Optimal. Leaf size=668 \[ -\frac {a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {\left (a^2 f \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (\sqrt {e^2-4 d f}+e\right )+c^2 d^3 \left (e-\sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (a^2 f \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (e-\sqrt {e^2-4 d f}\right )+c^2 d^3 \left (\sqrt {e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {a \sqrt {a+c x^2} \left (e^2-d f\right )}{d^3}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\sqrt {a+c x^2} \left (2 \left (a \left (e^2-d f\right )+c d^2\right )-c d e x\right )}{2 d^3}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {3 c \sqrt {a+c x^2}}{2 d}-\frac {3 \sqrt {a} c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 d} \]
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Rubi [A] time = 3.46, antiderivative size = 668, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 15, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6728, 266, 47, 50, 63, 208, 277, 195, 217, 206, 1020, 1068, 1080, 1034, 725} \begin {gather*} \frac {\left (a^2 f \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (\sqrt {e^2-4 d f}+e\right )+c^2 d^3 \left (e-\sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\left (a^2 f \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (e-\sqrt {e^2-4 d f}\right )+c^2 d^3 \left (\sqrt {e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {a \sqrt {a+c x^2} \left (e^2-d f\right )}{d^3}-\frac {\sqrt {a+c x^2} \left (2 \left (a \left (e^2-d f\right )+c d^2\right )-c d e x\right )}{2 d^3}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {3 c \sqrt {a+c x^2}}{2 d}-\frac {3 \sqrt {a} c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 195
Rule 206
Rule 208
Rule 217
Rule 266
Rule 277
Rule 725
Rule 1020
Rule 1034
Rule 1068
Rule 1080
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^{3/2}}{x^3 \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac {\left (a+c x^2\right )^{3/2}}{d x^3}-\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x^2}+\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{3/2}}{d^3 x}+\frac {\left (-e \left (e^2-2 d f\right )-f \left (e^2-d f\right ) x\right ) \left (a+c x^2\right )^{3/2}}{d^3 \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (-e \left (e^2-2 d f\right )-f \left (e^2-d f\right ) x\right ) \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx}{d^3}+\frac {\int \frac {\left (a+c x^2\right )^{3/2}}{x^3} \, dx}{d}-\frac {e \int \frac {\left (a+c x^2\right )^{3/2}}{x^2} \, dx}{d^2}+\frac {\left (e^2-d f\right ) \int \frac {\left (a+c x^2\right )^{3/2}}{x} \, dx}{d^3}\\ &=-\frac {\left (e^2-d f\right ) \left (a+c x^2\right )^{3/2}}{3 d^3}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}+\frac {\operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^2} \, dx,x,x^2\right )}{2 d}-\frac {(3 c e) \int \sqrt {a+c x^2} \, dx}{d^2}+\frac {\int \frac {\sqrt {a+c x^2} \left (-3 a e f \left (e^2-2 d f\right )+3 f (c d-a f) \left (e^2-d f\right ) x+3 c d e f^2 x^2\right )}{d+e x+f x^2} \, dx}{3 d^3 f}+\frac {\left (e^2-d f\right ) \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x} \, dx,x,x^2\right )}{2 d^3}\\ &=-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\left (2 \left (c d^2+a \left (e^2-d f\right )\right )-c d e x\right ) \sqrt {a+c x^2}}{2 d^3}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{4 d}-\frac {(3 a c e) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 d^2}-\frac {\int \frac {3 a c e f^3 \left (c d^2+2 a \left (e^2-2 d f\right )\right )-3 c f^3 \left (2 c^2 d^3+a c d \left (3 e^2-4 d f\right )-2 a^2 f \left (e^2-d f\right )\right ) x-9 a c^2 d e f^4 x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c d^3 f^3}+\frac {\left (a \left (e^2-d f\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^3}\\ &=\frac {3 c \sqrt {a+c x^2}}{2 d}+\frac {a \left (e^2-d f\right ) \sqrt {a+c x^2}}{d^3}-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\left (2 \left (c d^2+a \left (e^2-d f\right )\right )-c d e x\right ) \sqrt {a+c x^2}}{2 d^3}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}+\frac {(3 a c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 d}+\frac {(3 a c e) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 d^2}-\frac {(3 a c e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 d^2}-\frac {\int \frac {9 a c^2 d^2 e f^4+3 a c e f^4 \left (c d^2+2 a \left (e^2-2 d f\right )\right )+\left (9 a c^2 d e^2 f^4-3 c f^4 \left (2 c^2 d^3+a c d \left (3 e^2-4 d f\right )-2 a^2 f \left (e^2-d f\right )\right )\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{6 c d^3 f^4}+\frac {\left (a^2 \left (e^2-d f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^3}\\ &=\frac {3 c \sqrt {a+c x^2}}{2 d}+\frac {a \left (e^2-d f\right ) \sqrt {a+c x^2}}{d^3}-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\left (2 \left (c d^2+a \left (e^2-d f\right )\right )-c d e x\right ) \sqrt {a+c x^2}}{2 d^3}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac {3 a \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d^2}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 d}+\frac {(3 a c e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 d^2}+\frac {\left (a^2 \left (e^2-d f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^3}-\frac {\left (c^2 d^3 \left (e-\sqrt {e^2-4 d f}\right )+2 a c d^2 f \left (e+\sqrt {e^2-4 d f}\right )+a^2 f \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^3 \sqrt {e^2-4 d f}}+\frac {\left (2 a c d^2 f \left (e-\sqrt {e^2-4 d f}\right )+c^2 d^3 \left (e+\sqrt {e^2-4 d f}\right )+a^2 f \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{d^3 \sqrt {e^2-4 d f}}\\ &=\frac {3 c \sqrt {a+c x^2}}{2 d}+\frac {a \left (e^2-d f\right ) \sqrt {a+c x^2}}{d^3}-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\left (2 \left (c d^2+a \left (e^2-d f\right )\right )-c d e x\right ) \sqrt {a+c x^2}}{2 d^3}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac {3 \sqrt {a} c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 d}-\frac {a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}+\frac {\left (c^2 d^3 \left (e-\sqrt {e^2-4 d f}\right )+2 a c d^2 f \left (e+\sqrt {e^2-4 d f}\right )+a^2 f \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^3 \sqrt {e^2-4 d f}}-\frac {\left (2 a c d^2 f \left (e-\sqrt {e^2-4 d f}\right )+c^2 d^3 \left (e+\sqrt {e^2-4 d f}\right )+a^2 f \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{d^3 \sqrt {e^2-4 d f}}\\ &=\frac {3 c \sqrt {a+c x^2}}{2 d}+\frac {a \left (e^2-d f\right ) \sqrt {a+c x^2}}{d^3}-\frac {3 c e x \sqrt {a+c x^2}}{2 d^2}-\frac {\left (2 \left (c d^2+a \left (e^2-d f\right )\right )-c d e x\right ) \sqrt {a+c x^2}}{2 d^3}-\frac {\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{d^2 x}+\frac {\left (c^2 d^3 \left (e-\sqrt {e^2-4 d f}\right )+2 a c d^2 f \left (e+\sqrt {e^2-4 d f}\right )+a^2 f \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}-\frac {\left (2 a c d^2 f \left (e-\sqrt {e^2-4 d f}\right )+c^2 d^3 \left (e+\sqrt {e^2-4 d f}\right )+a^2 f \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^3 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}-\frac {3 \sqrt {a} c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 d}-\frac {a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^3}\\ \end {align*}
