Optimal. Leaf size=224 \[ \frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}+\frac {2 h (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]
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Rubi [A] time = 0.62, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310, 2445} \[ \frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}+\frac {2 h (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2399
Rule 2400
Rule 2445
Rubi steps
\begin {align*} \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname {Subst}\left (\frac {2 \int \frac {g+h x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(f g-e h) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname {Subst}\left (\frac {2 \int \left (\frac {f g-e h}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}+\frac {h (e+f x)}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right ) \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(f g-e h) \operatorname {Subst}\left (\int \frac {1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname {Subst}\left (\frac {(2 h) \int \frac {e+f x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(2 (f g-e h)) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left ((f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname {Subst}\left (\frac {(2 h) \operatorname {Subst}\left (\int \frac {x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(2 (f g-e h)) \operatorname {Subst}\left (\int \frac {1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname {Subst}\left (\frac {\left (2 h (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (2 (f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}+\frac {2 e^{-\frac {2 a}{b p q}} h (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 269, normalized size = 1.20 \[ -\frac {(e+f x) e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (-e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Ei}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )-2 h (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \text {Ei}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )+b f p q (g+h x) e^{\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}}\right )}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 328, normalized size = 1.46 \[ \frac {{\left ({\left ({\left (b f g - b e h\right )} p q \log \left (f x + e\right ) + a f g - a e h + {\left (b f g - b e h\right )} q \log \relax (d) + {\left (b f g - b e h\right )} \log \relax (c)\right )} e^{\left (\frac {b q \log \relax (d) + b \log \relax (c) + a}{b p q}\right )} \operatorname {log\_integral}\left ({\left (f x + e\right )} e^{\left (\frac {b q \log \relax (d) + b \log \relax (c) + a}{b p q}\right )}\right ) - {\left (b f^{2} h p q x^{2} + b e f g p q + {\left (b f^{2} g + b e f h\right )} p q x\right )} e^{\left (\frac {2 \, {\left (b q \log \relax (d) + b \log \relax (c) + a\right )}}{b p q}\right )} + 2 \, {\left (b h p q \log \left (f x + e\right ) + b h q \log \relax (d) + b h \log \relax (c) + a h\right )} \operatorname {log\_integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} e^{\left (\frac {2 \, {\left (b q \log \relax (d) + b \log \relax (c) + a\right )}}{b p q}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b q \log \relax (d) + b \log \relax (c) + a\right )}}{b p q}\right )}}{b^{3} f^{2} p^{3} q^{3} \log \left (f x + e\right ) + b^{3} f^{2} p^{2} q^{3} \log \relax (d) + b^{3} f^{2} p^{2} q^{2} \log \relax (c) + a b^{2} f^{2} p^{2} q^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 1968, normalized size = 8.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {h x +g}{\left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{2}}\, dx \]
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 \[ -\frac {f h x^{2} + e g + {\left (f g + e h\right )} x}{b^{2} f p q \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + a b f p q + {\left (f p q^{2} \log \relax (d) + f p q \log \relax (c)\right )} b^{2}} + \int \frac {2 \, f h x + f g + e h}{b^{2} f p q \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + a b f p q + {\left (f p q^{2} \log \relax (d) + f p q \log \relax (c)\right )} b^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {g+h\,x}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {g + h x}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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