Optimal. Leaf size=508 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 C d-B e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (C e (b d-a e)-c \left (2 C d^2-e (B d-A e)\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{a+b x+c x^2} \left (C d^2-e (B d-A e)\right )}{e \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.648922, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {1650, 843, 718, 424, 419} \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-C e (b d-a e)-c e (B d-A e)+2 c C d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{a+b x+c x^2} \left (C d^2-e (B d-A e)\right )}{e \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 C d-B e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{c e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \int \frac{-\frac{b d (C d-B e)+e (A c d-a C d+a B e)}{2 e}+\frac{1}{2} \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{(2 C d-B e) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{e^2}-\frac{\left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{e \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\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}}}{\sqrt{1-x^2}} \, 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 e \left (c d^2-b d e+a e^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}-\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} (2 C d-B 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 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 e \left (c d^2-b d e+a e^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 C d-B 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 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 7.38258, size = 772, normalized size = 1.52 \[ \frac{2 \left (-\frac{i (d+e x)^{3/2} \sqrt{1-\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+1} \left (\left (b \left (C d e \sqrt{e^2 \left (b^2-4 a c\right )}+a C e^3+A c e^3+B c d e^2\right )-A c e^2 \left (\sqrt{e^2 \left (b^2-4 a c\right )}+2 c d\right )+B c d e \sqrt{e^2 \left (b^2-4 a c\right )}-2 c C d^2 \sqrt{e^2 \left (b^2-4 a c\right )}-a C e^2 \sqrt{e^2 \left (b^2-4 a c\right )}-2 a B c e^3+2 a c C d e^2-b^2 C d e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a e^2-b d e+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}{\sqrt{d+e x}}\right ),-\frac{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}{\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d}\right )+\left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) \left (C e (a e-b d)+c e (A e-B d)+2 c C d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} c \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}+e^2 (a+x (b+c x)) \left (-\left (e (A e-B d)+C d^2\right )\right )+\frac{e^2 (a+x (b+c x)) \left (C e (a e-b d)+c e (A e-B d)+2 c C d^2\right )}{c}\right )}{e^3 \sqrt{d+e x} \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.376, size = 6053, normalized size = 11.9 \begin{align*} \text{output too large to display} \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{C x^{2} + B x + A}{\sqrt{c x^{2} + b x + 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{{\left (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c e^{2} x^{4} +{\left (2 \, c d e + b e^{2}\right )} x^{3} + a d^{2} +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} +{\left (b d^{2} + 2 \, a d e\right )} x}, 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{A + B x + C x^{2}}{\left (d + e x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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