3.159 \(\int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{17} x^{17/2} (A c+b B)+\frac{2}{13} A b x^{13/2}+\frac{2}{21} B c x^{21/2} \]

[Out]

(2*A*b*x^(13/2))/13 + (2*(b*B + A*c)*x^(17/2))/17 + (2*B*c*x^(21/2))/21

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Rubi [A]  time = 0.063964, 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 \[ \frac{2}{17} x^{17/2} (A c+b B)+\frac{2}{13} A b x^{13/2}+\frac{2}{21} B c x^{21/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(2*A*b*x^(13/2))/13 + (2*(b*B + A*c)*x^(17/2))/17 + (2*B*c*x^(21/2))/21

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Rubi in Sympy [A]  time = 7.49571, size = 41, normalized size = 1.05 \[ \frac{2 A b x^{\frac{13}{2}}}{13} + \frac{2 B c x^{\frac{21}{2}}}{21} + x^{\frac{17}{2}} \left (\frac{2 A c}{17} + \frac{2 B b}{17}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x**2+A)*(c*x**4+b*x**2),x)

[Out]

2*A*b*x**(13/2)/13 + 2*B*c*x**(21/2)/21 + x**(17/2)*(2*A*c/17 + 2*B*b/17)

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Mathematica [A]  time = 0.0203631, size = 33, normalized size = 0.85 \[ \frac{2 x^{13/2} \left (273 x^2 (A c+b B)+357 A b+221 B c x^4\right )}{4641} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(2*x^(13/2)*(357*A*b + 273*(b*B + A*c)*x^2 + 221*B*c*x^4))/4641

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{442\,Bc{x}^{4}+546\,A{x}^{2}c+546\,Bb{x}^{2}+714\,Ab}{4641}{x}^{{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2),x)

[Out]

2/4641*x^(13/2)*(221*B*c*x^4+273*A*c*x^2+273*B*b*x^2+357*A*b)

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Maxima [A]  time = 1.36186, size = 36, normalized size = 0.92 \[ \frac{2}{21} \, B c x^{\frac{21}{2}} + \frac{2}{17} \,{\left (B b + A c\right )} x^{\frac{17}{2}} + \frac{2}{13} \, A b x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)*x^(7/2),x, algorithm="maxima")

[Out]

2/21*B*c*x^(21/2) + 2/17*(B*b + A*c)*x^(17/2) + 2/13*A*b*x^(13/2)

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Fricas [A]  time = 0.218452, size = 43, normalized size = 1.1 \[ \frac{2}{4641} \,{\left (221 \, B c x^{10} + 273 \,{\left (B b + A c\right )} x^{8} + 357 \, A b x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)*x^(7/2),x, algorithm="fricas")

[Out]

2/4641*(221*B*c*x^10 + 273*(B*b + A*c)*x^8 + 357*A*b*x^6)*sqrt(x)

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Sympy [A]  time = 35.7526, size = 46, normalized size = 1.18 \[ \frac{2 A b x^{\frac{13}{2}}}{13} + \frac{2 A c x^{\frac{17}{2}}}{17} + \frac{2 B b x^{\frac{17}{2}}}{17} + \frac{2 B c x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(B*x**2+A)*(c*x**4+b*x**2),x)

[Out]

2*A*b*x**(13/2)/13 + 2*A*c*x**(17/2)/17 + 2*B*b*x**(17/2)/17 + 2*B*c*x**(21/2)/2
1

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GIAC/XCAS [A]  time = 0.206931, size = 39, normalized size = 1. \[ \frac{2}{21} \, B c x^{\frac{21}{2}} + \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{17} \, A c x^{\frac{17}{2}} + \frac{2}{13} \, A b x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)*x^(7/2),x, algorithm="giac")

[Out]

2/21*B*c*x^(21/2) + 2/17*B*b*x^(17/2) + 2/17*A*c*x^(17/2) + 2/13*A*b*x^(13/2)