Optimal. Leaf size=712 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (e (-2 a C e-3 b B e+8 b C d)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (2 \left (4 c d-\frac{b e}{2}\right ) \left (-a C e-A c e+b C d+B c d-\frac{2 c C d^2}{e}\right )+6 c (e (a B e-a C d+A c d)+b d (C d-B e))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^3 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{a+b x+c x^2} \left (e^2 x \left (-a C e-A c e+b C d+B c d-\frac{2 c C d^2}{e}\right )+e (b d-a e) (7 C d-3 B e)-c d \left (8 C d^2-e (4 B d-A e)\right )\right )}{3 e^3 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 1.27107, antiderivative size = 711, 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 = {1650, 812, 843, 718, 424, 419} \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (e (-2 a C e-3 b B e+8 b C d)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (2 \left (4 c d-\frac{b e}{2}\right ) \left (-a C e-A c e+b C d+B c d-\frac{2 c C d^2}{e}\right )+6 c (e (a B e-a C d+A c d)+b d (C d-B e))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^3 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{3 e (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{2 \sqrt{a+b x+c x^2} \left (-e x \left (-a C e-A c e+b C d+B c d-\frac{2 c C d^2}{e}\right )-(b d-a e) (7 C d-3 B e)-c d (4 B d-A e)+\frac{8 c C d^3}{e}\right )}{3 e^2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 812
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 )}{(d+e x)^{5/2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac{2 \int \frac{\left (-\frac{3 (b d (C d-B e)+e (A c d-a C d+a B e))}{2 e}+\frac{3}{2} \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{3/2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{2 \left (\frac{8 c C d^3}{e}-c d (4 B d-A e)-(b d-a e) (7 C d-3 B e)-e \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{3 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac{4 \int \frac{\frac{3}{4} \left (2 (2 b d-a e) \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right )+3 b (b d (C d-B e)+e (A c d-a C d+a B e))\right )+\frac{3}{4} \left (2 \left (4 c d-\frac{b e}{2}\right ) \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right )+6 c (b d (C d-B e)+e (A c d-a C d+a B e))\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{9 e^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{2 \left (\frac{8 c C d^3}{e}-c d (4 B d-A e)-(b d-a e) (7 C d-3 B e)-e \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{3 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac{\left (e (8 b C d-3 b B e-2 a C e)-2 c \left (8 C d^2-e (4 B d-A e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3 e^4}-\frac{\left (b C e^2 (b d-a e)+2 c^2 \left (8 C d^3-d e (4 B d-A e)\right )-c e \left (16 b C d^2-b e (7 B d-A e)-2 a e (7 C d-3 B e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{3 e^4 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{2 \left (\frac{8 c C d^3}{e}-c d (4 B d-A e)-(b d-a e) (7 C d-3 B e)-e \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{3 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (b C e^2 (b d-a e)+2 c^2 \left (8 C d^3-d e (4 B d-A e)\right )-c e \left (16 b C d^2-b e (7 B d-A e)-2 a e (7 C d-3 B 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 )}{3 c e^4 \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} \left (e (8 b C d-3 b B e-2 a C e)-2 c \left (8 C d^2-e (4 B d-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 )}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \left (\frac{8 c C d^3}{e}-c d (4 B d-A e)-(b d-a e) (7 C d-3 B e)-e \left (B c d+b C d-\frac{2 c C d^2}{e}-A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{3 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{3 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (b C e^2 (b d-a e)+2 c^2 \left (8 C d^3-d e (4 B d-A e)\right )-c e \left (16 b C d^2-b e (7 B d-A e)-2 a e (7 C d-3 B 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 )}{3 c e^4 \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} \left (e (8 b C d-3 b B e-2 a C e)-2 c \left (8 C d^2-e (4 B d-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 )}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 14.4505, size = 8456, normalized size = 11.88 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.452, size = 21038, normalized size = 29.6 \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}}{{\left (e x + d\right )}^{\frac{5}{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}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, 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}}}{\left (d + e x\right )^{\frac{5}{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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