268\) |
risch | \(-\frac {d x +c}{\left (a d -c b \right ) B \left (b x +a \right )^{2} g^{3} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )}+\frac {b c d \,{\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \,g^{3} B^{2} \left (a d -c b \right )^{3}}-\frac {2 c \,b^{2} {\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )}{e^{2} g^{3} B^{2} \left (a d -c b \right )^{3}}-\frac {a \,d^{2} {\mathrm e}^{-\frac {A}{B}} \expIntegral \left (1, -\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \,g^{3} B^{2} \left (a d -c b \right )^{3}}+\frac {2 d b a \,{\mathrm e}^{-\frac {2 A}{B}} \expIntegral \left (1, -2 \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )-\frac {2 A}{B}\right )}{e^{2} g^{3} B^{2} \left (a d -c b \right )^{3}}\) | \(338\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 550 vs.
\(2 (158) = 316\).
time = 0.36, size = 550, normalized size = 3.46 \begin {gather*} \frac {{\left (B b c^{2} - B a c d + {\left (B b c d - B a d^{2}\right )} x\right )} e^{\left (\frac {2 \, A}{B} + 2\right )} - 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b + {\left (B b^{3} x^{2} + 2 \, B a b^{2} x + B a^{2} b\right )} \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right )\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 ({\left (B b^{2} d x^{2} + 2 \, B a b d x + B a^{2} d\right )} e^{\left (\frac {A}{B} + 1\right )} \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + {\left (A b^{2} d x^{2} + 2 \, A a b d x + A a^{2} d\right )} e^{\left (\frac {A}{B} + 1\right )}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (\frac {A}{B} + 1\right )}}{b x + a}\right )}{{\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} g^{3} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} g^{3}\right )} e^{\left (\frac {2 \, A}{B} + 2\right )} \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + {\left ({\left (A B^{2} b^{4} c^{2} - 2 \, A B^{2} a b^{3} c d + A B^{2} a^{2} b^{2} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (A B^{2} a b^{3} c^{2} - 2 \, A B^{2} a^{2} b^{2} c d + A B^{2} a^{3} b d^{2}\right )} g^{3} x + {\left (A B^{2} a^{2} b^{2} c^{2} - 2 \, A B^{2} a^{3} b c d + A B^{2} a^{4} d^{2}\right )} g^{3}\right )} e^{\left (\frac {2 \, A}{B} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs.
\(2 (158) = 316\).
time = 5.14, size = 317, normalized size = 1.99 \begin {gather*} {\left (\frac {d {\rm Ei}\left (\frac {A}{B} + \log \left (\frac {d x e + c e}{b x + a}\right )\right ) e^{\left (-\frac {A}{B} + 1\right )}}{B^{2} b c g^{3} e - B^{2} a d g^{3} e} - \frac {2 \, b {\rm Ei}\left (\frac {2 \, A}{B} + 2 \, \log \left (\frac {d x e + c e}{b x + a}\right )\right ) e^{\left (-\frac {2 \, A}{B}\right )}}{B^{2} b c g^{3} e - B^{2} a d g^{3} e} - \frac {\frac {{\left (d x e + c e\right )} d e}{b x + a} - \frac {{\left (d x e + c e\right )}^{2} b}{{\left (b x + a\right )}^{2}}}{B^{2} b c g^{3} e \log \left (\frac {d x e + c e}{b x + a}\right ) - B^{2} a d g^{3} e \log \left (\frac {d x e + c e}{b x + a}\right ) + A B b c g^{3} e - A B a d g^{3} e}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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 (c+d\,x\right )}{a+b\,x}\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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