Optimal. Leaf size=1361 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.17, antiderivative size = 1361, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2517, 211,
2520, 12, 266, 6857, 5048, 4966, 2449, 2352, 2497} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \text {ArcTan}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}+1\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 266
Rule 2352
Rule 2449
Rule 2497
Rule 2517
Rule 2520
Rule 4966
Rule 5048
Rule 6857
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {h x} (f+g x)} \, dx &=\frac {2 \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{f+\frac {g x^2}{h}} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(8 b e p) \text {Subst}\left (\int \frac {\sqrt {h} x^3 \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt {f} \sqrt {g} \left (d+\frac {e x^4}{h^2}\right )} \, dx,x,\sqrt {h x}\right )}{h^3}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(8 b e p) \text {Subst}\left (\int \frac {x^3 \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} h^{5/2}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(8 b e p) \text {Subst}\left (\int \left (\frac {h^2 x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 \left (-\sqrt {-d} \sqrt {e} h+e x^2\right )}+\frac {h^2 x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 \left (\sqrt {-d} \sqrt {e} h+e x^2\right )}\right ) \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} h^{5/2}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \text {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{-\sqrt {-d} \sqrt {e} h+e x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \text {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt {-d} \sqrt {e} h+e x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \text {Subst}\left (\int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} x\right )}\right ) \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {(4 b e p) \text {Subst}\left (\int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{2 e^{3/4} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} x\right )}\right ) \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {\left (2 b \sqrt [4]{e} p\right ) \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {\left (2 b \sqrt [4]{e} p\right ) \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {\left (2 b \sqrt [4]{e} p\right ) \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {\left (2 b \sqrt [4]{e} p\right ) \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}{\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} x} \, dx,x,\sqrt {h x}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-4 \frac {(2 b p) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \text {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} x\right )}{\sqrt {f} \left (-i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g} \sqrt {-h}}{\sqrt {f} \sqrt {h}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \text {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} x\right )}{\sqrt {f} \left (-i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g}}{\sqrt {f}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \text {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} x\right )}{\sqrt {f} \left (i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g} \sqrt {-h}}{\sqrt {f} \sqrt {h}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}+\frac {(2 b p) \text {Subst}\left (\int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} x\right )}{\sqrt {f} \left (i \sqrt [4]{e}+\frac {\sqrt [4]{-d} \sqrt {g}}{\sqrt {f}}\right ) \sqrt {h} \left (1-\frac {i \sqrt {g} x}{\sqrt {f} \sqrt {h}}\right )}\right )}{1+\frac {g x^2}{f h}} \, dx,x,\sqrt {h x}\right )}{f h}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-4 \frac {(2 i b p) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {8 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {2 b p \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}+\frac {i b p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}-\frac {4 i b p \text {Li}_2\left (1-\frac {2}{1-\frac {i \sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}}\right )}{\sqrt {f} \sqrt {g} \sqrt {h}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 1297, normalized size = 0.95 \begin {gather*} \frac {\sqrt {x} \left (a \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{-\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (i \sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )-a \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-\sqrt [4]{e} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}-i \sqrt [4]{e} \sqrt {x}\right )}{i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+i \sqrt [4]{e} \sqrt {x}\right )}{-i \sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b p \log \left (\frac {\sqrt {g} \left (\sqrt [4]{-d}+\sqrt [4]{e} \sqrt {x}\right )}{-\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right ) \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )+b \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right ) \log \left (c \left (d+e x^2\right )^p\right )-b \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right ) \log \left (c \left (d+e x^2\right )^p\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )-b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-\sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}-i \sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+i \sqrt [4]{-d} \sqrt {g}}\right )+b p \text {Li}_2\left (\frac {\sqrt [4]{e} \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right )}{\sqrt [4]{e} \sqrt {-f}+\sqrt [4]{-d} \sqrt {g}}\right )\right )}{\sqrt {-f} \sqrt {g} \sqrt {h x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{\sqrt {h x}\, \left (g x +f \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{\left (f+g\,x\right )\,\sqrt {h\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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