Optimal. Leaf size=206 \[ \frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (f x)^{m+1} \left (c^2 d (m+2) (m+3)+e (m+1)^2\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},c^2 x^2\right )}{c^2 f (m+1)^2 (m+2) (m+3)}+\frac{d (f x)^{m+1} \left (a+b \text{sech}^{-1}(c x)\right )}{f (m+1)}+\frac{e (f x)^{m+3} \left (a+b \text{sech}^{-1}(c x)\right )}{f^3 (m+3)}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+1}}{c^2 f (m+2) (m+3)} \]
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Rubi [A] time = 0.179143, antiderivative size = 192, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {14, 6301, 12, 459, 364} \[ \frac{d (f x)^{m+1} \left (a+b \text{sech}^{-1}(c x)\right )}{f (m+1)}+\frac{e (f x)^{m+3} \left (a+b \text{sech}^{-1}(c x)\right )}{f^3 (m+3)}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} (f x)^{m+1} \left (\frac{e}{c^2 (m+2) (m+3)}+\frac{d}{(m+1)^2}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right )}{f}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} (f x)^{m+1}}{c^2 f (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 6301
Rule 12
Rule 459
Rule 364
Rubi steps
\begin{align*} \int (f x)^m \left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{d (f x)^{1+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f^3 (3+m)}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(f x)^m \left (d (3+m)+e (1+m) x^2\right )}{(1+m) (3+m) \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{d (f x)^{1+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(f x)^m \left (d (3+m)+e (1+m) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{3+4 m+m^2}\\ &=-\frac{b e (f x)^{1+m} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2 f (2+m) (3+m)}+\frac{d (f x)^{1+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac{\left (b \left (\frac{e (1+m)^2}{c^2 (2+m)}+d (3+m)\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(f x)^m}{\sqrt{1-c^2 x^2}} \, dx}{3+4 m+m^2}\\ &=-\frac{b e (f x)^{1+m} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{c^2 f (2+m) (3+m)}+\frac{d (f x)^{1+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f (1+m)}+\frac{e (f x)^{3+m} \left (a+b \text{sech}^{-1}(c x)\right )}{f^3 (3+m)}+\frac{b \left (\frac{e (1+m)^2}{c^2 (2+m)}+d (3+m)\right ) (f x)^{1+m} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};c^2 x^2\right )}{f (1+m) \left (3+4 m+m^2\right )}\\ \end{align*}
Mathematica [F] time = 0.116859, size = 0, normalized size = 0. \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text{sech}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.873, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \right ) \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \operatorname{arsech}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{m} \left (a + b \operatorname{asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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