Optimal. Leaf size=774 \[ -\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 g \sqrt{b^2-4 a c}}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}+\frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^2 e f g^2+105 d^3 g^3-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c} \]
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Rubi [A] time = 2.10553, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {941, 1653, 843, 718, 424, 419} \[ \frac{2 e \sqrt{f+g x} \sqrt{a+b x+c x^2} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (-\left (-90 d^2 g^2+12 d e f g+7 e^2 f^2\right )\right )\right )}{105 c^3 g^2}-\frac{2 \sqrt{2} e \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e g (-25 a e g-84 b d g+13 b e f)+24 b^2 e^2 g^2+c^2 \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e g \left (a e g (189 d g+19 e f)-b \left (-210 d^2 g^2-63 d e f g+9 e^2 f^2\right )\right )-8 b c e^2 g^2 (13 a e g+21 b d g+2 b e f)+48 b^3 e^3 g^3+c^3 \left (-\left (105 d^2 e f g^2+105 d^3 g^3-42 d e^2 f^2 g+8 e^3 f^3\right )\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{105 c^4 g^3 \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e^2 (f+g x)^{3/2} \sqrt{a+b x+c x^2} (-6 b e g+11 c d g+c e f)}{35 c^2 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 941
Rule 1653
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}-\frac{\int \frac{(d+e x) \left (-7 c d^2 f+e (b d f+4 a e f+a d g)-(c d (12 e f+7 d g)-e (5 b e f+2 b d g+5 a e g)) x-e (c e f+11 c d g-6 b e g) x^2\right )}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{7 c}\\ &=\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{2 \int \frac{-\frac{1}{2} g \left (6 b^2 e^3 f^2 g+b e f \left (18 a e^2 g^2-c \left (e^2 f^2+11 d e f g+5 d^2 g^2\right )\right )+c g \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e f g+5 d^2 g^2\right )\right )\right )-\frac{1}{2} g \left (6 b e^3 g^2 (5 b f+3 a g)-c^2 \left (2 e^3 f^3+22 d e^2 f^2 g-95 d^2 e f g^2-35 d^3 g^3\right )-c e g \left (a e g (23 e f+63 d g)-b \left (7 e^2 f^2-85 d e f g-10 d^2 g^2\right )\right )\right ) x-\frac{1}{2} e g^2 \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) x^2}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{35 c^2 g^3}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{4 \int \frac{\frac{1}{4} g^3 \left (24 b^3 e^3 f g^2+b^2 e^2 g \left (24 a e g^2-c f (5 e f+84 d g)\right )-b c e \left (6 a e g^2 (11 e f+14 d g)+c f \left (4 e^2 f^2-21 d e f g-105 d^2 g^2\right )\right )-c g \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )\right )+\frac{1}{4} g^3 \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) x}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 g^5}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{\left (e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 g^3}-\frac{\left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^3 g^3}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\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^4 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} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\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^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 e \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)-c^2 \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt{f+g x} \sqrt{a+b x+c x^2}}{105 c^3 g^2}+\frac{2 e (d+e x)^2 \sqrt{f+g x} \sqrt{a+b x+c x^2}}{7 c}+\frac{2 e^2 (c e f+11 c d g-6 b e g) (f+g x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 g^2}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 e^3 g^3-8 b c e^2 g^2 (2 b e f+21 b d g+13 a e g)-c^3 \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+c^2 e g \left (a e g (19 e f+189 d g)-b \left (9 e^2 f^2-63 d e f g-210 d^2 g^2\right )\right )\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^4 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} e \left (c f^2-b f g+a g^2\right ) \left (24 b^2 e^2 g^2+c e g (13 b e f-84 b d g-25 a e g)+c^2 \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\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^4 g^3 \sqrt{f+g x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 14.9232, size = 10649, normalized size = 13.76 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.395, size = 14978, normalized size = 19.4 \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 \frac{{\left (e x + d\right )}^{3} \sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}\,{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 (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}, 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 (d + e x\right )^{3} \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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