Optimal. Leaf size=98 \[ \frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.114727, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 18.3167, size = 95, normalized size = 0.97 \[ - \frac{2 \left (b + 2 c x\right ) \left (d + e x\right )^{2}}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (b e - 2 c d\right ) \left (4 a e - 2 b d + x \left (2 b e - 4 c d\right )\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.210937, size = 167, normalized size = 1.7 \[ \frac{2 \left (4 b \left (2 a^2 e^2+3 a c (d-e x)^2+2 c^2 d x^2 (3 d-2 e x)\right )+8 c \left (-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3\right )+b^2 \left (2 c x \left (3 d^2-12 d e x+e^2 x^2\right )-4 a e (d-3 e x)\right )+b^3 \left (-\left (d^2+6 d e x-3 e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.013, size = 215, normalized size = 2.2 \[{\frac{16\,a{c}^{2}{e}^{2}{x}^{3}+4\,{b}^{2}c{e}^{2}{x}^{3}-32\,b{c}^{2}de{x}^{3}+32\,{c}^{3}{d}^{2}{x}^{3}+24\,abc{e}^{2}{x}^{2}+6\,{b}^{3}{e}^{2}{x}^{2}-48\,{b}^{2}cde{x}^{2}+48\,b{c}^{2}{d}^{2}{x}^{2}+24\,a{b}^{2}{e}^{2}x-48\,abcdex+48\,a{c}^{2}{d}^{2}x-12\,{b}^{3}dex+12\,{b}^{2}c{d}^{2}x+16\,{e}^{2}{a}^{2}b-32\,{a}^{2}cde-8\,a{b}^{2}de+24\,abc{d}^{2}-2\,{b}^{3}{d}^{2}}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(c*x^2+b*x+a)^(5/2),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((e*x + d)^2/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.377228, size = 412, normalized size = 4.2 \[ \frac{2 \,{\left (8 \, a^{2} b e^{2} + 2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e +{\left (b^{2} c + 4 \, a c^{2}\right )} e^{2}\right )} x^{3} -{\left (b^{3} - 12 \, a b c\right )} d^{2} - 4 \,{\left (a b^{2} + 4 \, a^{2} c\right )} d e + 3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e +{\left (b^{3} + 4 \, a b c\right )} e^{2}\right )} x^{2} + 6 \,{\left (2 \, a b^{2} e^{2} +{\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} + 4 \, a b c\right )} d e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x + a)^(5/2),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((e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23168, size = 387, normalized size = 3.95 \[ \frac{{\left ({\left (\frac{2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2} + 4 \, a c^{2} e^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2} + 4 \, a b c e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (b^{2} c d^{2} + 4 \, a c^{2} d^{2} - b^{3} d e - 4 \, a b c d e + 2 \, a b^{2} e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d^{2} - 12 \, a b c d^{2} + 4 \, a b^{2} d e + 16 \, a^{2} c d e - 8 \, a^{2} b e^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]