Optimal. Leaf size=541 \[ -\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Rubi [A] time = 0.81, antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2472, 275, 205, 2416, 260, 2394, 2393, 2391} \[ -\frac {p \text {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 275
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2472
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{f+g x^4} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {\sqrt {g} x \log \left (c (d+e x)^p\right )}{2 \sqrt {-f} \left (\sqrt {-f} \sqrt {g}-g x^2\right )}-\frac {\sqrt {g} x \log \left (c (d+e x)^p\right )}{2 \sqrt {-f} \left (\sqrt {-f} \sqrt {g}+g x^2\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {g} \operatorname {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{\sqrt {-f} \sqrt {g}-g x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {-f}}-\frac {\sqrt {g} \operatorname {Subst}\left (\int \frac {x \log \left (c (d+e x)^p\right )}{\sqrt {-f} \sqrt {g}+g x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {-f}}\\ &=-\frac {\sqrt {g} \operatorname {Subst}\left (\int \left (-\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} x\right )}+\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-f}}-\frac {\sqrt {g} \operatorname {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt [4]{-f}-\sqrt [4]{g} x\right )}-\frac {\log \left (c (d+e x)^p\right )}{2 g^{3/4} \left (\sqrt [4]{-f}+\sqrt [4]{g} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-f}}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-\sqrt {-f}}-\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt [4]{-f}-\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-\sqrt {-f}}+\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt [4]{-f}+\sqrt [4]{g} x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt [4]{g}}\\ &=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} x\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} x\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} x\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \operatorname {Subst}\left (\int \frac {\log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} x\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{d+e x} \, dx,x,\sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}\\ &=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{g} x}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [4]{g} x}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{g} x}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [4]{g} x}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{2 \sqrt {-f} \sqrt {g}}\\ &=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}\\ \end {align*}
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Mathematica [C] time = 0.30, size = 422, normalized size = 0.78 \[ \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-i d \sqrt [4]{g}}\right )+\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )+p \text {Li}_2\left (-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )-p \text {Li}_2\left (\frac {i \sqrt [4]{g} \left (d+e \sqrt {x}\right )}{i \sqrt [4]{g} d+e \sqrt [4]{-f}}\right )-p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+i e \sqrt [4]{-f}}\right )+p \text {Li}_2\left (\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt [4]{-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (e \sqrt {x}+d \right )^{p}\right )}{g \,x^{2}+f}\, dx \]
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 \[ \int \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^p\right )}{g\,x^2+f} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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