Optimal. Leaf size=631 \[ \frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 a f \sqrt {h x}}{h}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {\sqrt {2} b d^{3/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} g p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b \sqrt [4]{d} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2} \]
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Rubi [A] time = 0.90, 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 = {2467, 2471, 2448, 321, 211, 1165, 628, 1162, 617, 204, 2455, 297} \[ \frac {2 g (h x)^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2}+\frac {2 a f \sqrt {h x}}{h}+\frac {2 b f \sqrt {h x} \log \left (c \left (d+e x^2\right )^p\right )}{h}+\frac {\sqrt {2} b d^{3/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b d^{3/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 e^{3/4} \sqrt {h}}-\frac {2 \sqrt {2} b d^{3/4} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 e^{3/4} \sqrt {h}}+\frac {2 \sqrt {2} b d^{3/4} g p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 e^{3/4} \sqrt {h}}-\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {\sqrt {2} b \sqrt [4]{d} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {2 \sqrt {2} b \sqrt [4]{d} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} \sqrt {h}}+\frac {2 \sqrt {2} b \sqrt [4]{d} f p \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} \sqrt {h}}-\frac {8 b f p \sqrt {h x}}{h}-\frac {8 b g p (h x)^{3/2}}{9 h^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 297
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2448
Rule 2455
Rule 2467
Rule 2471
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 \operatorname {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 \operatorname {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) \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^2}+\frac {(2 f) \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}\\ &=\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) \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}-\frac {(8 b e g p) \operatorname {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) \operatorname {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) \operatorname {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) \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^2}+\frac {(4 b d g p) \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^2}+\frac {(8 b d f p) \operatorname {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) \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}+\frac {(2 b d g p) \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}+\frac {\left (4 b \sqrt {d} f 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^2}+\frac {\left (4 b \sqrt {d} f 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^2}+\frac {\left (\sqrt {2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt {h}}+\frac {\left (\sqrt {2} b d^{3/4} 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 )}{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 ) \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}}+\frac {\left (2 b \sqrt {d} f 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}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} f 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} \sqrt {h}}-\frac {\left (\sqrt {2} b \sqrt [4]{d} f 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} \sqrt {h}}+\frac {\left (2 \sqrt {2} b d^{3/4} 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 )}{3 e^{3/4} \sqrt {h}}-\frac {\left (2 \sqrt {2} b d^{3/4} 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 )}{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 ) \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} \sqrt {h}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{d} f 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} \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 = 0.47, size = 344, normalized size = 0.55 \[ \frac {2 \sqrt {x} \left (\frac {1}{3} g x^{3/2} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )+a f \sqrt {x}+b f \sqrt {x} \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 b g p \left (2 \sqrt [4]{-d} e^{3/4} x^{3/2}-3 d \tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+3 d \tanh ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )\right )}{9 \sqrt [4]{-d} e^{3/4}}-\frac {b f p \left (\sqrt {2} \sqrt [4]{d} \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {d}+\sqrt {e} x\right )-\sqrt {2} \sqrt [4]{d} \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {d}+\sqrt {e} x\right )+2 \sqrt {2} \sqrt [4]{d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \sqrt {2} \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}+1\right )+8 \sqrt [4]{e} \sqrt {x}\right )}{2 \sqrt [4]{e}}\right )}{\sqrt {h x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.14, size = 1196, normalized size = 1.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 514, normalized size = 0.81 \[ \frac {6 \, \sqrt {h x} b g x \log \relax (c) + 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 \relax (c) + \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right ) \left (b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )+a \right )}{\sqrt {h x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 572, normalized size = 0.91 \[ \frac {2 \, b g x^{2} \log \left ({\left (e x^{2} + 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 (e x^{2} + d\right )}^{p} c\right )}{h} - \frac {{\left (\frac {8 \, \sqrt {h x} h^{2}}{e} - \frac {{\left (\frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x + \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}} e^{\frac {1}{4}}} - \frac {\sqrt {2} h^{4} \log \left (\sqrt {e} h x - \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}} e^{\frac {1}{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} \sqrt {e}\right )}}{2 \, \sqrt {\sqrt {d} \sqrt {e} h}}\right )}{\sqrt {\sqrt {d} \sqrt {e} 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} \sqrt {e}\right )}}{2 \, \sqrt {\sqrt {d} \sqrt {e} h}}\right )}{\sqrt {\sqrt {d} \sqrt {e} h} \sqrt {d}}\right )} d}{e}\right )} b e f p}{h^{3}} + \frac {2 \, \sqrt {h x} a f}{h} + \frac {{\left (\frac {3 \, d h^{4} {\left (\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} \sqrt {e}\right )}}{2 \, \sqrt {\sqrt {d} \sqrt {e} h}}\right )}{\sqrt {\sqrt {d} \sqrt {e} h} \sqrt {e}} + \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} \sqrt {e}\right )}}{2 \, \sqrt {\sqrt {d} \sqrt {e} h}}\right )}{\sqrt {\sqrt {d} \sqrt {e} h} \sqrt {e}} - \frac {\sqrt {2} \log \left (\sqrt {e} h x + \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}} e^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {e} h x - \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}} e^{\frac {3}{4}}}\right )}}{e} - \frac {8 \, \left (h x\right )^{\frac {3}{2}} h^{2}}{e}\right )} b e g p}{9 \, h^{4}} \]
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 \[ \int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{\sqrt {h\,x}} \,d x \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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