3.391 \(\int x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 c x^{9/2}+\frac{2}{17} b x^{17/2} (2 a d+b c)+\frac{2}{13} a x^{13/2} (a d+2 b c)+\frac{2}{21} b^2 d x^{21/2} \]

[Out]

(2*a^2*c*x^(9/2))/9 + (2*a*(2*b*c + a*d)*x^(13/2))/13 + (2*b*(b*c + 2*a*d)*x^(17
/2))/17 + (2*b^2*d*x^(21/2))/21

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Rubi [A]  time = 0.0932629, antiderivative size = 63, 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 \[ \frac{2}{9} a^2 c x^{9/2}+\frac{2}{17} b x^{17/2} (2 a d+b c)+\frac{2}{13} a x^{13/2} (a d+2 b c)+\frac{2}{21} b^2 d x^{21/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(a + b*x^2)^2*(c + d*x^2),x]

[Out]

(2*a^2*c*x^(9/2))/9 + (2*a*(2*b*c + a*d)*x^(13/2))/13 + (2*b*(b*c + 2*a*d)*x^(17
/2))/17 + (2*b^2*d*x^(21/2))/21

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Rubi in Sympy [A]  time = 12.0578, size = 63, normalized size = 1. \[ \frac{2 a^{2} c x^{\frac{9}{2}}}{9} + \frac{2 a x^{\frac{13}{2}} \left (a d + 2 b c\right )}{13} + \frac{2 b^{2} d x^{\frac{21}{2}}}{21} + \frac{2 b x^{\frac{17}{2}} \left (2 a d + b c\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c*x**(9/2)/9 + 2*a*x**(13/2)*(a*d + 2*b*c)/13 + 2*b**2*d*x**(21/2)/21 + 2
*b*x**(17/2)*(2*a*d + b*c)/17

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Mathematica [A]  time = 0.0347354, size = 53, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (1547 a^2 c+819 b x^4 (2 a d+b c)+1071 a x^2 (a d+2 b c)+663 b^2 d x^6\right )}{13923} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(1547*a^2*c + 1071*a*(2*b*c + a*d)*x^2 + 819*b*(b*c + 2*a*d)*x^4 + 66
3*b^2*d*x^6))/13923

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[{\frac{1326\,{b}^{2}d{x}^{6}+3276\,{x}^{4}abd+1638\,{b}^{2}c{x}^{4}+2142\,{x}^{2}{a}^{2}d+4284\,abc{x}^{2}+3094\,{a}^{2}c}{13923}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x^2+a)^2*(d*x^2+c),x)

[Out]

2/13923*x^(9/2)*(663*b^2*d*x^6+1638*a*b*d*x^4+819*b^2*c*x^4+1071*a^2*d*x^2+2142*
a*b*c*x^2+1547*a^2*c)

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Maxima [A]  time = 1.35182, size = 69, normalized size = 1.1 \[ \frac{2}{21} \, b^{2} d x^{\frac{21}{2}} + \frac{2}{17} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{17}{2}} + \frac{2}{9} \, a^{2} c x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*b^2*d*x^(21/2) + 2/17*(b^2*c + 2*a*b*d)*x^(17/2) + 2/9*a^2*c*x^(9/2) + 2/13
*(2*a*b*c + a^2*d)*x^(13/2)

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Fricas [A]  time = 0.216111, size = 76, normalized size = 1.21 \[ \frac{2}{13923} \,{\left (663 \, b^{2} d x^{10} + 819 \,{\left (b^{2} c + 2 \, a b d\right )} x^{8} + 1547 \, a^{2} c x^{4} + 1071 \,{\left (2 \, a b c + a^{2} d\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13923*(663*b^2*d*x^10 + 819*(b^2*c + 2*a*b*d)*x^8 + 1547*a^2*c*x^4 + 1071*(2*a
*b*c + a^2*d)*x^6)*sqrt(x)

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Sympy [A]  time = 70.9966, size = 80, normalized size = 1.27 \[ \frac{2 a^{2} c x^{\frac{9}{2}}}{9} + \frac{2 a^{2} d x^{\frac{13}{2}}}{13} + \frac{4 a b c x^{\frac{13}{2}}}{13} + \frac{4 a b d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c x^{\frac{17}{2}}}{17} + \frac{2 b^{2} d x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c*x**(9/2)/9 + 2*a**2*d*x**(13/2)/13 + 4*a*b*c*x**(13/2)/13 + 4*a*b*d*x**
(17/2)/17 + 2*b**2*c*x**(17/2)/17 + 2*b**2*d*x**(21/2)/21

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GIAC/XCAS [A]  time = 0.227682, size = 72, normalized size = 1.14 \[ \frac{2}{21} \, b^{2} d x^{\frac{21}{2}} + \frac{2}{17} \, b^{2} c x^{\frac{17}{2}} + \frac{4}{17} \, a b d x^{\frac{17}{2}} + \frac{4}{13} \, a b c x^{\frac{13}{2}} + \frac{2}{13} \, a^{2} d x^{\frac{13}{2}} + \frac{2}{9} \, a^{2} c x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*b^2*d*x^(21/2) + 2/17*b^2*c*x^(17/2) + 4/17*a*b*d*x^(17/2) + 4/13*a*b*c*x^(
13/2) + 2/13*a^2*d*x^(13/2) + 2/9*a^2*c*x^(9/2)