Optimal. Leaf size=178 \[ -\frac{2 a^3 A}{5 x^{5/2}}-\frac{2 a^2 (a B+3 A b)}{3 x^{3/2}}+\frac{6}{5} c x^{5/2} \left (a B c+A b c+b^2 B\right )-\frac{6 a \left (A \left (a c+b^2\right )+a b B\right )}{\sqrt{x}}+\frac{2}{3} x^{3/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+2 \sqrt{x} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{2}{9} B c^3 x^{9/2} \]
[Out]
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Rubi [A] time = 0.277916, antiderivative size = 178, 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 \[ -\frac{2 a^3 A}{5 x^{5/2}}-\frac{2 a^2 (a B+3 A b)}{3 x^{3/2}}+\frac{6}{5} c x^{5/2} \left (a B c+A b c+b^2 B\right )-\frac{6 a \left (A \left (a c+b^2\right )+a b B\right )}{\sqrt{x}}+\frac{2}{3} x^{3/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+2 \sqrt{x} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{7} c^2 x^{7/2} (A c+3 b B)+\frac{2}{9} B c^3 x^{9/2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 39.019, size = 190, normalized size = 1.07 \[ - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} - \frac{2 a^{2} \left (A b + \frac{B a}{3}\right )}{x^{\frac{3}{2}}} - \frac{6 a \left (A a c + A b^{2} + B a b\right )}{\sqrt{x}} + \frac{2 c^{2} x^{\frac{7}{2}} \left (A c + 3 B b\right )}{7} + \frac{6 c x^{\frac{5}{2}} \left (A b c + B a c + B b^{2}\right )}{5} + x^{\frac{3}{2}} \left (2 A a c^{2} + 2 A b^{2} c + 4 B a b c + \frac{2 B b^{3}}{3}\right ) + \sqrt{x} \left (12 A a b c + 2 A b^{3} + 6 B a^{2} c + 6 B a b^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.125615, size = 169, normalized size = 0.95 \[ \frac{2 \left (-21 a^3 (3 A+5 B x)-315 a^2 x (A (b+3 c x)+3 B x (b-c x))+63 a x^2 \left (5 A \left (-3 b^2+6 b c x+c^2 x^2\right )+B x \left (15 b^2+10 b c x+3 c^2 x^2\right )\right )+x^3 \left (9 A \left (35 b^3+35 b^2 c x+21 b c^2 x^2+5 c^3 x^3\right )+B x \left (105 b^3+189 b^2 c x+135 b c^2 x^2+35 c^3 x^3\right )\right )\right )}{315 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^(7/2),x]
[Out]
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Maple [A] time = 0.01, size = 192, normalized size = 1.1 \[ -{\frac{-70\,B{c}^{3}{x}^{7}-90\,A{c}^{3}{x}^{6}-270\,B{x}^{6}b{c}^{2}-378\,A{x}^{5}b{c}^{2}-378\,aB{c}^{2}{x}^{5}-378\,B{x}^{5}{b}^{2}c-630\,aA{c}^{2}{x}^{4}-630\,A{x}^{4}{b}^{2}c-1260\,B{x}^{4}abc-210\,B{x}^{4}{b}^{3}-3780\,A{x}^{3}abc-630\,A{b}^{3}{x}^{3}-1890\,{a}^{2}Bc{x}^{3}-1890\,B{x}^{3}a{b}^{2}+1890\,{a}^{2}Ac{x}^{2}+1890\,A{x}^{2}a{b}^{2}+1890\,B{x}^{2}{a}^{2}b+630\,A{a}^{2}bx+210\,{a}^{3}Bx+126\,A{a}^{3}}{315}{x}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^3/x^(7/2),x)
[Out]
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Maxima [A] time = 0.725859, size = 225, normalized size = 1.26 \[ \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{3}{2}} + 2 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{3} + 45 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279724, size = 224, normalized size = 1.26 \[ \frac{2 \,{\left (35 \, B c^{3} x^{7} + 45 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 189 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 105 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 63 \, A a^{3} + 315 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 945 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{315 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 22.1244, size = 275, normalized size = 1.54 \[ - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} - \frac{2 A a^{2} b}{x^{\frac{3}{2}}} - \frac{6 A a^{2} c}{\sqrt{x}} - \frac{6 A a b^{2}}{\sqrt{x}} + 12 A a b c \sqrt{x} + 2 A a c^{2} x^{\frac{3}{2}} + 2 A b^{3} \sqrt{x} + 2 A b^{2} c x^{\frac{3}{2}} + \frac{6 A b c^{2} x^{\frac{5}{2}}}{5} + \frac{2 A c^{3} x^{\frac{7}{2}}}{7} - \frac{2 B a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 B a^{2} b}{\sqrt{x}} + 6 B a^{2} c \sqrt{x} + 6 B a b^{2} \sqrt{x} + 4 B a b c x^{\frac{3}{2}} + \frac{6 B a c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{3} x^{\frac{3}{2}}}{3} + \frac{6 B b^{2} c x^{\frac{5}{2}}}{5} + \frac{6 B b c^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**3/x**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.271993, size = 259, normalized size = 1.46 \[ \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{6}{7} \, B b c^{2} x^{\frac{7}{2}} + \frac{2}{7} \, A c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B b^{2} c x^{\frac{5}{2}} + \frac{6}{5} \, B a c^{2} x^{\frac{5}{2}} + \frac{6}{5} \, A b c^{2} x^{\frac{5}{2}} + \frac{2}{3} \, B b^{3} x^{\frac{3}{2}} + 4 \, B a b c x^{\frac{3}{2}} + 2 \, A b^{2} c x^{\frac{3}{2}} + 2 \, A a c^{2} x^{\frac{3}{2}} + 6 \, B a b^{2} \sqrt{x} + 2 \, A b^{3} \sqrt{x} + 6 \, B a^{2} c \sqrt{x} + 12 \, A a b c \sqrt{x} - \frac{2 \,{\left (45 \, B a^{2} b x^{2} + 45 \, A a b^{2} x^{2} + 45 \, A a^{2} c x^{2} + 5 \, B a^{3} x + 15 \, A a^{2} b x + 3 \, A a^{3}\right )}}{15 \, x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^(7/2),x, algorithm="giac")
[Out]