3.355 \(\int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{\sqrt{x}} \, dx\)

Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(9/2))/9
+ (2*b^2*B*x^(13/2))/13

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Rubi [A]  time = 0.0878789, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(A + B*x^2))/Sqrt[x],x]

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(5/2))/5 + (2*b*(A*b + 2*a*B)*x^(9/2))/9
+ (2*b^2*B*x^(13/2))/13

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Rubi in Sympy [A]  time = 12.6494, size = 61, normalized size = 1. \[ 2 A a^{2} \sqrt{x} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{2 a x^{\frac{5}{2}} \left (2 A b + B a\right )}{5} + \frac{2 b x^{\frac{9}{2}} \left (A b + 2 B a\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 2*B*b**2*x**(13/2)/13 + 2*a*x**(5/2)*(2*A*b + B*a)/5 + 2*b*x*
*(9/2)*(A*b + 2*B*a)/9

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Mathematica [A]  time = 0.0328795, size = 53, normalized size = 0.87 \[ \frac{2}{585} \sqrt{x} \left (585 a^2 A+65 b x^4 (2 a B+A b)+117 a x^2 (a B+2 A b)+45 b^2 B x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(A + B*x^2))/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(585*a^2*A + 117*a*(2*A*b + a*B)*x^2 + 65*b*(A*b + 2*a*B)*x^4 + 45*b^
2*B*x^6))/585

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Maple [A]  time = 0.009, size = 56, normalized size = 0.9 \[{\frac{90\,{b}^{2}B{x}^{6}+130\,A{b}^{2}{x}^{4}+260\,{x}^{4}abB+468\,aAb{x}^{2}+234\,{x}^{2}{a}^{2}B+1170\,{a}^{2}A}{585}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*x^(1/2)*(45*B*b^2*x^6+65*A*b^2*x^4+130*B*a*b*x^4+234*A*a*b*x^2+117*B*a^2*x
^2+585*A*a^2)

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Maxima [A]  time = 1.33669, size = 69, normalized size = 1.13 \[ \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{9}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/sqrt(x),x, algorithm="maxima")

[Out]

2/13*B*b^2*x^(13/2) + 2/9*(2*B*a*b + A*b^2)*x^(9/2) + 2*A*a^2*sqrt(x) + 2/5*(B*a
^2 + 2*A*a*b)*x^(5/2)

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Fricas [A]  time = 0.217327, size = 72, normalized size = 1.18 \[ \frac{2}{585} \,{\left (45 \, B b^{2} x^{6} + 65 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 585 \, A a^{2} + 117 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/sqrt(x),x, algorithm="fricas")

[Out]

2/585*(45*B*b^2*x^6 + 65*(2*B*a*b + A*b^2)*x^4 + 585*A*a^2 + 117*(B*a^2 + 2*A*a*
b)*x^2)*sqrt(x)

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Sympy [A]  time = 6.71726, size = 78, normalized size = 1.28 \[ 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{5}{2}}}{5} + \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5} + \frac{4 B a b x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(B*x**2+A)/x**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(5/2)/5 + 2*A*b**2*x**(9/2)/9 + 2*B*a**2*x**(5/2)/
5 + 4*B*a*b*x**(9/2)/9 + 2*B*b**2*x**(13/2)/13

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GIAC/XCAS [A]  time = 0.209922, size = 72, normalized size = 1.18 \[ \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{9} \, B a b x^{\frac{9}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} + 2 \, A a^{2} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^2/sqrt(x),x, algorithm="giac")

[Out]

2/13*B*b^2*x^(13/2) + 4/9*B*a*b*x^(9/2) + 2/9*A*b^2*x^(9/2) + 2/5*B*a^2*x^(5/2)
+ 4/5*A*a*b*x^(5/2) + 2*A*a^2*sqrt(x)