3.280 \(\int x^{7/2} \left (a+b x^2\right )^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{2}{9} a^3 x^{9/2}+\frac{6}{13} a^2 b x^{13/2}+\frac{6}{17} a b^2 x^{17/2}+\frac{2}{21} b^3 x^{21/2} \]

[Out]

(2*a^3*x^(9/2))/9 + (6*a^2*b*x^(13/2))/13 + (6*a*b^2*x^(17/2))/17 + (2*b^3*x^(21
/2))/21

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Rubi [A]  time = 0.0425014, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2}{9} a^3 x^{9/2}+\frac{6}{13} a^2 b x^{13/2}+\frac{6}{17} a b^2 x^{17/2}+\frac{2}{21} b^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(a + b*x^2)^3,x]

[Out]

(2*a^3*x^(9/2))/9 + (6*a^2*b*x^(13/2))/13 + (6*a*b^2*x^(17/2))/17 + (2*b^3*x^(21
/2))/21

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Rubi in Sympy [A]  time = 6.35518, size = 49, normalized size = 0.96 \[ \frac{2 a^{3} x^{\frac{9}{2}}}{9} + \frac{6 a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(b*x**2+a)**3,x)

[Out]

2*a**3*x**(9/2)/9 + 6*a**2*b*x**(13/2)/13 + 6*a*b**2*x**(17/2)/17 + 2*b**3*x**(2
1/2)/21

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Mathematica [A]  time = 0.0143093, size = 41, normalized size = 0.8 \[ \frac{2 x^{9/2} \left (1547 a^3+3213 a^2 b x^2+2457 a b^2 x^4+663 b^3 x^6\right )}{13923} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(a + b*x^2)^3,x]

[Out]

(2*x^(9/2)*(1547*a^3 + 3213*a^2*b*x^2 + 2457*a*b^2*x^4 + 663*b^3*x^6))/13923

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Maple [A]  time = 0.008, size = 38, normalized size = 0.8 \[{\frac{1326\,{b}^{3}{x}^{6}+4914\,a{b}^{2}{x}^{4}+6426\,{a}^{2}b{x}^{2}+3094\,{a}^{3}}{13923}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x^2+a)^3,x)

[Out]

2/13923*x^(9/2)*(663*b^3*x^6+2457*a*b^2*x^4+3213*a^2*b*x^2+1547*a^3)

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Maxima [A]  time = 1.34791, size = 47, normalized size = 0.92 \[ \frac{2}{21} \, b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, a b^{2} x^{\frac{17}{2}} + \frac{6}{13} \, a^{2} b x^{\frac{13}{2}} + \frac{2}{9} \, a^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^(7/2),x, algorithm="maxima")

[Out]

2/21*b^3*x^(21/2) + 6/17*a*b^2*x^(17/2) + 6/13*a^2*b*x^(13/2) + 2/9*a^3*x^(9/2)

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Fricas [A]  time = 0.211878, size = 54, normalized size = 1.06 \[ \frac{2}{13923} \,{\left (663 \, b^{3} x^{10} + 2457 \, a b^{2} x^{8} + 3213 \, a^{2} b x^{6} + 1547 \, a^{3} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^(7/2),x, algorithm="fricas")

[Out]

2/13923*(663*b^3*x^10 + 2457*a*b^2*x^8 + 3213*a^2*b*x^6 + 1547*a^3*x^4)*sqrt(x)

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Sympy [A]  time = 70.7934, size = 49, normalized size = 0.96 \[ \frac{2 a^{3} x^{\frac{9}{2}}}{9} + \frac{6 a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(b*x**2+a)**3,x)

[Out]

2*a**3*x**(9/2)/9 + 6*a**2*b*x**(13/2)/13 + 6*a*b**2*x**(17/2)/17 + 2*b**3*x**(2
1/2)/21

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GIAC/XCAS [A]  time = 0.206874, size = 47, normalized size = 0.92 \[ \frac{2}{21} \, b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, a b^{2} x^{\frac{17}{2}} + \frac{6}{13} \, a^{2} b x^{\frac{13}{2}} + \frac{2}{9} \, a^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^(7/2),x, algorithm="giac")

[Out]

2/21*b^3*x^(21/2) + 6/17*a*b^2*x^(17/2) + 6/13*a^2*b*x^(13/2) + 2/9*a^3*x^(9/2)