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

Optimal. Leaf size=37 \[ \frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2} \]

[Out]

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

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Rubi [A]  time = 0.0481667, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{5} x^{5/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{9} b B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.97293, size = 39, normalized size = 1.05 \[ 2 A a \sqrt{x} + \frac{2 B b x^{\frac{9}{2}}}{9} + x^{\frac{5}{2}} \left (\frac{2 A b}{5} + \frac{2 B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0169575, size = 33, normalized size = 0.89 \[ \frac{2}{45} \sqrt{x} \left (9 x^2 (a B+A b)+45 a A+5 b B x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(45*a*A + 9*(A*b + a*B)*x^2 + 5*b*B*x^4))/45

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Maple [A]  time = 0.006, size = 32, normalized size = 0.9 \[{\frac{10\,bB{x}^{4}+18\,A{x}^{2}b+18\,B{x}^{2}a+90\,Aa}{45}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*x^(1/2)*(5*B*b*x^4+9*A*b*x^2+9*B*a*x^2+45*A*a)

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Maxima [A]  time = 1.34509, size = 36, normalized size = 0.97 \[ \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{5} \,{\left (B a + A b\right )} x^{\frac{5}{2}} + 2 \, A a \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222465, size = 39, normalized size = 1.05 \[ \frac{2}{45} \,{\left (5 \, B b x^{4} + 9 \,{\left (B a + A b\right )} x^{2} + 45 \, A a\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(5*B*b*x^4 + 9*(B*a + A*b)*x^2 + 45*A*a)*sqrt(x)

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Sympy [A]  time = 2.22367, size = 44, normalized size = 1.19 \[ 2 A a \sqrt{x} + \frac{2 A b x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{5}{2}}}{5} + \frac{2 B b x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213575, size = 39, normalized size = 1.05 \[ \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{5} \, B a x^{\frac{5}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} + 2 \, A a \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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