Optimal. Leaf size=604 \[ -\frac{e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]
[Out]
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Rubi [A] time = 10.9386, antiderivative size = 604, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 6.3641, size = 690, normalized size = 1.14 \[ \sqrt{d+e x} \left (\frac{-3 a b c e^2+4 a c^2 d e-2 a c^2 e^2 x+b^3 e^2-2 b^2 c d e+b^2 c e^2 x+b c^2 d^2-2 b c^2 d e x+2 c^3 d^2 x}{\left (4 a c-b^2\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e^3}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}\right )-\frac{\left (-2 c^3 d^2 e \sqrt{b^2-4 a c}+2 b c^2 d e^2 \sqrt{b^2-4 a c}+10 a c^2 e^3 \sqrt{b^2-4 a c}-3 b^2 c e^3 \sqrt{b^2-4 a c}-16 a b c^2 e^3+32 a c^3 d e^2+3 b^3 c e^3-2 b^2 c^2 d e^2-12 b c^3 d^2 e+8 c^4 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )^2}-\frac{\left (-2 c^3 d^2 e \sqrt{b^2-4 a c}+2 b c^2 d e^2 \sqrt{b^2-4 a c}+10 a c^2 e^3 \sqrt{b^2-4 a c}-3 b^2 c e^3 \sqrt{b^2-4 a c}+16 a b c^2 e^3-32 a c^3 d e^2-3 b^3 c e^3+2 b^2 c^2 d e^2+12 b c^3 d^2 e-8 c^4 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^2),x]
[Out]
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Maple [B] time = 0.107, size = 10471, normalized size = 17.3 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{2}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^2*(e*x + d)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x + a)^2*(e*x + d)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**2,x)
[Out]
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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(1/((c*x^2 + b*x + a)^2*(e*x + d)^(3/2)),x, algorithm="giac")
[Out]