Optimal. Leaf size=216 \[ \frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{c \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.109953, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {768, 718, 419} \[ \frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 768
Rule 718
Rule 419
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \sqrt{d+e x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{d+e x}}{\sqrt{a+b x+c x^2}}+e \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 \sqrt{d+e x}}{\sqrt{a+b x+c x^2}}+\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} e \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{c \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x}}{\sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} e \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.48523, size = 318, normalized size = 1.47 \[ \frac{-2 \sqrt{d+e x}+\frac{i (d+e x) \sqrt{2-\frac{4 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a e^2-b d e+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}{\sqrt{d+e x}}\right ),-\frac{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}{\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d}\right )}{\sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}}{\sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 705, normalized size = 3.3 \begin{align*}{\frac{1}{ \left ( ce{x}^{3}+be{x}^{2}+cd{x}^{2}+aex+bdx+ad \right ) c} \left ( -\sqrt{2}{\it EllipticF} \left ( \sqrt{2}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}},\sqrt{-{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) ^{-1}}} \right ) be\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}}\sqrt{{e \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) ^{-1}}}\sqrt{{e \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}}+2\,\sqrt{2}{\it EllipticF} \left ( \sqrt{2}\sqrt{-{\frac{ \left ( ex+d \right ) c}{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}}},\sqrt{-{\frac{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}{e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd}}} \right ) cd\sqrt{-{\frac{ \left ( ex+d \right ) c}{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}}}\sqrt{{\frac{ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) e}{e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd}}}\sqrt{{\frac{ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) e}{e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd}}}-\sqrt{2}{\it EllipticF} \left ( \sqrt{2}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}},\sqrt{-{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) ^{-1}}} \right ) e\sqrt{-4\,ac+{b}^{2}}\sqrt{-{ \left ( ex+d \right ) c \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}}\sqrt{{e \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) ^{-1}}}\sqrt{{e \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) ^{-1}}}-2\,cex-2\,cd \right ) \sqrt{ex+d}\sqrt{c{x}^{2}+bx+a}} \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{{\left (2 \, c x + b\right )} \sqrt{e x + d}}{{\left (c x^{2} + b x + 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} + b x + a}{\left (2 \, c x + b\right )} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} 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{\left (b + 2 c x\right ) \sqrt{d + e x}}{\left (a + b x + 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{{\left (2 \, c x + b\right )} \sqrt{e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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