Optimal. Leaf size=652 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c g (-10 a e g-7 b d g+b e f)+4 b^2 e g^2-2 c^2 f (4 e f-7 d g)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 g \sqrt{b^2-4 a c}}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^3 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\left (-3 c g (b f-2 a g)-2 b^2 g^2+8 c^2 f^2\right ) (-4 b e g+7 c d g+c e f)-5 c g (2 c f-b g) (7 c d f-e (a g+3 b f))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (-c g (-5 a e g+7 b d g+2 b e f)+4 b^2 e g^2-3 c g x (-4 b e g+7 c d g+c e f)+c^2 f (4 e f-7 d g)\right )}{105 c^2 g^2}+\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
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Rubi [A] time = 1.11233, antiderivative size = 652, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c g (-10 a e g-7 b d g+b e f)+4 b^2 e g^2-2 c^2 f (4 e f-7 d g)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\left (-3 c g (b f-2 a g)-2 b^2 g^2+8 c^2 f^2\right ) (-4 b e g+7 c d g+c e f)-5 c g (2 c f-b g) (7 c d f-e (a g+3 b f))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (-c g (-5 a e g+7 b d g+2 b e f)+4 b^2 e g^2-3 c g x (-4 b e g+7 c d g+c e f)+c^2 f (4 e f-7 d g)\right )}{105 c^2 g^2}+\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (d+e x) \sqrt{f+g x} \sqrt{a+b x+c x^2} \, dx &=\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (\frac{1}{2} (7 c d f-3 b e f-a e g)+\frac{1}{2} (c e f+7 c d g-4 b e g) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{f+g x}} \, dx}{7 c}\\ &=-\frac{2 \sqrt{f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 g^2}+\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{4 \int \frac{\frac{1}{4} \left (5 c g (b f-2 a g) (7 c d f-e (3 b f+a g))-2 (c e f+7 c d g-4 b e g) \left (\frac{1}{2} b f (4 c f-b g)-a g \left (c f+\frac{b g}{2}\right )\right )\right )-\frac{1}{4} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 g^2}\\ &=-\frac{2 \sqrt{f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 g^2}+\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (\left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2 g^3}+\frac{\left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^2 g^3}\\ &=-\frac{2 \sqrt{f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 g^2}+\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \sqrt{f+g 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} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}}{\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 g^3 \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{a+b x+c x^2}}+\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right ) \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \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} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}} \, 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 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt{a+b x+c x^2}}{105 c^2 g^2}+\frac{2 e \sqrt{f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \sqrt{f+g 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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right ) \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \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} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 13.9134, size = 8432, normalized size = 12.93 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.371, size = 10711, normalized size = 16.4 \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 \sqrt{c x^{2} + b x + a}{\left (e x + d\right )} \sqrt{g x + f}\,{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 (\sqrt{c x^{2} + b x + a}{\left (e x + d\right )} \sqrt{g x + f}, 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 \left (d + e x\right ) \sqrt{f + g x} \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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