Optimal. Leaf size=315 \[ \frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{4 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]
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Rubi [A] time = 0.463358, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5226, 1574, 958, 719, 419, 933, 168, 538, 537, 844, 424} \[ \frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{4 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]
Antiderivative was successfully verified.
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Rule 5226
Rule 1574
Rule 958
Rule 719
Rule 419
Rule 933
Rule 168
Rule 538
Rule 537
Rule 844
Rule 424
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{(2 b) \int \frac{(d+e x)^{3/2}}{\sqrt{1-\frac{1}{c^2 x^2}} x^2} \, dx}{3 c e}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{(d+e x)^{3/2}}{x \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \left (\frac{2 d e}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}}+\frac{d^2}{x \sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}}+\frac{e^2 x}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{\left (4 b d \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b d^2 \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b e \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{x}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{\left (2 b \sqrt{-\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 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}{\sqrt{d+e x} \sqrt{-\frac{1}{c^2}+x^2}} \, dx}{3 c \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b d^2 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (8 b d \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 )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{8 b d \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 )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (4 b d^2 \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 )}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{d+e x} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{2 e x^2}{c \left (d+\frac{e}{c}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{d+e x}{d+\frac{e}{c}}}}-\frac{\left (4 b d \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 )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{4 b \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b d \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 )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (4 b d^2 \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 )}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 (d+e x)^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{4 b \sqrt{d+e x} \sqrt{1-c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b d \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 )}{3 c^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b d^2 \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 )}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 6.01815, size = 277, normalized size = 0.88 \[ \frac{2 \left (\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left ((2 c d-e) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right ),\frac{c d+e}{c d-e}\right )+(e-c d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )-c d \Pi \left (\frac{e}{c d}+1;i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right )|\frac{c d+e}{c d-e}\right )\right )}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+a (d+e x)^{3/2}+b \sec ^{-1}(c x) (d+e x)^{3/2}\right )}{3 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.255, size = 388, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{e} \left ( 1/3\, \left ( ex+d \right ) ^{3/2}a+b \left ( 1/3\, \left ( ex+d \right ) ^{3/2}{\rm arcsec} \left (cx\right )-2/3\,{\frac{1}{{c}^{2}x} \left ( 2\,d{\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) c-{\it EllipticE} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) cd-d{\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 ) c+{\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) e-{\it EllipticE} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) e \right ) \sqrt{-{\frac{ \left ( ex+d \right ) c-dc-e}{dc+e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc+e}{dc-e}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}{\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}}}}}}} \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x + d}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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