Optimal. Leaf size=109 \[ -\frac{2 a^2 A}{5 x^{5/2}}+2 \sqrt{x} \left (2 a B c+2 A b c+b^2 B\right )-\frac{2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{\sqrt{x}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+\frac{2}{5} B c^2 x^{5/2} \]
[Out]
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Rubi [A] time = 0.143145, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 a^2 A}{5 x^{5/2}}+2 \sqrt{x} \left (2 a B c+2 A b c+b^2 B\right )-\frac{2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{\sqrt{x}}-\frac{2 a (a B+2 A b)}{3 x^{3/2}}+\frac{2}{3} c x^{3/2} (A c+2 b B)+\frac{2}{5} B c^2 x^{5/2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 19.7268, size = 114, normalized size = 1.05 \[ - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} + \frac{2 B c^{2} x^{\frac{5}{2}}}{5} - \frac{2 a \left (2 A b + B a\right )}{3 x^{\frac{3}{2}}} + \frac{2 c x^{\frac{3}{2}} \left (A c + 2 B b\right )}{3} + \sqrt{x} \left (4 A b c + 4 B a c + 2 B b^{2}\right ) - \frac{4 A a c + 2 A b^{2} + 4 B a b}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**2/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.076003, size = 95, normalized size = 0.87 \[ \frac{-2 a^2 (3 A+5 B x)-20 a x (A (b+3 c x)+3 B x (b-c x))+2 x^2 \left (5 A \left (-3 b^2+6 b c x+c^2 x^2\right )+B x \left (15 b^2+10 b c x+3 c^2 x^2\right )\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^(7/2),x]
[Out]
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Maple [A] time = 0.01, size = 102, normalized size = 0.9 \[ -{\frac{-6\,B{c}^{2}{x}^{5}-10\,A{c}^{2}{x}^{4}-20\,B{x}^{4}bc-60\,A{x}^{3}bc-60\,aBc{x}^{3}-30\,B{b}^{2}{x}^{3}+60\,aAc{x}^{2}+30\,A{b}^{2}{x}^{2}+60\,B{x}^{2}ab+20\,aAbx+10\,{a}^{2}Bx+6\,A{a}^{2}}{15}{x}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^2/x^(7/2),x)
[Out]
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Maxima [A] time = 0.719184, size = 127, normalized size = 1.17 \[ \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{3}{2}} + 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 5 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277649, size = 126, normalized size = 1.16 \[ \frac{2 \,{\left (3 \, B c^{2} x^{5} + 5 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} - 3 \, A a^{2} - 15 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 5 \,{\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.99523, size = 151, normalized size = 1.39 \[ - \frac{2 A a^{2}}{5 x^{\frac{5}{2}}} - \frac{4 A a b}{3 x^{\frac{3}{2}}} - \frac{4 A a c}{\sqrt{x}} - \frac{2 A b^{2}}{\sqrt{x}} + 4 A b c \sqrt{x} + \frac{2 A c^{2} x^{\frac{3}{2}}}{3} - \frac{2 B a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 B a b}{\sqrt{x}} + 4 B a c \sqrt{x} + 2 B b^{2} \sqrt{x} + \frac{4 B b c x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} x^{\frac{5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**2/x**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.268511, size = 138, normalized size = 1.27 \[ \frac{2}{5} \, B c^{2} x^{\frac{5}{2}} + \frac{4}{3} \, B b c x^{\frac{3}{2}} + \frac{2}{3} \, A c^{2} x^{\frac{3}{2}} + 2 \, B b^{2} \sqrt{x} + 4 \, B a c \sqrt{x} + 4 \, A b c \sqrt{x} - \frac{2 \,{\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 30 \, A a c x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)/x^(7/2),x, algorithm="giac")
[Out]