3.163 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{\sqrt{x}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{5} A b x^{5/2}+\frac{2}{13} B c x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0605734, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2}{9} x^{9/2} (A c+b B)+\frac{2}{5} A b x^{5/2}+\frac{2}{13} B c x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*(b*x^2 + c*x^4))/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.49306, size = 41, normalized size = 1.05 \[ \frac{2 A b x^{\frac{5}{2}}}{5} + \frac{2 B c x^{\frac{13}{2}}}{13} + x^{\frac{9}{2}} \left (\frac{2 A c}{9} + \frac{2 B b}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)/x**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0184208, size = 33, normalized size = 0.85 \[ \frac{2}{585} x^{5/2} \left (65 x^2 (A c+b B)+117 A b+45 B c x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/Sqrt[x],x]

[Out]

(2*x^(5/2)*(117*A*b + 65*(b*B + A*c)*x^2 + 45*B*c*x^4))/585

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Maple [A]  time = 0.006, size = 32, normalized size = 0.8 \[{\frac{90\,Bc{x}^{4}+130\,A{x}^{2}c+130\,Bb{x}^{2}+234\,Ab}{585}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)/x^(1/2),x)

[Out]

2/585*x^(5/2)*(45*B*c*x^4+65*A*c*x^2+65*B*b*x^2+117*A*b)

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Maxima [A]  time = 1.37022, size = 36, normalized size = 0.92 \[ \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{9} \,{\left (B b + A c\right )} x^{\frac{9}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/sqrt(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216066, size = 43, normalized size = 1.1 \[ \frac{2}{585} \,{\left (45 \, B c x^{6} + 65 \,{\left (B b + A c\right )} x^{4} + 117 \, A b x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/sqrt(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.16389, size = 46, normalized size = 1.18 \[ \frac{2 A b x^{\frac{5}{2}}}{5} + \frac{2 A c x^{\frac{9}{2}}}{9} + \frac{2 B b x^{\frac{9}{2}}}{9} + \frac{2 B c x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)/x**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207963, size = 39, normalized size = 1. \[ \frac{2}{13} \, B c x^{\frac{13}{2}} + \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{9} \, A c x^{\frac{9}{2}} + \frac{2}{5} \, A b x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)/sqrt(x),x, algorithm="giac")

[Out]

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