Optimal. Leaf size=90 \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.066327, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 9.94835, size = 88, normalized size = 0.98 \[ \frac{4 \left (2 b + 4 c x\right ) \left (2 A c - B b\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 \left (A b - 2 B a + x \left (2 A c - B b\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(c*x**2+b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.122737, size = 99, normalized size = 1.1 \[ -\frac{2 \left (B \left (8 a^2 c+2 a b (b+6 c x)+b x \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )+A (b+2 c x) \left (-4 c \left (3 a+2 c x^2\right )+b^2-8 b c x\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(a + b*x + c*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.009, size = 132, normalized size = 1.5 \[{\frac{32\,A{c}^{3}{x}^{3}-16\,B{x}^{3}b{c}^{2}+48\,Ab{c}^{2}{x}^{2}-24\,B{x}^{2}{b}^{2}c+48\,Axa{c}^{2}+12\,A{b}^{2}cx-24\,Bxabc-6\,Bx{b}^{3}+24\,Aabc-2\,A{b}^{3}-16\,B{a}^{2}c-4\,a{b}^{2}B}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(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((B*x + A)/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.431578, size = 331, normalized size = 3.68 \[ -\frac{2 \,{\left (2 \, B a b^{2} + A b^{3} + 8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 12 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c + 3 \,{\left (B b^{3} - 8 \, A a c^{2} + 2 \,{\left (2 \, B a b - A b^{2}\right )} c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(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((B*x+A)/(c*x**2+b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.290688, size = 293, normalized size = 3.26 \[ -\frac{{\left (4 \,{\left (\frac{2 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (B b^{3} + 4 \, B a b c - 2 \, A b^{2} c - 8 \, A a c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{2 \, B a b^{2} + A b^{3} + 8 \, B a^{2} c - 12 \, A a b c}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]