Optimal. Leaf size=298 \[ \frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]
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Rubi [A] time = 0.209321, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {737, 12, 844, 719, 424, 419} \[ \frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}+\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]
Antiderivative was successfully verified.
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Rule 737
Rule 12
Rule 844
Rule 719
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx &=\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{\int \frac{e x}{2 \sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{a}\\ &=\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{e \int \frac{x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{2 a}\\ &=\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{\int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{2 a}+\frac{d \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{2 a}\\ &=\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{\left (\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} \sqrt{c} \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (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} \sqrt{c} \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{\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 )}{\sqrt{-a} \sqrt{c} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}+\frac{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 )}{\sqrt{-a} \sqrt{c} \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.05554, size = 408, normalized size = 1.37 \[ \frac{\sqrt{d+e x} \left (\frac{\sqrt{a} \sqrt{d+e x} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{c} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{e \left (a+c x^2\right )}{c (d+e x)}-\frac{i \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}+x\right )}{a \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.268, size = 649, normalized size = 2.2 \begin{align*}{\frac{1}{ce \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ) a}\sqrt{ex+d}\sqrt{c{x}^{2}+a} \left ({\it EllipticE} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) a{e}^{2}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}+{\it EllipticE} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) c{d}^{2}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}-{\it EllipticF} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) a{e}^{2}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}-{\it EllipticF} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) de\sqrt{-ac}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}+c{e}^{2}{x}^{2}+cdex \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{e x + d}}{{\left (c x^{2} + a\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}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{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{d + e x}}{\left (a + c x^{2}\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{e x + d}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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