Optimal. Leaf size=89 \[ -\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]
[Out]
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Rubi [A] time = 0.104692, antiderivative size = 89, 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 \[ -\frac{2 a^4}{d \sqrt{d x}}+\frac{8 a^3 b (d x)^{3/2}}{3 d^3}+\frac{12 a^2 b^2 (d x)^{7/2}}{7 d^5}+\frac{8 a b^3 (d x)^{11/2}}{11 d^7}+\frac{2 b^4 (d x)^{15/2}}{15 d^9} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 25.8596, size = 87, normalized size = 0.98 \[ - \frac{2 a^{4}}{d \sqrt{d x}} + \frac{8 a^{3} b \left (d x\right )^{\frac{3}{2}}}{3 d^{3}} + \frac{12 a^{2} b^{2} \left (d x\right )^{\frac{7}{2}}}{7 d^{5}} + \frac{8 a b^{3} \left (d x\right )^{\frac{11}{2}}}{11 d^{7}} + \frac{2 b^{4} \left (d x\right )^{\frac{15}{2}}}{15 d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**2/(d*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0215086, size = 55, normalized size = 0.62 \[ \frac{2 x \left (-1155 a^4+1540 a^3 b x^2+990 a^2 b^2 x^4+420 a b^3 x^6+77 b^4 x^8\right )}{1155 (d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/(d*x)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 52, normalized size = 0.6 \[ -{\frac{ \left ( -154\,{b}^{4}{x}^{8}-840\,a{b}^{3}{x}^{6}-1980\,{a}^{2}{b}^{2}{x}^{4}-3080\,{a}^{3}b{x}^{2}+2310\,{a}^{4} \right ) x}{1155} \left ( dx \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^4+2*a*b*x^2+a^2)^2/(d*x)^(3/2),x)
[Out]
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Maxima [A] time = 0.693294, size = 103, normalized size = 1.16 \[ -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \left (d x\right )^{\frac{15}{2}} b^{4} + 420 \, \left (d x\right )^{\frac{11}{2}} a b^{3} d^{2} + 990 \, \left (d x\right )^{\frac{7}{2}} a^{2} b^{2} d^{4} + 1540 \, \left (d x\right )^{\frac{3}{2}} a^{3} b d^{6}}{d^{8}}\right )}}{1155 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/(d*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260165, size = 72, normalized size = 0.81 \[ \frac{2 \,{\left (77 \, b^{4} x^{8} + 420 \, a b^{3} x^{6} + 990 \, a^{2} b^{2} x^{4} + 1540 \, a^{3} b x^{2} - 1155 \, a^{4}\right )}}{1155 \, \sqrt{d x} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/(d*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.82865, size = 88, normalized size = 0.99 \[ - \frac{2 a^{4}}{d^{\frac{3}{2}} \sqrt{x}} + \frac{8 a^{3} b x^{\frac{3}{2}}}{3 d^{\frac{3}{2}}} + \frac{12 a^{2} b^{2} x^{\frac{7}{2}}}{7 d^{\frac{3}{2}}} + \frac{8 a b^{3} x^{\frac{11}{2}}}{11 d^{\frac{3}{2}}} + \frac{2 b^{4} x^{\frac{15}{2}}}{15 d^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**4+2*a*b*x**2+a**2)**2/(d*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.264769, size = 120, normalized size = 1.35 \[ -\frac{2 \,{\left (\frac{1155 \, a^{4}}{\sqrt{d x}} - \frac{77 \, \sqrt{d x} b^{4} d^{119} x^{7} + 420 \, \sqrt{d x} a b^{3} d^{119} x^{5} + 990 \, \sqrt{d x} a^{2} b^{2} d^{119} x^{3} + 1540 \, \sqrt{d x} a^{3} b d^{119} x}{d^{120}}\right )}}{1155 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/(d*x)^(3/2),x, algorithm="giac")
[Out]