Optimal. Leaf size=949 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.26132, antiderivative size = 949, normalized size of antiderivative = 1., 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 = {2467, 2476, 2448, 321, 211, 1165, 628, 1162, 617, 204, 2455, 297} \[ -\frac{2 \sqrt{2} b \sqrt [4]{e} p \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \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 d^{3/4} g^2 p \tan ^{-1}\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}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2467
Rule 2476
Rule 2448
Rule 321
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 2455
Rule 297
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 \operatorname{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 \operatorname{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 ) \operatorname{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) \operatorname{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 ) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.829528, size = 436, normalized size = 0.46 \[ \frac{2 x^{3/2} \left (-\frac{f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt{x}}+\frac{1}{3} g^2 x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+2 a f g \sqrt{x}+2 b f g \sqrt{x} \log \left (c \left (d+e x^2\right )^p\right )-\frac{2 b g^2 p \left (2 \sqrt [4]{-d} e^{3/4} x^{3/2}-3 d \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )+3 d \tanh ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )\right )}{9 \sqrt [4]{-d} e^{3/4}}+\frac{2 b \sqrt [4]{e} f^2 p \left (\tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac{d \sqrt [4]{e} \sqrt{x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac{b f g p \left (\sqrt{2} \sqrt [4]{d} \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )-\sqrt{2} \sqrt [4]{d} \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )+2 \sqrt{2} \sqrt [4]{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )-2 \sqrt{2} \sqrt [4]{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )+8 \sqrt [4]{e} \sqrt{x}\right )}{\sqrt [4]{e}}\right )}{(h x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.332, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) ^{2} \left ( a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) \left ( hx \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42991, size = 4340, normalized size = 4.57 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.7384, size = 869, normalized size = 0.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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