Optimal. Leaf size=305 \[ \frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}} \]
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Rubi [A] time = 0.183042, antiderivative size = 305, 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 = {733, 844, 719, 424, 419} \[ \frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 733
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^2}}{(d+e x)^{3/2}} \, dx &=-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}}+\frac{(2 c) \int \frac{x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{e}\\ &=-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}}+\frac{(2 c) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{e^2}-\frac{(2 c d) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{e^2}\\ &=-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}}+\frac{\left (4 a \sqrt{c} \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 )}{\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 \sqrt{c} d \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 )}{\sqrt{-a} e^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{c} \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 )}{e^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{4 \sqrt{-a} \sqrt{c} d \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 )}{e^2 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.30891, size = 419, normalized size = 1.37 \[ \frac{2 \left (-\frac{2 \sqrt{a} \sqrt{c} e (d+e x)^{3/2} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{2 \sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+e^2 \left (a+c x^2\right )\right )}{e^3 \sqrt{a+c x^2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.323, size = 646, normalized size = 2.1 \begin{align*} -2\,{\frac{\sqrt{ex+d}\sqrt{c{x}^{2}+a}}{ \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ){e}^{3}} \left ( 2\,{\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 ) a{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}}}+2\,{\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{d}^{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}}}-2\,{\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 ) a{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}}}-2\,{\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 ) de\sqrt{-ac}\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}}}+c{e}^{2}{x}^{2}+a{e}^{2} \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}}{{\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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, 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}}}{\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{\sqrt{c x^{2} + 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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