Optimal. Leaf size=515 \[ \frac {\sqrt {g} \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {h e^2+f^2 g}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{\sqrt {h} e+\sqrt {h e^2+f^2 g}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+f^2 g}}+1\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{\sqrt {e^2 h+f^2 g}+e \sqrt {h}}+1\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}} \]
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Rubi [A] time = 1.20, antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2406, 215, 2404, 12, 5799, 5561, 2190, 2279, 2391, 2445} \[ -\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \text {PolyLog}\left (2,-\frac {f \sqrt {g} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+f^2 g}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \text {PolyLog}\left (2,-\frac {f \sqrt {g} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{\sqrt {e^2 h+f^2 g}+e \sqrt {h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+f^2 g}}+1\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (\frac {f \sqrt {g} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}}{\sqrt {e^2 h+f^2 g}+e \sqrt {h}}+1\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {b \sqrt {g} p q \sqrt {\frac {h x^2}{g}+1} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 2190
Rule 2279
Rule 2391
Rule 2404
Rule 2406
Rule 2445
Rule 5561
Rule 5799
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {g+h x^2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname {Subst}\left (\frac {\sqrt {1+\frac {h x^2}{g}} \int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {1+\frac {h x^2}{g}}} \, dx}{\sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\operatorname {Subst}\left (\frac {\left (b f p q \sqrt {1+\frac {h x^2}{g}}\right ) \int \frac {\sqrt {g} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{\sqrt {h} (e+f x)} \, dx}{\sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\operatorname {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}{e+f x} \, dx}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\operatorname {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \frac {x \cosh (x)}{\frac {e \sqrt {h}}{\sqrt {g}}+f \sinh (x)} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\operatorname {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \frac {e^x x}{e^x f+\frac {e \sqrt {h}}{\sqrt {g}}-\frac {\sqrt {f^2 g+e^2 h}}{\sqrt {g}}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {\left (b f \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \frac {e^x x}{e^x f+\frac {e \sqrt {h}}{\sqrt {g}}+\frac {\sqrt {f^2 g+e^2 h}}{\sqrt {g}}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}+\operatorname {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {e^x f}{\frac {e \sqrt {h}}{\sqrt {g}}-\frac {\sqrt {f^2 g+e^2 h}}{\sqrt {g}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \log \left (1+\frac {e^x f}{\frac {e \sqrt {h}}{\sqrt {g}}+\frac {\sqrt {f^2 g+e^2 h}}{\sqrt {g}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}+\operatorname {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f x}{\frac {e \sqrt {h}}{\sqrt {g}}-\frac {\sqrt {f^2 g+e^2 h}}{\sqrt {g}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {\left (b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f x}{\frac {e \sqrt {h}}{\sqrt {g}}+\frac {\sqrt {f^2 g+e^2 h}}{\sqrt {g}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )}\right )}{\sqrt {h} \sqrt {g+h x^2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )^2}{2 \sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \log \left (1+\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}+\frac {\sqrt {g} \sqrt {1+\frac {h x^2}{g}} \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}-\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}-\frac {b \sqrt {g} p q \sqrt {1+\frac {h x^2}{g}} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {g}}\right )} f \sqrt {g}}{e \sqrt {h}+\sqrt {f^2 g+e^2 h}}\right )}{\sqrt {h} \sqrt {g+h x^2}}\\ \end {align*}
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Mathematica [F] time = 3.79, size = 0, normalized size = 0.00 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {g+h x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {h x^{2} + g} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt {h x^{2} + g} a}{h x^{2} + g}, x\right ) \]
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 \[ \int \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\sqrt {h \,x^{2}+g}}\, dx \]
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 \[ b \int \frac {q \log \relax (d) + \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \relax (c)}{\sqrt {h x^{2} + g}}\,{d x} + \frac {a \operatorname {arsinh}\left (\frac {h x}{\sqrt {g h}}\right )}{\sqrt {h}} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {h\,x^2+g}} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {g + h x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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