Optimal. Leaf size=212 \[ \frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
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Rubi [A] time = 0.303554, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5226, 1574, 944, 719, 419, 933, 168, 538, 537} \[ \frac{2 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 5226
Rule 1574
Rule 944
Rule 719
Rule 419
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{a+b \sec ^{-1}(c x)}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{(2 b) \int \frac{\sqrt{d+e x}}{\sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{c e}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{x \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{c \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b d \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 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{\left (2 b d \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{\left (4 b \sqrt{\frac{d+e x}{d+\frac{e}{c}}} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1-\frac{2 e x^2}{c \left (d+\frac{e}{c}\right )}}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (4 b d \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 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (4 b d \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 \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{\frac{c (d+e x)}{c d+e}} \sqrt{1-c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b d \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 [C] time = 2.65761, size = 212, normalized size = 1. \[ \frac{2 \left (\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right ),\frac{c d+e}{c d-e}\right )-\Pi \left (\frac{e}{c d}+1;i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )\right )}{c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+a \sqrt{d+e x}+b \sec ^{-1}(c x) \sqrt{d+e x}\right )}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.257, size = 254, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{e} \left ( a\sqrt{ex+d}+b \left ( \sqrt{ex+d}{\rm arcsec} \left (cx\right )-2\,{\frac{1}{cx}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc+e}{dc-e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc-e}{dc+e}}} \left ({\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) -{\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 ) \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcsec}\left (c x\right ) + a}{\sqrt{e x + d}}, 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 \frac{a + b \operatorname{asec}{\left (c x \right )}}{\sqrt{d + e x}}\, 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}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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