Optimal. Leaf size=39 \[ \frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
[Out]
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Rubi [A] time = 0.055534, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 13.7624, size = 36, normalized size = 0.92 \[ \frac{2 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{7 d^{8} \left (b + 2 c x\right )^{7} \left (- 4 a c + b^{2}\right )} \]
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)**8,x)
[Out]
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Mathematica [A] time = 0.200414, size = 38, normalized size = 0.97 \[ \frac{2 (a+x (b+c x))^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]
[Out]
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Maple [A] time = 0.008, size = 38, normalized size = 1. \[ -{\frac{2}{7\, \left ( 2\,cx+b \right ) ^{7}{d}^{8} \left ( 4\,ac-{b}^{2} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \]
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)^8,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)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.10928, size = 365, normalized size = 9.36 \[ \frac{2 \,{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt{c x^{2} + b x + a}}{7 \,{\left (128 \,{\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{8} x^{7} + 448 \,{\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{8} x^{6} + 672 \,{\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{8} x^{5} + 560 \,{\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{8} x^{4} + 280 \,{\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{8} x^{3} + 84 \,{\left (b^{7} c^{2} - 4 \, a b^{5} c^{3}\right )} d^{8} x^{2} + 14 \,{\left (b^{8} c - 4 \, a b^{6} c^{2}\right )} d^{8} x +{\left (b^{9} - 4 \, a b^{7} c\right )} d^{8}\right )}} \]
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)^8,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)**(5/2)/(2*c*d*x+b*d)**8,x)
[Out]
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GIAC/XCAS [A] time = 1.14012, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]
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)^8,x, algorithm="giac")
[Out]