3.666 \(\int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx\)

Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]

[Out]

(2*a^2*(d*x)^(5/2))/(5*d) + (4*a*b*(d*x)^(9/2))/(9*d^3) + (2*b^2*(d*x)^(13/2))/(
13*d^5)

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Rubi [A]  time = 0.0382421, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*a^2*(d*x)^(5/2))/(5*d) + (4*a*b*(d*x)^(9/2))/(9*d^3) + (2*b^2*(d*x)^(13/2))/(
13*d^5)

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Rubi in Sympy [A]  time = 15.5899, size = 48, normalized size = 0.94 \[ \frac{2 a^{2} \left (d x\right )^{\frac{5}{2}}}{5 d} + \frac{4 a b \left (d x\right )^{\frac{9}{2}}}{9 d^{3}} + \frac{2 b^{2} \left (d x\right )^{\frac{13}{2}}}{13 d^{5}} \]

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),x)

[Out]

2*a**2*(d*x)**(5/2)/(5*d) + 4*a*b*(d*x)**(9/2)/(9*d**3) + 2*b**2*(d*x)**(13/2)/(
13*d**5)

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Mathematica [A]  time = 0.0182458, size = 33, normalized size = 0.65 \[ \frac{2}{585} x (d x)^{3/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(2*x*(d*x)^(3/2)*(117*a^2 + 130*a*b*x^2 + 45*b^2*x^4))/585

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Maple [A]  time = 0.01, size = 30, normalized size = 0.6 \[{\frac{2\,x \left ( 45\,{b}^{2}{x}^{4}+130\,ab{x}^{2}+117\,{a}^{2} \right ) }{585} \left ( dx \right ) ^{{\frac{3}{2}}}} \]

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),x)

[Out]

2/585*x*(45*b^2*x^4+130*a*b*x^2+117*a^2)*(d*x)^(3/2)

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Maxima [A]  time = 0.681239, size = 55, normalized size = 1.08 \[ \frac{2 \,{\left (45 \, \left (d x\right )^{\frac{13}{2}} b^{2} + 130 \, \left (d x\right )^{\frac{9}{2}} a b d^{2} + 117 \, \left (d x\right )^{\frac{5}{2}} a^{2} d^{4}\right )}}{585 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(3/2),x, algorithm="maxima")

[Out]

2/585*(45*(d*x)^(13/2)*b^2 + 130*(d*x)^(9/2)*a*b*d^2 + 117*(d*x)^(5/2)*a^2*d^4)/
d^5

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Fricas [A]  time = 0.256475, size = 46, normalized size = 0.9 \[ \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 130 \, a b d x^{4} + 117 \, a^{2} d x^{2}\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(3/2),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d*x^6 + 130*a*b*d*x^4 + 117*a^2*d*x^2)*sqrt(d*x)

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Sympy [A]  time = 3.81795, size = 49, normalized size = 0.96 \[ \frac{2 a^{2} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a b d^{\frac{3}{2}} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} \]

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),x)

[Out]

2*a**2*d**(3/2)*x**(5/2)/5 + 4*a*b*d**(3/2)*x**(9/2)/9 + 2*b**2*d**(3/2)*x**(13/
2)/13

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GIAC/XCAS [A]  time = 0.26309, size = 57, normalized size = 1.12 \[ \frac{2}{13} \, \sqrt{d x} b^{2} d x^{6} + \frac{4}{9} \, \sqrt{d x} a b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{2} d x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(3/2),x, algorithm="giac")

[Out]

2/13*sqrt(d*x)*b^2*d*x^6 + 4/9*sqrt(d*x)*a*b*d*x^4 + 2/5*sqrt(d*x)*a^2*d*x^2