Integrand size = 23, antiderivative size = 631 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b c^2 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
[Out]
Time = 1.03 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {277, 198, 197, 5346, 12, 6874, 425, 21, 438, 437, 435, 483, 597, 538, 432, 430, 482, 434} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {c^2 x^2-1}}{3 d^3 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {b c \sqrt {c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt {d+e x^2}}{d^3 \sqrt {c^2 x^2} \left (c^2 d+e\right )}-\frac {b c e \sqrt {c^2 x^2-1}}{d^2 \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rule 12
Rule 21
Rule 197
Rule 198
Rule 277
Rule 425
Rule 430
Rule 432
Rule 434
Rule 435
Rule 437
Rule 438
Rule 482
Rule 483
Rule 538
Rule 597
Rule 5346
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-3 d^2-12 d e x^2-8 e^2 x^4}{3 d^3 x^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-3 d^2-12 d e x^2-8 e^2 x^4}{x^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \left (-\frac {12 d e}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 d^2}{x^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}-\frac {8 e^2 x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}\right ) \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e 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^2 \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{d \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{d^2 \sqrt {c^2 x^2}}+\frac {\left (8 b c e^2 x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e 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+2 e-c^2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {c^2 d+c^2 e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x\right ) \int \frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}} \, dx}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(8 b c e x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}+\frac {(b c x) \int \frac {-c^2 d e-c^2 e \left (c^2 d+2 e\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d^3 \left (c^2 d+e\right ) \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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {\left (b c^3 x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d^2 \sqrt {c^2 x^2}}-\frac {\left (b c^3 \left (c^2 d+2 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {\left (8 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 \left (c^2 d+e\right ) \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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (8 b c 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^3 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {\left (b c^3 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (8 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 \left (c^2 d+e\right ) \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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 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}{d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {\left (8 b c 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^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (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 (b c^3 \left (c^2 d+2 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}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 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}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = -\frac {b c e \sqrt {-1+c^2 x^2}}{d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c e^2 x^2 \sqrt {-1+c^2 x^2}}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {b c \left (c^2 d+2 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{d x \left (d+e x^2\right )^{3/2}}-\frac {4 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 \left (d+e x^2\right )^{3/2}}-\frac {8 e x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b c^2 \left (c^2 d+2 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d^3 \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (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*}
Result contains complex when optimal does not.
Time = 7.59 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\frac {-a \left (c^2 d+e\right ) \left (3 d^2+12 d e x^2+8 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (3 c^2 d \left (d+e x^2\right )+e \left (3 d+2 e x^2\right )\right )-b \left (c^2 d+e\right ) \left (3 d^2+12 d e x^2+8 e^2 x^4\right ) \sec ^{-1}(c x)}{3 d^3 \left (c^2 d+e\right ) x \left (d+e x^2\right )^{3/2}}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (3 c^2 d+2 e\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (3 c^4 d^2+11 c^2 d e+8 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{3 \sqrt {-c^2} d^3 \left (c^2 d+e\right ) \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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none
Time = 0.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=-\frac {{\left (3 \, a c^{3} d^{4} + 3 \, a c d^{3} e + 8 \, {\left (a c^{3} d^{2} e^{2} + a c d e^{3}\right )} x^{4} + 12 \, {\left (a c^{3} d^{3} e + a c d^{2} e^{2}\right )} x^{2} + {\left (3 \, b c^{3} d^{4} + 3 \, b c d^{3} e + 8 \, {\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{4} + 12 \, {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (3 \, b c^{3} d^{4} + 3 \, b c d^{3} e + {\left (3 \, b c^{3} d^{2} e^{2} + 2 \, b c d e^{3}\right )} x^{4} + {\left (6 \, b c^{3} d^{3} e + 5 \, b c d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left ({\left (3 \, b c^{6} d^{2} e^{2} + 2 \, b c^{4} d e^{3}\right )} x^{5} + 2 \, {\left (3 \, b c^{6} d^{3} e + 2 \, b c^{4} d^{2} e^{2}\right )} x^{3} + {\left (3 \, b c^{6} d^{4} + 2 \, b c^{4} d^{3} e\right )} x\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (3 \, b c^{6} d^{2} e^{2} + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d e^{3} + 8 \, b e^{4}\right )} x^{5} + 2 \, {\left (3 \, b c^{6} d^{3} e + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d^{2} e^{2} + 8 \, b d e^{3}\right )} x^{3} + {\left (3 \, b c^{6} d^{4} + {\left (2 \, b c^{4} + 9 \, b c^{2}\right )} d^{3} e + 8 \, b d^{2} e^{2}\right )} x\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{3 \, {\left ({\left (c^{3} d^{5} e^{2} + c d^{4} e^{3}\right )} x^{5} + 2 \, {\left (c^{3} d^{6} e + c d^{5} e^{2}\right )} x^{3} + {\left (c^{3} d^{7} + c d^{6} e\right )} x\right )}} \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^2 \left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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