Optimal. Leaf size=749 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (-c e \left (2 a e (9 C d-5 B e)-b \left (32 C d^2-5 e (5 B d-3 A e)\right )\right )+b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{15 c^2 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e (-6 a C e-5 b B e+8 b C d)+c^2 \left (-\left (48 C d^2-10 e (4 B d-3 A e)\right )\right )+2 b^2 C e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac{6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\right )}{15 c e^3 \left (a e^2-b d e+c d^2\right )}-\frac{2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 1.51845, antiderivative size = 746, 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, 814, 843, 718, 424, 419} \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (c e \left (-2 a e (9 C d-5 B e)-5 b e (5 B d-3 A e)+32 b C d^2\right )+b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e (-6 a C e-5 b B e+8 b C d)+c^2 \left (-\left (48 C d^2-10 e (4 B d-3 A e)\right )\right )+2 b^2 C e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac{6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^3 \left (a e^2-b d e+c d^2\right )}-\frac{2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1650
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 )}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{2 \int \frac{\left (-\frac{3 b C d^2-b e (3 B d-2 A e)+e (A c d-a C d+a B e)}{2 e}+\frac{1}{2} \left (5 B c d+b C d-\frac{6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{c d^2-b d e+a e^2}\\ &=-\frac{2 \sqrt{d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac{6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}+\frac{4 \int \frac{-\frac{\left (c d^2-b d e+a e^2\right ) \left (b^2 C d e+a b C e^2+2 a c e (6 C d-5 B e)-b c \left (24 C d^2-5 e (4 B d-3 A e)\right )\right )}{4 e}-\frac{\left (c d^2-b d e+a e^2\right ) \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) x}{4 e}}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 c e^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \sqrt{d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac{6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x}}-\frac{\left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 c e^4}+\frac{\left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 c e^4}\\ &=-\frac{2 \sqrt{d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac{6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{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 (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 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 )}{15 c^2 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 (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B 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 )}{15 c^2 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac{6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt{a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac{2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{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 (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 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 )}{15 c^2 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 (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B 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 )}{15 c^2 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 14.1107, size = 13240, normalized size = 17.68 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.379, size = 8221, normalized size = 11. \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{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}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, 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{3}{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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