Integrand size = 31, antiderivative size = 1002 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{5/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{5/4} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}} \]
[Out]
Time = 0.86 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {2517, 2521, 2498, 327, 217, 1179, 642, 1176, 631, 210, 2505, 303, 308} \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac {2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) (h x)^{5/2}}{5 h^3}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}+\frac {4 f g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) (h x)^{3/2}}{3 h^2}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}+\frac {2 b f^2 \log \left (c \left (e x^2+d\right )^p\right ) \sqrt {h x}}{h}+\frac {2 a f^2 \sqrt {h x}}{h}-\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b d^{5/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b d^{5/4} g^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{5 e^{5/4} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}+\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}} \]
[In]
[Out]
Rule 210
Rule 217
Rule 303
Rule 308
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2498
Rule 2505
Rule 2517
Rule 2521
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \left (f+\frac {g x^2}{h}\right )^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {2 \text {Subst}\left (\int \left (f^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )+\frac {2 f g x^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h}+\frac {g^2 x^4 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h^2}\right ) \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {\left (2 g^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {(4 f g) \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right ) \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (2 f^2\right ) \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 )}{h} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}+\frac {\left (2 b f^2\right ) \text {Subst}\left (\int \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right ) \, dx,x,\sqrt {h x}\right )}{h}-\frac {\left (8 b e g^2 p\right ) \text {Subst}\left (\int \frac {x^8}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{5 h^5}-\frac {(16 b e f g p) \text {Subst}\left (\int \frac {x^6}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^4} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac {\left (8 b e g^2 p\right ) \text {Subst}\left (\int \left (-\frac {d h^4}{e^2}+\frac {h^2 x^4}{e}+\frac {d^2 h^4}{e^2 \left (d+\frac {e x^4}{h^2}\right )}\right ) \, dx,x,\sqrt {h x}\right )}{5 h^5}-\frac {\left (8 b e f^2 p\right ) \text {Subst}\left (\int \frac {x^4}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {(16 b d f g p) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^2} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac {(8 b d f g p) \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 )}{3 \sqrt {e} h^2}+\frac {(8 b d f g p) \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 )}{3 \sqrt {e} h^2}+\frac {\left (8 b d f^2 p\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h}-\frac {\left (8 b d^2 g^2 p\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{5 e h} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}+\frac {(4 b d f g p) \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 )}{3 e}+\frac {(4 b d f g p) \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 )}{3 e}+\frac {\left (4 b \sqrt {d} f^2 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 )}{h^2}+\frac {\left (4 b \sqrt {d} f^2 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 )}{h^2}-\frac {\left (4 b d^{3/2} g^2 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 )}{5 e h^2}-\frac {\left (4 b d^{3/2} g^2 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 )}{5 e h^2}+\frac {\left (2 \sqrt {2} b d^{3/4} f g 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 )}{3 e^{3/4} \sqrt {h}}+\frac {\left (2 \sqrt {2} b d^{3/4} f g 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 )}{3 e^{3/4} \sqrt {h}} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\left (2 b \sqrt {d} f^2 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}}+\frac {\left (2 b \sqrt {d} f^2 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}}-\frac {\left (2 b d^{3/2} g^2 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 )}{5 e^{3/2}}-\frac {\left (2 b d^{3/2} g^2 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 )}{5 e^{3/2}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}+\frac {\left (4 \sqrt {2} b d^{3/4} f g 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 )}{3 e^{3/4} \sqrt {h}}-\frac {\left (4 \sqrt {2} b d^{3/4} f g 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 )}{3 e^{3/4} \sqrt {h}}+\frac {\left (\sqrt {2} b d^{5/4} g^2 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 )}{5 e^{5/4} \sqrt {h}}+\frac {\left (\sqrt {2} b d^{5/4} g^2 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 )}{5 e^{5/4} \sqrt {h}} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac {4 \sqrt {2} b d^{3/4} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {\left (2 \sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {\left (2 \sqrt {2} b d^{5/4} g^2 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 )}{5 e^{5/4} \sqrt {h}}+\frac {\left (2 \sqrt {2} b d^{5/4} g^2 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 )}{5 e^{5/4} \sqrt {h}} \\ & = \frac {2 a f^2 \sqrt {h x}}{h}-\frac {8 b f^2 p \sqrt {h x}}{h}+\frac {8 b d g^2 p \sqrt {h x}}{5 e h}-\frac {16 b f g p (h x)^{3/2}}{9 h^2}-\frac {8 b g^2 p (h x)^{5/2}}{25 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {4 \sqrt {2} b d^{3/4} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{5/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {4 \sqrt {2} b d^{3/4} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{5/4} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 e^{5/4} \sqrt {h}}+\frac {2 b f^2 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {4 f g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 g^2 (h x)^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^3}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 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} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{5/4} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 e^{5/4} \sqrt {h}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 628, normalized size of antiderivative = 0.63 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\frac {2 \sqrt {x} \left (a f^2 \sqrt {x}-4 b f^2 p \sqrt {x}+\frac {2}{3} a f g x^{3/2}-\frac {8}{9} b f g p x^{3/2}-\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}+\frac {\sqrt {2} b \sqrt [4]{d} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{e}}-\frac {4 b (-d)^{3/4} f g p \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}+\frac {4 b (-d)^{3/4} f g p \text {arctanh}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )}{3 e^{3/4}}-\frac {b \sqrt [4]{d} f^2 p \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}+\frac {b \sqrt [4]{d} f^2 p \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt {2} \sqrt [4]{e}}-\frac {b g^2 p \left (-40 d \sqrt [4]{e} \sqrt {x}+8 e^{5/4} x^{5/2}-10 \sqrt {2} d^{5/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+10 \sqrt {2} d^{5/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-5 \sqrt {2} d^{5/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )+5 \sqrt {2} d^{5/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{50 e^{5/4}}+b f^2 \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} b f g x^{3/2} \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^{5/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{\sqrt {h x}} \]
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\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\sqrt {h x}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 2178 vs. \(2 (708) = 1416\).
Time = 0.43 (sec) , antiderivative size = 2178, normalized size of antiderivative = 2.17 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.32 (sec) , antiderivative size = 1165, normalized size of antiderivative = 1.16 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \]
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Time = 0.39 (sec) , antiderivative size = 885, normalized size of antiderivative = 0.88 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{\sqrt {h\,x}} \,d x \]
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