Optimal. Leaf size=225 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]
[Out]
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Rubi [A] time = 0.489402, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (-2 a e+x (2 c d-b e)+b d)}{8 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 61.9289, size = 209, normalized size = 0.93 \[ - \frac{3 \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{128 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{64 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{8 \left (d + e x\right )^{4} \left (a e^{2} - b d e + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 1.17253, size = 331, normalized size = 1.47 \[ \frac{-2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (2 (d+e x)^2 \left (4 c e (5 a e-6 b d)+b^2 e^2+24 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )-(d+e x)^3 (2 c d-b e) \left (4 c e (5 a e-2 b d)-3 b^2 e^2+8 c^2 d^2\right )-24 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2+16 \left (e (a e-b d)+c d^2\right )^3\right )-3 e^3 \left (b^2-4 a c\right )^2 (d+e x)^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+3 e^3 \left (b^2-4 a c\right )^2 (d+e x)^4 \log (d+e x)}{128 e^3 (d+e x)^4 \left (e (a e-b d)+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.033, size = 15932, normalized size = 70.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(3/2)/(e*x+d)^5,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)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 4.96749, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^5,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)**5,x)
[Out]
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GIAC/XCAS [A] time = 11.5552, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^5,x, algorithm="giac")
[Out]