Optimal. Leaf size=109 \[ \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {191, 5228, 12, 421, 419} \[ \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 419
Rule 421
Rule 5228
Rubi steps
\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {1}{d \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{d \sqrt {c^2 x^2}}\\ &=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b c 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 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {\left (b c 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 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b 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 )}{d \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 113, normalized size = 1.04 \[ \frac {x \left (a+b \sec ^{-1}(c x)\right )}{d \sqrt {d+e x^2}}-\frac {b c x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d \left (c^3 x^2-c\right ) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.31, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsec}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ b \int \frac {\arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} + \frac {a x}{\sqrt {e x^{2} + d} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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