Optimal. Leaf size=53 \[ \frac{2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac{2 (a+b x)^{9/2}}{9 b^3}-\frac{4 a (a+b x)^{7/2}}{7 b^3} \]
[Out]
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Rubi [A] time = 0.0381957, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac{2 (a+b x)^{9/2}}{9 b^3}-\frac{4 a (a+b x)^{7/2}}{7 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.0538, size = 49, normalized size = 0.92 \[ \frac{2 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{4 a \left (a + b x\right )^{\frac{7}{2}}}{7 b^{3}} + \frac{2 \left (a + b x\right )^{\frac{9}{2}}}{9 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.025047, size = 35, normalized size = 0.66 \[ \frac{2 (a+b x)^{5/2} \left (8 a^2-20 a b x+35 b^2 x^2\right )}{315 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 32, normalized size = 0.6 \[{\frac{70\,{b}^{2}{x}^{2}-40\,abx+16\,{a}^{2}}{315\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.34642, size = 55, normalized size = 1.04 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}}}{9 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{7}{2}} a}{7 \, b^{3}} + \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21479, size = 72, normalized size = 1.36 \[ \frac{2 \,{\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt{b x + a}}{315 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.68273, size = 733, normalized size = 13.83 \[ \frac{16 a^{\frac{25}{2}} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{25}{2}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{40 a^{\frac{23}{2}} b x \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{23}{2}} b x}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{30 a^{\frac{21}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{21}{2}} b^{2} x^{2}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{110 a^{\frac{19}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{19}{2}} b^{3} x^{3}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{380 a^{\frac{17}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{516 a^{\frac{15}{2}} b^{5} x^{5} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{310 a^{\frac{13}{2}} b^{6} x^{6} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{70 a^{\frac{11}{2}} b^{7} x^{7} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236799, size = 153, normalized size = 2.89 \[ \frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} a}{b^{14}} + \frac{35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}}{b^{26}}\right )}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*x^2,x, algorithm="giac")
[Out]