Optimal. Leaf size=307 \[ -\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.919856, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,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((c*x**2+b*x+a)**(3/2)/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 2.37311, size = 334, normalized size = 1.09 \[ \frac{-\frac{2 e \sqrt{a+x (b+c x)} \left (\frac{(d+e x)^2 \left (4 c e (8 a e-11 b d)+3 b^2 e^2+44 c^2 d^2\right )}{e (a e-b d)+c d^2}+8 \left (e (a e-b d)+c d^2\right )-14 (d+e x) (2 c d-b e)\right )}{(d+e x)^3}-\frac{3 (b e-2 c d) \log (d+e x) \left (4 c e (2 b d-3 a e)+b^2 e^2-8 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{3 (b e-2 c d) \left (4 c e (2 b d-3 a e)+b^2 e^2-8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+48 c^{3/2} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{48 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.026, size = 10401, normalized size = 33.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(3/2)/(e*x+d)^4,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^4,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((c*x^2 + b*x + a)^(3/2)/(e*x + d)^4,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((c*x**2+b*x+a)**(3/2)/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 17.0627, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^4,x, algorithm="giac")
[Out]