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Mathematica [C] time = 3.22, size = 904, normalized size = 1.35 \begin {gather*} \frac {\frac {6 c d^2 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {c x^2}{a}+1\right ) \left (c x^2+a\right )^{5/2}}{a^2}-5 \left (e^2-\frac {\left (e^2-3 d f\right ) e}{\sqrt {e^2-4 d f}}-d f\right ) \left (c x^2+a\right )^{3/2}-5 \left (e^2+\frac {\left (e^2-3 d f\right ) e}{\sqrt {e^2-4 d f}}-d f\right ) \left (c x^2+a\right )^{3/2}+\frac {30 a d e \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {c x^2}{a}\right ) \sqrt {c x^2+a}}{x \sqrt {\frac {c x^2}{a}+1}}+\frac {15 \left (-e^2-\frac {\left (e^2-3 d f\right ) e}{\sqrt {e^2-4 d f}}+d f\right ) \left (\frac {2 \sqrt {c} \left (\sqrt {e^2-4 d f}-e\right ) \sqrt {c x^2+a} \left (\sqrt {c} \sqrt {\frac {c x^2}{a}+1} x+\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )\right )}{\sqrt {\frac {c x^2}{a}+1}}+\frac {2 \left (2 a f^2+c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )\right ) \left (2 \sqrt {c x^2+a} f+\sqrt {c} \left (\sqrt {e^2-4 d f}-e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+a}}\right )-\sqrt {2 c e^2-2 c \sqrt {e^2-4 d f} e+4 a f^2-4 c d f} \tanh ^{-1}\left (\frac {2 a f+c \left (\sqrt {e^2-4 d f}-e\right ) x}{\sqrt {4 a f^2-2 c \left (-e^2+\sqrt {e^2-4 d f} e+2 d f\right )} \sqrt {c x^2+a}}\right )\right )}{f^2}\right )}{8 f}-\frac {15 \left (-e^2+\frac {\left (e^2-3 d f\right ) e}{\sqrt {e^2-4 d f}}+d f\right ) \left (\frac {2 \sqrt {c} \left (e+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+a} \left (\sqrt {c} \sqrt {\frac {c x^2}{a}+1} x+\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )\right )}{\sqrt {\frac {c x^2}{a}+1}}+\frac {2 \left (2 a f^2+c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )\right ) \left (-2 \sqrt {c x^2+a} f+\sqrt {c} \left (e+\sqrt {e^2-4 d f}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+a}}\right )+\sqrt {4 a f^2+2 c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )} \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {4 a f^2+2 c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )} \sqrt {c x^2+a}}\right )\right )}{f^2}\right )}{8 f}+10 \left (e^2-d f\right ) \left (\sqrt {c x^2+a} \left (c x^2+4 a\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c x^2+a}}{\sqrt {a}}\right )\right )}{30 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.95, size = 624, normalized size = 0.93 \begin {gather*} -\frac {\text {RootSum}\left [\text {$\#$1}^4 f-2 \text {$\#$1}^3 \sqrt {c} e-2 \text {$\#$1}^2 a f+4 \text {$\#$1}^2 c d+2 \text {$\#$1} a \sqrt {c} e+a^2 f\&,\frac {-\text {$\#$1}^2 a^2 d f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+\text {$\#$1}^2 a^2 e^2 f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-\text {$\#$1}^2 c^2 d^3 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+2 \text {$\#$1}^2 a c d^2 f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a^3 d f^2 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-a^3 e^2 f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-2 a^2 c d^2 f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+4 \text {$\#$1} a^2 \sqrt {c} d e f \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1} a^2 \sqrt {c} e^3 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )-4 \text {$\#$1} a c^{3/2} d^2 e \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )+a c^2 d^3 \log \left (-\text {$\#$1}+\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 \text {$\#$1}^3 f-3 \text {$\#$1}^2 \sqrt {c} e-2 \text {$\#$1} a f+4 \text {$\#$1} c d+a \sqrt {c} e}\&\right ]}{d^3}+\frac {2 a^{3/2} \left (d f-e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{d^3}-\frac {a \sqrt {a+c x^2} (d-2 e x)}{2 d^2 x^2}+\frac {3 \sqrt {a} c \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 10298, normalized size = 15.42 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e x + d\right )} x^{3}}\,{d x} \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 (c\,x^2+a\right )}^{3/2}}{x^3\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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