Optimal. Leaf size=668 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e (-10 a C e-7 b B e+8 b C d)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )+4 b^2 C e^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left (-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right ) (4 b C e-7 B c e+6 c C d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e} \]
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Rubi [A] time = 1.18846, antiderivative size = 668, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1653, 814, 843, 718, 424, 419} \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e (-10 a C e-7 b B e+8 b C d)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )+4 b^2 C e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left (-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right ) (4 b C e-7 B c e+6 c C d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2} \left (A+B x+C x^2\right )}{\sqrt{d+e x}} \, dx &=\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}+\frac{2 \int \frac{\left (-\frac{1}{2} e (3 b C d-7 A c e+a C e)-\frac{1}{2} e (6 c C d-7 B c e+4 b C e) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{7 c e^2}\\ &=-\frac{2 \sqrt{d+e x} (5 c e (3 b C d-7 A c e+a C e)-(4 c d-b e) (6 c C d-7 B c e+4 b C e)+3 c e (6 c C d-7 B c e+4 b C e) x) \sqrt{a+b x+c x^2}}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac{4 \int \frac{-\frac{1}{4} e \left (5 c e (b d-2 a e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right )\right )-\frac{1}{4} e \left (5 c e (2 c d-b e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 e^4}\\ &=-\frac{2 \sqrt{d+e x} (5 c e (3 b C d-7 A c e+a C e)-(4 c d-b e) (6 c C d-7 B c e+4 b C e)+3 c e (6 c C d-7 B c e+4 b C e) x) \sqrt{a+b x+c x^2}}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}+\frac{\left (5 c e (2 c d-b e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^2 e^4}+\frac{\left (\left (c d^2-b d e+a e^2\right ) \left (4 b^2 C e^2+c e (8 b C d-7 b B e-10 a C e)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 e^4}\\ &=-\frac{2 \sqrt{d+e x} (5 c e (3 b C d-7 A c e+a C e)-(4 c d-b e) (6 c C d-7 B c e+4 b C e)+3 c e (6 c C d-7 B c e+4 b C e) x) \sqrt{a+b x+c x^2}}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (5 c e (2 c d-b e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )\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 )}{105 c^3 e^4 \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} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 C e^2+c e (8 b C d-7 b B e-10 a C e)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )\right ) \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 )}{105 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (5 c e (3 b C d-7 A c e+a C e)-(4 c d-b e) (6 c C d-7 B c e+4 b C e)+3 c e (6 c C d-7 B c e+4 b C e) x) \sqrt{a+b x+c x^2}}{105 c^2 e^3}+\frac{2 C \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (5 c e (2 c d-b e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )\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 )}{105 c^3 e^4 \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} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 C e^2+c e (8 b C d-7 b B e-10 a C e)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )\right ) \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 )}{105 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 14.4392, size = 9965, normalized size = 14.92 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.426, size = 12761, normalized size = 19.1 \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{{\left (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + 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{{\left (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + 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{\left (A + B x + C x^{2}\right ) \sqrt{a + b x + 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(-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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