Optimal. Leaf size=701 \[ \frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {2 b c \sqrt {c^2 x^2-1} \left (c^2 d-e\right ) \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}-\frac {2 b c^2 x \sqrt {1-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
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Rubi [A] time = 1.39, antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {271, 191, 5238, 12, 6742, 421, 419, 480, 583, 524, 427, 426, 424, 493} \[ \frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {2 b c \sqrt {c^2 x^2-1} \left (c^2 d-e\right ) \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {2 b c^2 x \sqrt {1-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 271
Rule 419
Rule 421
Rule 424
Rule 426
Rule 427
Rule 480
Rule 493
Rule 524
Rule 583
Rule 5238
Rule 6742
Rubi steps
\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}\\ &=-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \left (\frac {8 e^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {4 d e}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\right ) \, dx}{3 d^3 \sqrt {c^2 x^2}}\\ &=-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {1}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d \sqrt {c^2 x^2}}-\frac {(4 b c e x) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}\\ &=-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 \left (c^2 d-e\right )-c^2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {c^2 e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-c^2 d e+2 c^2 \left (c^2 d-e\right ) e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2}}+\frac {\left (4 b c^3 e^2 x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (2 b c^3 \left (c^2 d-e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2}}+\frac {\left (b c^3 \left (2 c^2 d-e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}+\frac {\left (4 b c^3 e x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {\left (4 b c^3 e x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}\\ &=\frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (2 b c^3 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (4 b c^3 e x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (b c^3 \left (2 c^2 d-e\right ) x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{9 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (4 b c^3 e x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (2 b c^3 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (4 b c^3 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 \left (2 c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (4 b c^3 e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {2 b c^2 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 \left (2 c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [C] time = 0.74, size = 292, normalized size = 0.42 \[ \frac {-3 a \left (d^2-4 d e x^2-8 e^2 x^4\right )+b c x \sqrt {1-\frac {1}{c^2 x^2}} \left (d^2 \left (2 c^2 x^2+1\right )+d e x^2 \left (2 c^2 x^2-13\right )-14 e^2 x^4\right )-3 b \sec ^{-1}(c x) \left (d^2-4 d e x^2-8 e^2 x^4\right )}{9 d^3 x^3 \sqrt {d+e x^2}}-\frac {i b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {e x^2}{d}+1} \left (2 c^2 d \left (c^2 d-7 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+\left (-2 c^4 d^2+13 c^2 d e+24 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{9 \sqrt {-c^2} d^3 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{e^{2} x^{8} + 2 \, d e x^{6} + d^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.46, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsec}\left (c x \right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ \frac {1}{3} \, a {\left (\frac {8 \, e^{2} x}{\sqrt {e x^{2} + d} d^{3}} + \frac {4 \, e}{\sqrt {e x^{2} + d} d^{2} x} - \frac {1}{\sqrt {e x^{2} + d} d x^{3}}\right )} - \frac {{\left (\frac {3 \, {\left (8 \, e^{2} x^{4} \log \relax (c) + 4 \, d e x^{2} \log \relax (c) - d^{2} \log \relax (c) + {\left (8 \, e^{2} x^{4} + 4 \, d e x^{2} - d^{2}\right )} \log \relax (x)\right )} {\left (e x^{2} + d\right )} d^{3} x^{3}}{d^{3} e x^{5} + d^{4} x^{3}} - {\left (8 \, e^{2} x^{4} + 4 \, d e x^{2} - d^{2}\right )} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{3 \, \sqrt {e x^{2} + d} d^{3} x^{3}} \]
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 \[ \int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^4\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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