Optimal. Leaf size=119 \[ -\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
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Rubi [A] time = 0.219172, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5226, 1574, 933, 168, 538, 537} \[ -\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 5226
Rule 1574
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{(2 b) \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c e}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{\left (4 b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{d+\frac{e}{c}-\frac{e x^2}{c}}} \, dx,x,\sqrt{1-c x}\right )}{c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{\left (4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{c \left (d+\frac{e}{c}\right )}}} \, dx,x,\sqrt{1-c x}\right )}{c e \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 \left (a+b \sec ^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c e \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.267054, size = 124, normalized size = 1.04 \[ -\frac{2 \left (\left (c^2 x^2-1\right ) \left (a+b \sec ^{-1}(c x)\right )+2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )\right )}{e \left (c^2 x^2-1\right ) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.247, size = 217, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{e} \left ( -{\frac{a}{\sqrt{ex+d}}}+b \left ( -{\frac{{\rm arcsec} \left (cx\right )}{\sqrt{ex+d}}}-2\,{\frac{1}{cdx}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc+e}{dc-e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc-e}{dc+e}}}{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},{\frac{dc-e}{dc}},{\sqrt{{\frac{c}{dc+e}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2} \left ( ex+d \right ) ^{2}-2\,d{c}^{2} \left ( ex+d \right ) +{c}^{2}{d}^{2}-{e}^{2}}{{c}^{2}{e}^{2}{x}^{2}}}}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}} \right ) \right ) } \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{a + b \operatorname{asec}{\left (c x \right )}}{\left (d + e x\right )^{\frac{3}{2}}}\, 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 \frac{b \operatorname{arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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