Optimal. Leaf size=432 \[ \frac {h 2^{-n} (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \Gamma \left (n+1,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{f^3}+\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f^3}+\frac {h^2 3^{-n-1} (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \Gamma \left (n+1,-\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{f^3} \]
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Rubi [A] time = 0.96, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2401, 2389, 2300, 2181, 2390, 2310, 2445} \[ \frac {h 2^{-n} (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text {Gamma}\left (n+1,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{f^3}+\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f^3}+\frac {h^2 3^{-n-1} (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text {Gamma}\left (n+1,-\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{f^3} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2300
Rule 2310
Rule 2389
Rule 2390
Rule 2401
Rule 2445
Rubi steps
\begin {align*} \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx &=\operatorname {Subst}\left (\int (g+h x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {(f g-e h)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n}{f^2}+\frac {2 h (f g-e h) (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n}{f^2}+\frac {h^2 (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n}{f^2}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {h^2 \int (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(2 h (f g-e h)) \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n \, dx}{f^2},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)^2 \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {h^2 \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c d^q x^{p q}\right )\right )^n \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(2 h (f g-e h)) \operatorname {Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^n \, dx,x,e+f x\right )}{f^3},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)^2 \operatorname {Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^n \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {\left (h^2 (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \operatorname {Subst}\left (\int e^{\frac {3 x}{p q}} (a+b x)^n \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{f^3 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 (2 h (f g-e h) (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \operatorname {Subst}\left (\int e^{\frac {2 x}{p q}} (a+b x)^n \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{f^3 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)^2 (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \operatorname {Subst}\left (\int e^{\frac {x}{p q}} (a+b x)^n \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {3^{-1-n} e^{-\frac {3 a}{b p q}} h^2 (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \Gamma \left (1+n,-\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f^3}+\frac {2^{-n} e^{-\frac {2 a}{b p q}} h (f g-e h) (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \Gamma \left (1+n,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f^3}+\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \Gamma \left (1+n,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f^3}\\ \end {align*}
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Mathematica [A] time = 1.08, size = 326, normalized size = 0.75 \[ \frac {2^{-n} 3^{-n-1} (e+f x) e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \left (3^{n+1} e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \left (2^n e^{\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \Gamma \left (n+1,-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )+h (e+f x) \Gamma \left (n+1,-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )\right )+h^2 2^n (e+f x)^2 \Gamma \left (n+1,-\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )\right )}{f^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (h^{2} x^{2} + 2 \, g h x + g^{2}\right )} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (h x + g\right )}^{2} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \left (h x +g \right )^{2} \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\left (g+h\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^n \,d x \]
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 \[ \int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{n} \left (g + h x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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