Optimal. Leaf size=131 \[ \frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}} \]
[Out]
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Rubi [A] time = 0.152198, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 a^6 (d x)^{5/2}}{5 d}+\frac{4 a^5 b (d x)^{9/2}}{3 d^3}+\frac{30 a^4 b^2 (d x)^{13/2}}{13 d^5}+\frac{40 a^3 b^3 (d x)^{17/2}}{17 d^7}+\frac{10 a^2 b^4 (d x)^{21/2}}{7 d^9}+\frac{12 a b^5 (d x)^{25/2}}{25 d^{11}}+\frac{2 b^6 (d x)^{29/2}}{29 d^{13}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 36.0061, size = 129, normalized size = 0.98 \[ \frac{2 a^{6} \left (d x\right )^{\frac{5}{2}}}{5 d} + \frac{4 a^{5} b \left (d x\right )^{\frac{9}{2}}}{3 d^{3}} + \frac{30 a^{4} b^{2} \left (d x\right )^{\frac{13}{2}}}{13 d^{5}} + \frac{40 a^{3} b^{3} \left (d x\right )^{\frac{17}{2}}}{17 d^{7}} + \frac{10 a^{2} b^{4} \left (d x\right )^{\frac{21}{2}}}{7 d^{9}} + \frac{12 a b^{5} \left (d x\right )^{\frac{25}{2}}}{25 d^{11}} + \frac{2 b^{6} \left (d x\right )^{\frac{29}{2}}}{29 d^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.0297783, size = 77, normalized size = 0.59 \[ \frac{2 x (d x)^{3/2} \left (672945 a^6+2243150 a^5 b x^2+3882375 a^4 b^2 x^4+3958500 a^3 b^3 x^6+2403375 a^2 b^4 x^8+807534 a b^5 x^{10}+116025 b^6 x^{12}\right )}{3364725} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
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Maple [A] time = 0.011, size = 74, normalized size = 0.6 \[{\frac{2\,x \left ( 116025\,{b}^{6}{x}^{12}+807534\,a{b}^{5}{x}^{10}+2403375\,{a}^{2}{b}^{4}{x}^{8}+3958500\,{a}^{3}{b}^{3}{x}^{6}+3882375\,{a}^{4}{b}^{2}{x}^{4}+2243150\,{a}^{5}b{x}^{2}+672945\,{a}^{6} \right ) }{3364725} \left ( dx \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2)^3,x)
[Out]
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Maxima [A] time = 0.705166, size = 142, normalized size = 1.08 \[ \frac{2 \,{\left (116025 \, \left (d x\right )^{\frac{29}{2}} b^{6} + 807534 \, \left (d x\right )^{\frac{25}{2}} a b^{5} d^{2} + 2403375 \, \left (d x\right )^{\frac{21}{2}} a^{2} b^{4} d^{4} + 3958500 \, \left (d x\right )^{\frac{17}{2}} a^{3} b^{3} d^{6} + 3882375 \, \left (d x\right )^{\frac{13}{2}} a^{4} b^{2} d^{8} + 2243150 \, \left (d x\right )^{\frac{9}{2}} a^{5} b d^{10} + 672945 \, \left (d x\right )^{\frac{5}{2}} a^{6} d^{12}\right )}}{3364725 \, d^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256682, size = 111, normalized size = 0.85 \[ \frac{2}{3364725} \,{\left (116025 \, b^{6} d x^{14} + 807534 \, a b^{5} d x^{12} + 2403375 \, a^{2} b^{4} d x^{10} + 3958500 \, a^{3} b^{3} d x^{8} + 3882375 \, a^{4} b^{2} d x^{6} + 2243150 \, a^{5} b d x^{4} + 672945 \, a^{6} d x^{2}\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 19.3357, size = 131, normalized size = 1. \[ \frac{2 a^{6} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a^{5} b d^{\frac{3}{2}} x^{\frac{9}{2}}}{3} + \frac{30 a^{4} b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} + \frac{40 a^{3} b^{3} d^{\frac{3}{2}} x^{\frac{17}{2}}}{17} + \frac{10 a^{2} b^{4} d^{\frac{3}{2}} x^{\frac{21}{2}}}{7} + \frac{12 a b^{5} d^{\frac{3}{2}} x^{\frac{25}{2}}}{25} + \frac{2 b^{6} d^{\frac{3}{2}} x^{\frac{29}{2}}}{29} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.265365, size = 149, normalized size = 1.14 \[ \frac{2}{29} \, \sqrt{d x} b^{6} d x^{14} + \frac{12}{25} \, \sqrt{d x} a b^{5} d x^{12} + \frac{10}{7} \, \sqrt{d x} a^{2} b^{4} d x^{10} + \frac{40}{17} \, \sqrt{d x} a^{3} b^{3} d x^{8} + \frac{30}{13} \, \sqrt{d x} a^{4} b^{2} d x^{6} + \frac{4}{3} \, \sqrt{d x} a^{5} b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{6} d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(3/2),x, algorithm="giac")
[Out]