Optimal. Leaf size=147 \[ \frac{5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{7/2} d^3}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{64 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2} \]
[Out]
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Rubi [A] time = 0.290239, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{5 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{128 c^{7/2} d^3}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}{64 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2}}{48 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 c d^3 (b+2 c x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 68.2208, size = 141, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{4 c d^{3} \left (b + 2 c x\right )^{2}} + \frac{5 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{48 c^{2} d^{3}} - \frac{5 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{3} d^{3}} + \frac{5 \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{128 c^{\frac{7}{2}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**3,x)
[Out]
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Mathematica [A] time = 0.603423, size = 165, normalized size = 1.12 \[ \frac{-15 \left (4 a c-b^2\right )^{3/2} \log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )+2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-\frac{3 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}+56 a c-12 b^2+8 b c x+8 c^2 x^2\right )+15 \left (4 a c-b^2\right )^{3/2} \log (b+2 c x)}{384 c^{7/2} d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^3,x]
[Out]
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Maple [B] time = 0.016, size = 840, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^3,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)^(5/2)/(2*c*d*x + b*d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.424852, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-\frac{b^{2} - 4 \, a c}{c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt{c x^{2} + b x + a} c \sqrt{-\frac{b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \,{\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 80 \, a b^{2} c - 48 \, a^{2} c^{2} - 8 \,{\left (b^{2} c^{2} - 28 \, a c^{3}\right )} x^{2} - 8 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}}, \frac{15 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{\frac{b^{2} - 4 \, a c}{c}} \arctan \left (-\frac{b^{2} - 4 \, a c}{2 \, \sqrt{c x^{2} + b x + a} c \sqrt{\frac{b^{2} - 4 \, a c}{c}}}\right ) + 2 \,{\left (32 \, c^{4} x^{4} + 64 \, b c^{3} x^{3} - 15 \, b^{4} + 80 \, a b^{2} c - 48 \, a^{2} c^{2} - 8 \,{\left (b^{2} c^{2} - 28 \, a c^{3}\right )} x^{2} - 8 \,{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{a^{2} \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{c^{2} x^{4} \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{2 a b x \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{2 a c x^{2} \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac{2 b c x^{3} \sqrt{a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.669429, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(5/2)/(2*c*d*x + b*d)^3,x, algorithm="giac")
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