Integrand size = 31, antiderivative size = 968 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{7/4} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}+\frac {4 \sqrt {2} b e^{5/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{3/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}-\frac {4 \sqrt {2} b e^{5/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{3/4} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac {\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}} \]
[Out]
Time = 0.80 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2517, 2526, 2505, 331, 217, 1179, 642, 1176, 631, 210, 303} \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\frac {2 \sqrt {2} b e^{7/4} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{7 d^{7/4} h^{9/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{7 h (h x)^{7/2}}+\frac {\sqrt {2} b e^{7/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{7 d^{7/4} h^{9/2}}-\frac {\sqrt {2} b e^{7/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{7 d^{7/4} h^{9/2}}-\frac {8 b e p f^2}{21 d h^3 (h x)^{3/2}}+\frac {4 \sqrt {2} b e^{5/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{5 d^{5/4} h^{9/2}}-\frac {4 \sqrt {2} b e^{5/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{5 d^{5/4} h^{9/2}}-\frac {4 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{5 h^2 (h x)^{5/2}}-\frac {2 \sqrt {2} b e^{5/4} 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 ) f}{5 d^{5/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} 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 ) f}{5 d^{5/4} h^{9/2}}-\frac {16 b e g p f}{5 d h^4 \sqrt {h x}}-\frac {2 \sqrt {2} b e^{3/4} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{3/4} g^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 d^{3/4} h^{9/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}+\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}} \]
[In]
[Out]
Rule 210
Rule 217
Rule 303
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2505
Rule 2517
Rule 2526
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\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 )}{x^8} \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {f^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^8}+\frac {2 f g \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h x^6}+\frac {g^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h^2 x^4}\right ) \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {\left (2 g^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^4} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {(4 f g) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^6} \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^8} \, dx,x,\sqrt {h x}\right )}{h} \\ & = -\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac {\left (8 b e g^2 p\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^5}+\frac {(16 b e f g p) \text {Subst}\left (\int \frac {1}{x^2 \left (d+\frac {e x^4}{h^2}\right )} \, dx,x,\sqrt {h x}\right )}{5 h^4}+\frac {\left (8 b e f^2 p\right ) \text {Subst}\left (\int \frac {1}{x^4 \left (d+\frac {e x^4}{h^2}\right )} \, dx,x,\sqrt {h x}\right )}{7 h^3} \\ & = -\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac {\left (16 b e^2 f g p\right ) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{5 d h^6}+\frac {\left (4 b e 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 )}{3 \sqrt {d} h^6}+\frac {\left (4 b e 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 )}{3 \sqrt {d} h^6}-\frac {\left (8 b e^2 f^2 p\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{7 d h^5} \\ & = -\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac {\left (4 b e^2 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 )}{7 d^{3/2} h^6}-\frac {\left (4 b e^2 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 )}{7 d^{3/2} h^6}+\frac {\left (8 b e^{3/2} f g 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 d h^6}-\frac {\left (8 b e^{3/2} f g 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 d h^6}-\frac {\left (\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\left (\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}+\frac {\left (2 b \sqrt {e} 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 )}{3 \sqrt {d} h^4}+\frac {\left (2 b \sqrt {e} 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 )}{3 \sqrt {d} h^4} \\ & = -\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}-\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}+\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}+\frac {\left (\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac {\left (\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac {\left (2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac {\left (2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac {\left (2 \sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\left (2 \sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\left (2 b e^{3/2} 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 )}{7 d^{3/2} h^4}-\frac {\left (2 b e^{3/2} 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 )}{7 d^{3/2} h^4}-\frac {(4 b e 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 )}{5 d h^4}-\frac {(4 b e 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 )}{5 d h^4} \\ & = -\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}-\frac {2 \sqrt {2} b e^{3/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{3/4} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac {\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\left (2 \sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac {\left (2 \sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac {\left (4 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac {\left (4 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}} \\ & = -\frac {8 b e f^2 p}{21 d h^3 (h x)^{3/2}}-\frac {16 b e f g p}{5 d h^4 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{7/4} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}+\frac {4 \sqrt {2} b e^{5/4} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{3/4} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{7/4} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{7 d^{7/4} h^{9/2}}-\frac {4 \sqrt {2} b e^{5/4} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{3/4} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{9/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{7 h (h x)^{7/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h^2 (h x)^{5/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3 (h x)^{3/2}}+\frac {\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}-\frac {2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}-\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}}-\frac {\sqrt {2} b e^{7/4} 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 )}{7 d^{7/4} h^{9/2}}+\frac {2 \sqrt {2} b e^{5/4} 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 )}{5 d^{5/4} h^{9/2}}+\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{9/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.30 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\frac {x \left (-40 b e f^2 p x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},-\frac {e x^2}{d}\right )-336 b e f g p x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )-35 \sqrt {2} b \sqrt [4]{d} e^{3/4} g^2 p x^{7/2} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )-30 d f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-84 d f g x \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )-70 d g^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{105 d (h x)^{9/2}} \]
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\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {9}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 2283 vs. \(2 (672) = 1344\).
Time = 0.43 (sec) , antiderivative size = 2283, normalized size of antiderivative = 2.36 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 1110, normalized size of antiderivative = 1.15 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\text {Too large to display} \]
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Time = 0.44 (sec) , antiderivative size = 688, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=-\frac {\frac {2 \, {\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p + 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e h} + \frac {2 \, {\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p + 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e h} + \frac {{\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p - 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e h} - \frac {{\left (15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b e^{2} f^{2} h p - 35 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e g^{2} h p - 42 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b f g p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e h} + \frac {2 \, {\left (35 \, b g^{2} h^{4} p x^{2} + 42 \, b f g h^{4} p x + 15 \, b f^{2} h^{4} p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{3} x^{3}} + \frac {2 \, {\left (168 \, b e f g h^{4} p x^{3} - 35 \, b d g^{2} h^{4} p x^{2} \log \left (h^{2}\right ) + 20 \, b e f^{2} h^{4} p x^{2} - 42 \, b d f g h^{4} p x \log \left (h^{2}\right ) + 35 \, b d g^{2} h^{4} x^{2} \log \left (c\right ) + 35 \, a d g^{2} h^{4} x^{2} - 15 \, b d f^{2} h^{4} p \log \left (h^{2}\right ) + 42 \, b d f g h^{4} x \log \left (c\right ) + 42 \, a d f g h^{4} x + 15 \, b d f^{2} h^{4} \log \left (c\right ) + 15 \, a d f^{2} h^{4}\right )}}{\sqrt {h x} d h^{3} x^{3}}}{105 \, h^{5}} \]
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Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{9/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{9/2}} \,d x \]
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