Optimal. Leaf size=631 \[ \frac {2 a f \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}-\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 e^{3/4} \sqrt {h}} \]
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Rubi [A]
time = 0.59, antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2517, 2521,
2498, 327, 217, 1179, 642, 1176, 631, 210, 2505, 303} \begin {gather*} \frac {2 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 a f \sqrt {h x}}{h}-\frac {2 \sqrt {2} b d^{3/4} g p \text {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^{3/4} g p \text {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 {2 \sqrt {2} b \sqrt [4]{d} f p \text {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 \sqrt [4]{d} f p \text {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 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {\sqrt {2} b d^{3/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 303
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2498
Rule 2505
Rule 2517
Rule 2521
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {h x}} \, dx &=\frac {2 \text {Subst}\left (\int \left (f+\frac {g x^2}{h}\right ) \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 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )+\frac {g x^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h}\right ) \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {(2 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 {(2 f) \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 \sqrt {h x}}{h}+\frac {2 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 b f) \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 {(8 b e 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 \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}-\frac {(8 b e f p) \text {Subst}\left (\int \frac {x^4}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {(8 b d 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 \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}-\frac {(4 b d 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 {(4 b d 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 p) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h}\\ &=\frac {2 a f \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 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 b d 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 {(2 b d 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 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 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 (\sqrt {2} b d^{3/4} 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 (\sqrt {2} b d^{3/4} 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 \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {\sqrt {2} b d^{3/4} 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^{3/4} 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 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 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 (\sqrt {2} b \sqrt [4]{d} f 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 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 (2 \sqrt {2} b d^{3/4} 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 (2 \sqrt {2} b d^{3/4} 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 {2 a f \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}-\frac {2 \sqrt {2} b d^{3/4} 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^{3/4} 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 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}-\frac {\sqrt {2} b \sqrt [4]{d} f 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 {\sqrt {2} b d^{3/4} 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 \sqrt [4]{d} f 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 {\sqrt {2} b d^{3/4} 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 \sqrt {2} b \sqrt [4]{d} f 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 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 {2 a f \sqrt {h x}}{h}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2}-\frac {2 \sqrt {2} b \sqrt [4]{d} f 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 {2 \sqrt {2} b d^{3/4} 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 \sqrt [4]{d} f 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 {2 \sqrt {2} b d^{3/4} 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 \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}-\frac {\sqrt {2} b \sqrt [4]{d} f 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 {\sqrt {2} b d^{3/4} 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 \sqrt [4]{d} f 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 {\sqrt {2} b d^{3/4} 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}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 344, normalized size = 0.55 \begin {gather*} \frac {2 \sqrt {x} \left (a f \sqrt {x}-\frac {2 b g 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 {b f 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 )}{2 \sqrt [4]{e}}+b f \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )\right )}{\sqrt {h x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\sqrt {h x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 538, normalized size = 0.85 \begin {gather*} \frac {2 \, b g x^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{3 \, \sqrt {h x}} + \frac {2 \, a g x^{2}}{3 \, \sqrt {h x}} + \frac {2 \, \sqrt {h x} b f \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{h} - \frac {{\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 p e}{h^{3}} + \frac {2 \, \sqrt {h x} a f}{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 p e}{9 \, h^{4}} \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. 1263 vs.
\(2 (423) = 846\).
time = 0.44, size = 1263, normalized size = 2.00 \begin {gather*} -\frac {2 \, {\left (3 \, h \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} + h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}} \log \left (-32 \, {\left (b^{3} d^{2} g^{4} p^{3} - 81 \, b^{3} f^{4} p^{3} e^{2}\right )} \sqrt {h x} + 32 \, {\left (3 \, b^{2} d f g^{2} h p^{2} e - 27 \, b^{2} f^{3} h p^{2} e^{2} - g h^{2} \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e^{2}\right )} \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} + h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}}\right ) - 3 \, h \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} + h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}} \log \left (-32 \, {\left (b^{3} d^{2} g^{4} p^{3} - 81 \, b^{3} f^{4} p^{3} e^{2}\right )} \sqrt {h x} - 32 \, {\left (3 \, b^{2} d f g^{2} h p^{2} e - 27 \, b^{2} f^{3} h p^{2} e^{2} - g h^{2} \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e^{2}\right )} \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} + h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}}\right ) + 3 \, h \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} - h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}} \log \left (-32 \, {\left (b^{3} d^{2} g^{4} p^{3} - 81 \, b^{3} f^{4} p^{3} e^{2}\right )} \sqrt {h x} + 32 \, {\left (3 \, b^{2} d f g^{2} h p^{2} e - 27 \, b^{2} f^{3} h p^{2} e^{2} + g h^{2} \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e^{2}\right )} \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} - h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}}\right ) - 3 \, h \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} - h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}} \log \left (-32 \, {\left (b^{3} d^{2} g^{4} p^{3} - 81 \, b^{3} f^{4} p^{3} e^{2}\right )} \sqrt {h x} - 32 \, {\left (3 \, b^{2} d f g^{2} h p^{2} e - 27 \, b^{2} f^{3} h p^{2} e^{2} + g h^{2} \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e^{2}\right )} \sqrt {-\frac {{\left (6 \, b^{2} d f g p^{2} - h \sqrt {-\frac {{\left (b^{4} d^{3} g^{4} p^{4} - 18 \, b^{4} d^{2} f^{2} g^{2} p^{4} e + 81 \, b^{4} d f^{4} p^{4} e^{2}\right )} e^{\left (-3\right )}}{h^{2}}} e\right )} e^{\left (-1\right )}}{h}}\right ) + {\left (36 \, b f p - 9 \, a f + {\left (4 \, b g p - 3 \, a g\right )} x - 3 \, {\left (b g p x + 3 \, b f p\right )} \log \left (x^{2} e + d\right ) - 3 \, {\left (b g x + 3 \, b f\right )} \log \left (c\right )\right )} \sqrt {h x}\right )}}{9 \, h} \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 = 5.66, size = 514, normalized size = 0.81 \begin {gather*} \frac {6 \, \sqrt {h x} b g x \log \left (c\right ) + 9 \, {\left ({\left (2 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \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 (-\frac {5}{4}\right )} + 2 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \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 (-\frac {5}{4}\right )} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {5}{4}\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 ) - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {5}{4}\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 ) - 8 \, \sqrt {h x} e^{\left (-1\right )}\right )} e + 2 \, \sqrt {h x} \log \left (x^{2} e + d\right )\right )} b f p + 6 \, \sqrt {h x} a g x + 18 \, \sqrt {h x} b f \log \left (c\right ) + \frac {{\left (6 \, \sqrt {h x} h x \log \left (x^{2} e + d\right ) - {\left (8 \, \sqrt {h x} h x e^{\left (-1\right )} - 6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} \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 (-\frac {7}{4}\right )} - 6 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} \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 (-\frac {7}{4}\right )} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} e^{\left (-\frac {7}{4}\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 ) - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} e^{\left (-\frac {7}{4}\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 )\right )} e\right )} b g p}{h} + 18 \, \sqrt {h x} a f}{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 )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{\sqrt {h\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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