Optimal. Leaf size=949 \[ \frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {4 \sqrt {2} b \sqrt [4]{d} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {4 \sqrt {2} b \sqrt [4]{d} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}+\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}} \]
[Out]
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Rubi [A]
time = 0.81, antiderivative size = 949, normalized size of antiderivative = 1.00, number of steps
used = 36, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2517, 2526,
2498, 327, 217, 1179, 642, 1176, 631, 210, 2505, 303} \begin {gather*} -\frac {2 \sqrt {2} b \sqrt [4]{e} p \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{\sqrt [4]{d} h^{3/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{h \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} 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}{\sqrt [4]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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}{\sqrt [4]{d} h^{3/2}}-\frac {4 \sqrt {2} b \sqrt [4]{d} g p \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {4 \sqrt {2} b \sqrt [4]{d} g p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{\sqrt [4]{e} h^{3/2}}+\frac {4 b g \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right ) f}{h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} 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}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} 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}{\sqrt [4]{e} h^{3/2}}-\frac {16 b g p \sqrt {h x} f}{h^2}+\frac {4 a g \sqrt {h x} f}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b d^{3/4} g^2 p \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b d^{3/4} g^2 p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} h^{3/2}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 217
Rule 303
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2498
Rule 2505
Rule 2517
Rule 2526
Rubi steps
\begin {align*} \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{3/2}} \, dx &=\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^2} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {2 f g \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h}+\frac {f^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^2}+\frac {g^2 x^2 \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^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^3}+\frac {(4 f g) \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^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^2} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {(4 b f g) \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^2}-\frac {\left (8 b e g^2 p\right ) \text {Subst}\left (\int \frac {x^6}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^5}+\frac {\left (8 b e f^2 p\right ) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}-\frac {(16 b e f g p) \text {Subst}\left (\int \frac {x^4}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^4}-\frac {\left (4 b \sqrt {e} 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^3}+\frac {\left (4 b \sqrt {e} 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^3}+\frac {\left (8 b d g^2 p\right ) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^3}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}-\frac {\left (4 b d 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 {e} h^3}+\frac {\left (4 b d 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 {e} h^3}+\frac {(16 b d f g p) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {\left (\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {\left (2 b 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 )}{h}+\frac {\left (2 b 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 )}{h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {\left (8 b \sqrt {d} 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 )}{h^3}+\frac {\left (8 b \sqrt {d} 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 )}{h^3}+\frac {\left (2 \sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {\left (\sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {\left (\sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {\left (2 b d 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 e h}+\frac {\left (2 b d 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 e h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}-\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}+\frac {\left (2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\left (2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {\left (4 b \sqrt {d} f g 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} h}+\frac {\left (4 b \sqrt {d} f g 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} h}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}+\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {\left (4 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}-\frac {\left (4 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}\\ &=\frac {4 a f g \sqrt {h x}}{h^2}-\frac {16 b f g p \sqrt {h x}}{h^2}-\frac {8 b g^2 p (h x)^{3/2}}{9 h^3}-\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}-\frac {4 \sqrt {2} b \sqrt [4]{d} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}-\frac {2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{3/2}}+\frac {4 \sqrt {2} b \sqrt [4]{d} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} h^{3/2}}+\frac {2 \sqrt {2} b d^{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 e^{3/4} h^{3/2}}+\frac {4 b f g \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h^2}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h \sqrt {h x}}+\frac {2 g^2 (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^3}+\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}-\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}+\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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]{d} h^{3/2}}+\frac {2 \sqrt {2} b \sqrt [4]{d} 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 )}{\sqrt [4]{e} h^{3/2}}-\frac {\sqrt {2} b d^{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 e^{3/4} h^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 436, normalized size = 0.46 \begin {gather*} \frac {2 x^{3/2} \left (2 a f g \sqrt {x}-\frac {2 b g^2 p \left (2 \sqrt [4]{-d} e^{3/4} x^{3/2}-3 d \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+3 d \tanh ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )\right )}{9 \sqrt [4]{-d} e^{3/4}}+\frac {2 b \sqrt [4]{e} f^2 p \left (\tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b f 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 f g \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}+\frac {1}{3} g^2 x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{(h x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 793, normalized size = 0.84 \begin {gather*} \frac {2 \, b g^{2} x^{3} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{3 \, \left (h x\right )^{\frac {3}{2}}} - \frac {{\left (\frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}}} - \frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}} - \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}}\right )} b f^{2} p e}{h} + \frac {2 \, a g^{2} x^{3}}{3 \, \left (h x\right )^{\frac {3}{2}}} + \frac {4 \, b f g x^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{\left (h x\right )^{\frac {3}{2}}} + \frac {4 \, a f g x^{2}}{\left (h x\right )^{\frac {3}{2}}} - \frac {2 \, b f^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{\sqrt {h x} h} - \frac {2 \, {\left (8 \, \sqrt {h x} h^{2} e^{\left (-1\right )} - {\left (\frac {\sqrt {2} h^{4} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} h^{4} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} h^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}} + \frac {2 \, \sqrt {2} h^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}}\right )} d e^{\left (-1\right )}\right )} b f g p e}{h^{4}} - \frac {2 \, a f^{2}}{\sqrt {h x} h} - \frac {{\left (3 \, {\left (\frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}}} - \frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}} - \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}}\right )} d h^{4} e^{\left (-1\right )} + 8 \, \left (h x\right )^{\frac {3}{2}} h^{2} e^{\left (-1\right )}\right )} b g^{2} p e}{9 \, h^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2334 vs.
\(2 (646) = 1292\).
time = 5.42, size = 2334, normalized size = 2.46 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.28, size = 649, normalized size = 0.68 \begin {gather*} \frac {\frac {6 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} + \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{4}\right )} + 2 \, \sqrt {h x}\right )} e^{\frac {1}{4}}}{2 \, \left (d h^{2}\right )^{\frac {1}{4}}}\right ) e^{\left (-3\right )}}{d h^{2}} + \frac {6 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} + \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{4}\right )} - 2 \, \sqrt {h x}\right )} e^{\frac {1}{4}}}{2 \, \left (d h^{2}\right )^{\frac {1}{4}}}\right ) e^{\left (-3\right )}}{d h^{2}} + \frac {3 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} - \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} e^{\left (-3\right )} \log \left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\left (-\frac {1}{4}\right )} + h x + \sqrt {d h^{2}} e^{\left (-\frac {1}{2}\right )}\right )}{d h^{2}} - \frac {3 \, {\left (6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d f g h p e^{\frac {11}{4}} - \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b d g^{2} p e^{\frac {9}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f^{2} p e^{\frac {13}{4}}\right )} e^{\left (-3\right )} \log \left (-\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\left (-\frac {1}{4}\right )} + h x + \sqrt {d h^{2}} e^{\left (-\frac {1}{2}\right )}\right )}{d h^{2}} + \frac {2 \, {\left (3 \, b g^{2} h^{2} p x^{2} \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b g^{2} h^{2} p x^{2} \log \left (h^{2}\right ) - 4 \, b g^{2} h^{2} p x^{2} + 18 \, b f g h^{2} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 18 \, b f g h^{2} p x \log \left (h^{2}\right ) + 3 \, b g^{2} h^{2} x^{2} \log \left (c\right ) - 72 \, b f g h^{2} p x + 3 \, a g^{2} h^{2} x^{2} - 9 \, b f^{2} h^{2} p \log \left (h^{2} x^{2} e + d h^{2}\right ) + 9 \, b f^{2} h^{2} p \log \left (h^{2}\right ) + 18 \, b f g h^{2} x \log \left (c\right ) + 18 \, a f g h^{2} x - 9 \, b f^{2} h^{2} \log \left (c\right ) - 9 \, a f^{2} h^{2}\right )}}{\sqrt {h x} h^{2}}}{9 \, h} \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 {{\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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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