3.667 \(\int \sqrt{d x} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx\)

Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5} \]

[Out]

(2*a^2*(d*x)^(3/2))/(3*d) + (4*a*b*(d*x)^(7/2))/(7*d^3) + (2*b^2*(d*x)^(11/2))/(
11*d^5)

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Rubi [A]  time = 0.0391867, 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 \[ \frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*(d*x)^(3/2))/(3*d) + (4*a*b*(d*x)^(7/2))/(7*d^3) + (2*b^2*(d*x)^(11/2))/(
11*d^5)

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Rubi in Sympy [A]  time = 15.5266, size = 48, normalized size = 0.94 \[ \frac{2 a^{2} \left (d x\right )^{\frac{3}{2}}}{3 d} + \frac{4 a b \left (d x\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{2 b^{2} \left (d x\right )^{\frac{11}{2}}}{11 d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)*(d*x)**(1/2),x)

[Out]

2*a**2*(d*x)**(3/2)/(3*d) + 4*a*b*(d*x)**(7/2)/(7*d**3) + 2*b**2*(d*x)**(11/2)/(
11*d**5)

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Mathematica [A]  time = 0.0133564, size = 33, normalized size = 0.65 \[ \frac{2}{231} x \sqrt{d x} \left (77 a^2+66 a b x^2+21 b^2 x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[d*x]*(77*a^2 + 66*a*b*x^2 + 21*b^2*x^4))/231

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Maple [A]  time = 0.009, size = 30, normalized size = 0.6 \[{\frac{2\,x \left ( 21\,{b}^{2}{x}^{4}+66\,ab{x}^{2}+77\,{a}^{2} \right ) }{231}\sqrt{dx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)*(d*x)^(1/2),x)

[Out]

2/231*x*(21*b^2*x^4+66*a*b*x^2+77*a^2)*(d*x)^(1/2)

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Maxima [A]  time = 0.691685, size = 55, normalized size = 1.08 \[ \frac{2 \,{\left (21 \, \left (d x\right )^{\frac{11}{2}} b^{2} + 66 \, \left (d x\right )^{\frac{7}{2}} a b d^{2} + 77 \, \left (d x\right )^{\frac{3}{2}} a^{2} d^{4}\right )}}{231 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231*(21*(d*x)^(11/2)*b^2 + 66*(d*x)^(7/2)*a*b*d^2 + 77*(d*x)^(3/2)*a^2*d^4)/d^
5

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Fricas [A]  time = 0.256995, size = 39, normalized size = 0.76 \[ \frac{2}{231} \,{\left (21 \, b^{2} x^{5} + 66 \, a b x^{3} + 77 \, a^{2} x\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231*(21*b^2*x^5 + 66*a*b*x^3 + 77*a^2*x)*sqrt(d*x)

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Sympy [A]  time = 1.25467, size = 49, normalized size = 0.96 \[ \frac{2 a^{2} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{4 a b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{2 b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} \]

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)**(1/2),x)

[Out]

2*a**2*sqrt(d)*x**(3/2)/3 + 4*a*b*sqrt(d)*x**(7/2)/7 + 2*b**2*sqrt(d)*x**(11/2)/
11

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GIAC/XCAS [A]  time = 0.261993, size = 61, normalized size = 1.2 \[ \frac{2 \,{\left (21 \, \sqrt{d x} b^{2} d x^{5} + 66 \, \sqrt{d x} a b d x^{3} + 77 \, \sqrt{d x} a^{2} d x\right )}}{231 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231*(21*sqrt(d*x)*b^2*d*x^5 + 66*sqrt(d*x)*a*b*d*x^3 + 77*sqrt(d*x)*a^2*d*x)/d