Optimal. Leaf size=458 \[ -\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2} (2 c d-b e)}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.950163, antiderivative size = 458, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{e \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2} (2 c d-b e)}{\sqrt{d+e x} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
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Rubi in Sympy [A] time = 170.793, size = 428, normalized size = 0.93 \[ - \frac{2 \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{\sqrt{d + e x} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{4 \sqrt{2} \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{- 4 a c + b^{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{e \sqrt{d + e x} \sqrt{a + b x + c x^{2}}} + \frac{\sqrt{2} \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} \sqrt{d + e x} \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{\frac{b + 2 c x + \sqrt{- 4 a c + b^{2}}}{\sqrt{- 4 a c + b^{2}}}}}{2} \right )}\middle | \frac{2 e \sqrt{- 4 a c + b^{2}}}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}\right )}{e \sqrt{\frac{c \left (- d - e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
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Mathematica [C] time = 1.78095, size = 541, normalized size = 1.18 \[ \frac{i (d+e x) \sqrt{1-\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{4 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+2} \left (\left (-2 c d \sqrt{e^2 \left (b^2-4 a c\right )}+b e \sqrt{e^2 \left (b^2-4 a c\right )}+4 a c e^2-b^2 e^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )-(b e-2 c d) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 e^2 \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right ) \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
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Maple [B] time = 0.069, size = 1863, normalized size = 4.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, c x + b}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{2 \, c x + b}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b + 2 c x}{\left (d + e x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)),x, algorithm="giac")
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