Optimal. Leaf size=1680 \[ \frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {8 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {4 i b \sqrt {f} \sqrt {h} p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.03, antiderivative size = 1680, normalized size of antiderivative = 1.00, number of steps
used = 39, number of rules used = 20, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.645, Rules used = {2517, 2526,
2498, 327, 217, 1179, 642, 1176, 631, 210, 211, 2520, 12, 266, 6857, 5048, 4966, 2449, 2352, 2497}
\begin {gather*} \frac {2 \sqrt {h x} a}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {8 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {4 i b \sqrt {f} \sqrt {h} p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {8 b p \sqrt {h x}}{g} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 210
Rule 211
Rule 217
Rule 266
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2352
Rule 2449
Rule 2497
Rule 2498
Rule 2517
Rule 2520
Rule 2526
Rule 4966
Rule 5048
Rule 6857
Rubi steps
\begin {align*} \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{f+\frac {g x^2}{h}} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {h \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{g}-\frac {f h \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{g \left (f+\frac {g x^2}{h}\right )}\right ) \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 \text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{g}-\frac {(2 f) \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 )}{g}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right ) \, dx,x,\sqrt {h x}\right )}{g}+\frac {(8 b e f 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 )}{g h^2}\\ &=\frac {2 a \sqrt {h x}}{g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {(8 b e p) \text {Subst}\left (\int \frac {x^4}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{g h^2}+\frac {\left (8 b e \sqrt {f} p\right ) \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 )}{g^{3/2} h^{3/2}}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}+\frac {(8 b d p) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{g}+\frac {\left (8 b e \sqrt {f} p\right ) \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 )}{g^{3/2} h^{3/2}}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}+\frac {\left (4 b \sqrt {d} p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h-\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{g h}+\frac {\left (4 b \sqrt {d} p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h+\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{g h}+\frac {\left (4 b e \sqrt {f} \sqrt {h} p\right ) \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 )}{g^{3/2}}+\frac {\left (4 b e \sqrt {f} \sqrt {h} p\right ) \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 )}{g^{3/2}}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}+\frac {\left (4 b e \sqrt {f} \sqrt {h} p\right ) \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 )}{g^{3/2}}+\frac {\left (4 b e \sqrt {f} \sqrt {h} p\right ) \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 )}{g^{3/2}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} \sqrt {h} p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {\left (\sqrt {2} b \sqrt [4]{d} \sqrt {h} p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\left (2 b \sqrt {d} h p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {e} g}+\frac {\left (2 b \sqrt {d} h p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt {e} g}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {\left (2 b \sqrt [4]{e} \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {\left (2 b \sqrt [4]{e} \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {\left (2 b \sqrt [4]{e} \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {\left (2 b \sqrt [4]{e} \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {\left (2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {8 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+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 )}{g}-\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 )}{g}-\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 )}{g}-\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 )}{g}-\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 )}{g}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {8 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+4 \frac {\left (2 i b \sqrt {f} \sqrt {h} p\right ) \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 )}{g^{3/2}}\\ &=\frac {2 a \sqrt {h x}}{g}-\frac {8 b p \sqrt {h x}}{g}-\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 \sqrt {2} b \sqrt [4]{d} \sqrt {h} p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {2 b \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{g}-\frac {2 \sqrt {f} \sqrt {h} \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 )}{g^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} g}-\frac {8 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {2 b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} 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 )}{g^{3/2}}+\frac {4 i b \sqrt {f} \sqrt {h} p \text {Li}_2\left (1-\frac {2}{1-\frac {i \sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}}\right )}{g^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.91, size = 1471, normalized size = 0.88 \begin {gather*} \frac {\sqrt {h x} \left (2 a \sqrt {g} \sqrt {x}-\frac {b \sqrt {g} p \left (8 \sqrt [4]{e} \sqrt {x}+2 \sqrt {2} \sqrt [4]{d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \sqrt {2} \sqrt [4]{d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\sqrt {2} \sqrt [4]{d} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\sqrt {2} \sqrt [4]{d} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{\sqrt [4]{e}}+2 b \sqrt {g} \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\sqrt {-f} \log \left (\sqrt {-f}-\sqrt {g} \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-\sqrt {-f} \log \left (\sqrt {-f}+\sqrt {g} \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-b \sqrt {-f} p \left (\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 )+\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 )+\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 )+\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 )+\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 )+\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 )+\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 )+\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 )+b \sqrt {-f} p \left (\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 )+\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 )+\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 )+\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 )+\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 )+\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 )+\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 )+\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 )\right )}{g^{3/2} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {h x}\, \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{g x +f}\, dx\]
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 \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 {\sqrt {h\,x}\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{f+g\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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