Optimal. Leaf size=176 \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]
[Out]
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Rubi [A] time = 0.27205, antiderivative size = 176, 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 \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 38.7972, size = 201, normalized size = 1.14 \[ - \frac{2 A a^{3}}{\sqrt{x}} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} + 2 a^{2} \sqrt{x} \left (3 A b + B a\right ) + 2 a x^{\frac{3}{2}} \left (A a c + A b^{2} + B a b\right ) + \frac{2 c^{2} x^{\frac{11}{2}} \left (A c + 3 B b\right )}{11} + \frac{2 c x^{\frac{9}{2}} \left (A b c + B a c + B b^{2}\right )}{3} + x^{\frac{7}{2}} \left (\frac{6 A a c^{2}}{7} + \frac{6 A b^{2} c}{7} + \frac{12 B a b c}{7} + \frac{2 B b^{3}}{7}\right ) + x^{\frac{5}{2}} \left (\frac{12 A a b c}{5} + \frac{2 A b^{3}}{5} + \frac{6 B a^{2} c}{5} + \frac{6 B a b^{2}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.244813, size = 176, normalized size = 1. \[ -\frac{2 a^3 A}{\sqrt{x}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{3} c x^{9/2} \left (a B c+A b c+b^2 B\right )+2 a x^{3/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{7} x^{7/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{5} x^{5/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{11} c^2 x^{11/2} (A c+3 b B)+\frac{2}{13} B c^3 x^{13/2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 192, normalized size = 1.1 \[ -{\frac{-2310\,B{c}^{3}{x}^{7}-2730\,A{c}^{3}{x}^{6}-8190\,B{x}^{6}b{c}^{2}-10010\,A{x}^{5}b{c}^{2}-10010\,aB{c}^{2}{x}^{5}-10010\,B{x}^{5}{b}^{2}c-12870\,aA{c}^{2}{x}^{4}-12870\,A{x}^{4}{b}^{2}c-25740\,B{x}^{4}abc-4290\,B{x}^{4}{b}^{3}-36036\,A{x}^{3}abc-6006\,A{b}^{3}{x}^{3}-18018\,{a}^{2}Bc{x}^{3}-18018\,B{x}^{3}a{b}^{2}-30030\,{a}^{2}Ac{x}^{2}-30030\,A{x}^{2}a{b}^{2}-30030\,B{x}^{2}{a}^{2}b-90090\,A{a}^{2}bx-30030\,{a}^{3}Bx+30030\,A{a}^{3}}{15015}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^3/x^(3/2),x)
[Out]
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Maxima [A] time = 0.720326, size = 224, normalized size = 1.27 \[ \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{11}{2}} + \frac{2}{3} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{5} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{5}{2}} + 2 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{3}{2}} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.327639, size = 224, normalized size = 1.27 \[ \frac{2 \,{\left (1155 \, B c^{3} x^{7} + 1365 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 5005 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 2145 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 15015 \, A a^{3} + 3003 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 15015 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 15015 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15015 \, \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 36.0011, size = 284, normalized size = 1.61 \[ - \frac{2 A a^{3}}{\sqrt{x}} + 6 A a^{2} b \sqrt{x} + 2 A a^{2} c x^{\frac{3}{2}} + 2 A a b^{2} x^{\frac{3}{2}} + \frac{12 A a b c x^{\frac{5}{2}}}{5} + \frac{6 A a c^{2} x^{\frac{7}{2}}}{7} + \frac{2 A b^{3} x^{\frac{5}{2}}}{5} + \frac{6 A b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 A b c^{2} x^{\frac{9}{2}}}{3} + \frac{2 A c^{3} x^{\frac{11}{2}}}{11} + 2 B a^{3} \sqrt{x} + 2 B a^{2} b x^{\frac{3}{2}} + \frac{6 B a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{5}{2}}}{5} + \frac{12 B a b c x^{\frac{7}{2}}}{7} + \frac{2 B a c^{2} x^{\frac{9}{2}}}{3} + \frac{2 B b^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} c x^{\frac{9}{2}}}{3} + \frac{6 B b c^{2} x^{\frac{11}{2}}}{11} + \frac{2 B c^{3} x^{\frac{13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.281505, size = 261, normalized size = 1.48 \[ \frac{2}{13} \, B c^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B b c^{2} x^{\frac{11}{2}} + \frac{2}{11} \, A c^{3} x^{\frac{11}{2}} + \frac{2}{3} \, B b^{2} c x^{\frac{9}{2}} + \frac{2}{3} \, B a c^{2} x^{\frac{9}{2}} + \frac{2}{3} \, A b c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{3} x^{\frac{7}{2}} + \frac{12}{7} \, B a b c x^{\frac{7}{2}} + \frac{6}{7} \, A b^{2} c x^{\frac{7}{2}} + \frac{6}{7} \, A a c^{2} x^{\frac{7}{2}} + \frac{6}{5} \, B a b^{2} x^{\frac{5}{2}} + \frac{2}{5} \, A b^{3} x^{\frac{5}{2}} + \frac{6}{5} \, B a^{2} c x^{\frac{5}{2}} + \frac{12}{5} \, A a b c x^{\frac{5}{2}} + 2 \, B a^{2} b x^{\frac{3}{2}} + 2 \, A a b^{2} x^{\frac{3}{2}} + 2 \, A a^{2} c x^{\frac{3}{2}} + 2 \, B a^{3} \sqrt{x} + 6 \, A a^{2} b \sqrt{x} - \frac{2 \, A a^{3}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(3/2),x, algorithm="giac")
[Out]