Optimal. Leaf size=322 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x}}{3 e} \]
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Rubi [A] time = 0.203577, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {735, 844, 719, 424, 419} \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x}}{3 e} \]
Antiderivative was successfully verified.
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Rule 735
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \sqrt{a+c x^2}}{3 e}+\frac{2 \int \frac{a e-c d x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{3 e}\\ &=\frac{2 \sqrt{d+e x} \sqrt{a+c x^2}}{3 e}+\frac{1}{3} \left (2 \left (a+\frac{c d^2}{e^2}\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx-\frac{(2 c d) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{3 e^2}\\ &=\frac{2 \sqrt{d+e x} \sqrt{a+c x^2}}{3 e}-\frac{\left (4 a \sqrt{c} d \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} e^2 \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (4 a \left (a+\frac{c d^2}{e^2}\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{3 \sqrt{-a} \sqrt{c} \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \sqrt{a+c x^2}}{3 e}+\frac{4 \sqrt{-a} \sqrt{c} d \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{4 \sqrt{-a} \left (a+\frac{c d^2}{e^2}\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{c} \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.98539, size = 456, normalized size = 1.42 \[ \frac{2 \sqrt{d+e x} \left (e^2 \left (a+c x^2\right )-\frac{2 \left (-\sqrt{a} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}+\sqrt{c} d (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{(d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{3 e^3 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.26, size = 688, normalized size = 2.1 \begin{align*} -{\frac{2}{3\,c \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ){e}^{3}}\sqrt{ex+d}\sqrt{c{x}^{2}+a} \left ( 2\,\sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}{\it EllipticF} \left ( \sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) \sqrt{-ac}a{e}^{3}+2\,\sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}{\it EllipticF} \left ( \sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) \sqrt{-ac}c{d}^{2}e-2\,\sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}{\it EllipticE} \left ( \sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) acd{e}^{2}-2\,\sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}{\it EllipticE} \left ( \sqrt{-{\frac{c \left ( ex+d \right ) }{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ){c}^{2}{d}^{3}-{x}^{3}{c}^{2}{e}^{3}-{x}^{2}{c}^{2}d{e}^{2}-xac{e}^{3}-ad{e}^{2}c \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}}{\sqrt{e x + d}}\,{d x} \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{\sqrt{c x^{2} + 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{\sqrt{a + c x^{2}}}{\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{\sqrt{c x^{2} + a}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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