Optimal. Leaf size=636 \[ -\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^2 d^2-a e (b d-10 c e)-4 b^2 e^2\right ) \left (a d^2-e (b d-c e)\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+2 a x+b}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 a^3 e^3 \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (4 a^2 d^2-a e (2 b d-5 c e)-3 a e x (a d-4 b e)+4 b^2 e^2\right )}{105 a^2 e^2}+\frac{\sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (-a^2 d e (5 b d-16 c e)+8 a^3 d^3-a b e^2 (5 b d+29 c e)+8 b^3 e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \left (a x^2+b x+c\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (a x^2+b x+c\right )}{7 a} \]
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Rubi [A] time = 0.993044, antiderivative size = 636, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {1573, 832, 814, 843, 718, 424, 419} \[ -\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (4 a^2 d^2-a e (2 b d-5 c e)-3 a e x (a d-4 b e)+4 b^2 e^2\right )}{105 a^2 e^2}-\frac{2 \sqrt{2} x \sqrt{b^2-4 a c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (8 a^2 d^2-a e (b d-10 c e)-4 b^2 e^2\right ) \left (a d^2-e (b d-c e)\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{\sqrt{2} x \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (-a^2 d e (5 b d-16 c e)+8 a^3 d^3-a b e^2 (5 b d+29 c e)+8 b^3 e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \left (a x^2+b x+c\right ) \sqrt{\frac{a (d+e x)}{2 a d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 x \sqrt{d+e x} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \left (a x^2+b x+c\right )}{7 a} \]
Antiderivative was successfully verified.
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Rule 1573
Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x^2 \sqrt{d+e x} \, dx &=\frac{\left (\sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int x \sqrt{d+e x} \sqrt{c+b x+a x^2} \, dx}{\sqrt{c+b x+a x^2}}\\ &=\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (c+b x+a x^2\right )}{7 a}+\frac{\left (2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\left (\frac{1}{2} (-3 b d-c e)+\frac{1}{2} (a d-4 b e) x\right ) \sqrt{c+b x+a x^2}}{\sqrt{d+e x}} \, dx}{7 a \sqrt{c+b x+a x^2}}\\ &=-\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (c+b x+a x^2\right )}{7 a}-\frac{\left (4 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\frac{1}{2} \left (-a^2 d^2 (2 b d-c e)-2 b^2 e^2 (b d+c e)+a e \left (b^2 d^2+9 b c d e+5 c^2 e^2\right )\right )-\frac{1}{4} \left (8 a^3 d^3+8 b^3 e^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)\right ) x}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{105 a^2 e^2 \sqrt{c+b x+a x^2}}\\ &=-\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (c+b x+a x^2\right )}{7 a}-\frac{\left (\left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c+b x+a x^2}} \, dx}{105 a^2 e^3 \sqrt{c+b x+a x^2}}-\frac{\left (4 \left (-\frac{1}{4} d \left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\right )+\frac{1}{2} e \left (-a^2 d^2 (2 b d-c e)-2 b^2 e^2 (b d+c e)+a e \left (b^2 d^2+9 b c d e+5 c^2 e^2\right )\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c+b x+a x^2}} \, dx}{105 a^2 e^3 \sqrt{c+b x+a x^2}}\\ &=-\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (c+b x+a x^2\right )}{7 a}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a 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 a 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 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{105 a^3 e^3 \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \left (c+b x+a x^2\right )}-\frac{\left (8 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{1}{4} d \left (-8 a^3 d^3-8 b^3 e^3+a^2 d e (5 b d-16 c e)+a b e^2 (5 b d+29 c e)\right )+\frac{1}{2} e \left (-a^2 d^2 (2 b d-c e)-2 b^2 e^2 (b d+c e)+a e \left (b^2 d^2+9 b c d e+5 c^2 e^2\right )\right )\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{a \left (c+b x+a 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 a d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{105 a^3 e^3 \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ &=-\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (4 a^2 d^2+4 b^2 e^2-a e (2 b d-5 c e)-3 a e (a d-4 b e) x\right )}{105 a^2 e^2}+\frac{2 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \left (c+b x+a x^2\right )}{7 a}+\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (8 a^3 d^3+8 b^3 e^3-a^2 d e (5 b d-16 c e)-a b e^2 (5 b d+29 c e)\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{d+e x} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (a d^2-b d e+c e^2\right ) \left (8 a^2 d^2-a b d e-4 b^2 e^2+10 a c e^2\right ) \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} x \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 a x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 a^3 e^3 \sqrt{d+e x} \left (c+b x+a x^2\right )}\\ \end{align*}
Mathematica [C] time = 13.0944, size = 5350, normalized size = 8.41 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.049, size = 6302, normalized size = 9.9 \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{e x + d} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} x^{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 (\sqrt{e x + d} x^{2} \sqrt{\frac{a x^{2} + b x + c}{x^{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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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