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

Optimal. Leaf size=91 \[ \frac{2 a^4 (d x)^{3/2}}{3 d}+\frac{8 a^3 b (d x)^{7/2}}{7 d^3}+\frac{12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a b^3 (d x)^{15/2}}{15 d^7}+\frac{2 b^4 (d x)^{19/2}}{19 d^9} \]

[Out]

(2*a^4*(d*x)^(3/2))/(3*d) + (8*a^3*b*(d*x)^(7/2))/(7*d^3) + (12*a^2*b^2*(d*x)^(1
1/2))/(11*d^5) + (8*a*b^3*(d*x)^(15/2))/(15*d^7) + (2*b^4*(d*x)^(19/2))/(19*d^9)

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Rubi [A]  time = 0.103946, antiderivative size = 91, 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 \[ \frac{2 a^4 (d x)^{3/2}}{3 d}+\frac{8 a^3 b (d x)^{7/2}}{7 d^3}+\frac{12 a^2 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a b^3 (d x)^{15/2}}{15 d^7}+\frac{2 b^4 (d x)^{19/2}}{19 d^9} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^4*(d*x)^(3/2))/(3*d) + (8*a^3*b*(d*x)^(7/2))/(7*d^3) + (12*a^2*b^2*(d*x)^(1
1/2))/(11*d^5) + (8*a*b^3*(d*x)^(15/2))/(15*d^7) + (2*b^4*(d*x)^(19/2))/(19*d^9)

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Rubi in Sympy [A]  time = 25.6687, size = 88, normalized size = 0.97 \[ \frac{2 a^{4} \left (d x\right )^{\frac{3}{2}}}{3 d} + \frac{8 a^{3} b \left (d x\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{12 a^{2} b^{2} \left (d x\right )^{\frac{11}{2}}}{11 d^{5}} + \frac{8 a b^{3} \left (d x\right )^{\frac{15}{2}}}{15 d^{7}} + \frac{2 b^{4} \left (d x\right )^{\frac{19}{2}}}{19 d^{9}} \]

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)**(1/2),x)

[Out]

2*a**4*(d*x)**(3/2)/(3*d) + 8*a**3*b*(d*x)**(7/2)/(7*d**3) + 12*a**2*b**2*(d*x)*
*(11/2)/(11*d**5) + 8*a*b**3*(d*x)**(15/2)/(15*d**7) + 2*b**4*(d*x)**(19/2)/(19*
d**9)

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Mathematica [A]  time = 0.0171594, size = 55, normalized size = 0.6 \[ \frac{2 x \sqrt{d x} \left (7315 a^4+12540 a^3 b x^2+11970 a^2 b^2 x^4+5852 a b^3 x^6+1155 b^4 x^8\right )}{21945} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[d*x]*(7315*a^4 + 12540*a^3*b*x^2 + 11970*a^2*b^2*x^4 + 5852*a*b^3*x^6
+ 1155*b^4*x^8))/21945

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Maple [A]  time = 0.01, size = 52, normalized size = 0.6 \[{\frac{2\,x \left ( 1155\,{b}^{4}{x}^{8}+5852\,a{b}^{3}{x}^{6}+11970\,{a}^{2}{b}^{2}{x}^{4}+12540\,{a}^{3}b{x}^{2}+7315\,{a}^{4} \right ) }{21945}\sqrt{dx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*x*(1155*b^4*x^8+5852*a*b^3*x^6+11970*a^2*b^2*x^4+12540*a^3*b*x^2+7315*a^
4)*(d*x)^(1/2)

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Maxima [A]  time = 0.675555, size = 99, normalized size = 1.09 \[ \frac{2 \,{\left (1155 \, \left (d x\right )^{\frac{19}{2}} b^{4} + 5852 \, \left (d x\right )^{\frac{15}{2}} a b^{3} d^{2} + 11970 \, \left (d x\right )^{\frac{11}{2}} a^{2} b^{2} d^{4} + 12540 \, \left (d x\right )^{\frac{7}{2}} a^{3} b d^{6} + 7315 \, \left (d x\right )^{\frac{3}{2}} a^{4} d^{8}\right )}}{21945 \, d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*(1155*(d*x)^(19/2)*b^4 + 5852*(d*x)^(15/2)*a*b^3*d^2 + 11970*(d*x)^(11/2
)*a^2*b^2*d^4 + 12540*(d*x)^(7/2)*a^3*b*d^6 + 7315*(d*x)^(3/2)*a^4*d^8)/d^9

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Fricas [A]  time = 0.256566, size = 69, normalized size = 0.76 \[ \frac{2}{21945} \,{\left (1155 \, b^{4} x^{9} + 5852 \, a b^{3} x^{7} + 11970 \, a^{2} b^{2} x^{5} + 12540 \, a^{3} b x^{3} + 7315 \, a^{4} x\right )} \sqrt{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*(1155*b^4*x^9 + 5852*a*b^3*x^7 + 11970*a^2*b^2*x^5 + 12540*a^3*b*x^3 + 7
315*a^4*x)*sqrt(d*x)

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Sympy [A]  time = 4.08404, size = 90, normalized size = 0.99 \[ \frac{2 a^{4} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{8 a^{3} b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{12 a^{2} b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} + \frac{8 a b^{3} \sqrt{d} x^{\frac{15}{2}}}{15} + \frac{2 b^{4} \sqrt{d} x^{\frac{19}{2}}}{19} \]

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)**(1/2),x)

[Out]

2*a**4*sqrt(d)*x**(3/2)/3 + 8*a**3*b*sqrt(d)*x**(7/2)/7 + 12*a**2*b**2*sqrt(d)*x
**(11/2)/11 + 8*a*b**3*sqrt(d)*x**(15/2)/15 + 2*b**4*sqrt(d)*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.264035, size = 107, normalized size = 1.18 \[ \frac{2 \,{\left (1155 \, \sqrt{d x} b^{4} d x^{9} + 5852 \, \sqrt{d x} a b^{3} d x^{7} + 11970 \, \sqrt{d x} a^{2} b^{2} d x^{5} + 12540 \, \sqrt{d x} a^{3} b d x^{3} + 7315 \, \sqrt{d x} a^{4} d x\right )}}{21945 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*(1155*sqrt(d*x)*b^4*d*x^9 + 5852*sqrt(d*x)*a*b^3*d*x^7 + 11970*sqrt(d*x)
*a^2*b^2*d*x^5 + 12540*sqrt(d*x)*a^3*b*d*x^3 + 7315*sqrt(d*x)*a^4*d*x)/d