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

Optimal. Leaf size=97 \[ \frac{2}{17} x^{17/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2} \]

[Out]

(2*a^2*c^2*x^(9/2))/9 + (4*a*c*(b*c + a*d)*x^(13/2))/13 + (2*(b^2*c^2 + 4*a*b*c*
d + a^2*d^2)*x^(17/2))/17 + (4*b*d*(b*c + a*d)*x^(21/2))/21 + (2*b^2*d^2*x^(25/2
))/25

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Rubi [A]  time = 0.14452, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2}{17} x^{17/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(9/2))/9 + (4*a*c*(b*c + a*d)*x^(13/2))/13 + (2*(b^2*c^2 + 4*a*b*c*
d + a^2*d^2)*x^(17/2))/17 + (4*b*d*(b*c + a*d)*x^(21/2))/21 + (2*b^2*d^2*x^(25/2
))/25

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Rubi in Sympy [A]  time = 22.1815, size = 102, normalized size = 1.05 \[ \frac{2 a^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 a c x^{\frac{13}{2}} \left (a d + b c\right )}{13} + \frac{2 b^{2} d^{2} x^{\frac{25}{2}}}{25} + \frac{4 b d x^{\frac{21}{2}} \left (a d + b c\right )}{21} + x^{\frac{17}{2}} \left (\frac{2 a^{2} d^{2}}{17} + \frac{8 a b c d}{17} + \frac{2 b^{2} c^{2}}{17}\right ) \]

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)**2,x)

[Out]

2*a**2*c**2*x**(9/2)/9 + 4*a*c*x**(13/2)*(a*d + b*c)/13 + 2*b**2*d**2*x**(25/2)/
25 + 4*b*d*x**(21/2)*(a*d + b*c)/21 + x**(17/2)*(2*a**2*d**2/17 + 8*a*b*c*d/17 +
2*b**2*c**2/17)

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Mathematica [A]  time = 0.0536611, size = 97, normalized size = 1. \[ \frac{2}{17} x^{17/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^2*x^(9/2))/9 + (4*a*c*(b*c + a*d)*x^(13/2))/13 + (2*(b^2*c^2 + 4*a*b*c*
d + a^2*d^2)*x^(17/2))/17 + (4*b*d*(b*c + a*d)*x^(21/2))/21 + (2*b^2*d^2*x^(25/2
))/25

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Maple [A]  time = 0.01, size = 97, normalized size = 1. \[{\frac{27846\,{b}^{2}{d}^{2}{x}^{8}+66300\,{x}^{6}ab{d}^{2}+66300\,{x}^{6}{b}^{2}cd+40950\,{x}^{4}{a}^{2}{d}^{2}+163800\,{x}^{4}abcd+40950\,{x}^{4}{b}^{2}{c}^{2}+107100\,{x}^{2}{a}^{2}cd+107100\,a{c}^{2}b{x}^{2}+77350\,{a}^{2}{c}^{2}}{348075}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/348075*x^(9/2)*(13923*b^2*d^2*x^8+33150*a*b*d^2*x^6+33150*b^2*c*d*x^6+20475*a^
2*d^2*x^4+81900*a*b*c*d*x^4+20475*b^2*c^2*x^4+53550*a^2*c*d*x^2+53550*a*b*c^2*x^
2+38675*a^2*c^2)

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Maxima [A]  time = 1.35629, size = 115, normalized size = 1.19 \[ \frac{2}{25} \, b^{2} d^{2} x^{\frac{25}{2}} + \frac{4}{21} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, a^{2} c^{2} x^{\frac{9}{2}} + \frac{4}{13} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*b^2*d^2*x^(25/2) + 4/21*(b^2*c*d + a*b*d^2)*x^(21/2) + 2/17*(b^2*c^2 + 4*a*
b*c*d + a^2*d^2)*x^(17/2) + 2/9*a^2*c^2*x^(9/2) + 4/13*(a*b*c^2 + a^2*c*d)*x^(13
/2)

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Fricas [A]  time = 0.210854, size = 122, normalized size = 1.26 \[ \frac{2}{348075} \,{\left (13923 \, b^{2} d^{2} x^{12} + 33150 \,{\left (b^{2} c d + a b d^{2}\right )} x^{10} + 20475 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{8} + 38675 \, a^{2} c^{2} x^{4} + 53550 \,{\left (a b c^{2} + a^{2} c d\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/348075*(13923*b^2*d^2*x^12 + 33150*(b^2*c*d + a*b*d^2)*x^10 + 20475*(b^2*c^2 +
4*a*b*c*d + a^2*d^2)*x^8 + 38675*a^2*c^2*x^4 + 53550*(a*b*c^2 + a^2*c*d)*x^6)*s
qrt(x)

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Sympy [A]  time = 125.044, size = 136, normalized size = 1.4 \[ \frac{2 a^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 a^{2} c d x^{\frac{13}{2}}}{13} + \frac{2 a^{2} d^{2} x^{\frac{17}{2}}}{17} + \frac{4 a b c^{2} x^{\frac{13}{2}}}{13} + \frac{8 a b c d x^{\frac{17}{2}}}{17} + \frac{4 a b d^{2} x^{\frac{21}{2}}}{21} + \frac{2 b^{2} c^{2} x^{\frac{17}{2}}}{17} + \frac{4 b^{2} c d x^{\frac{21}{2}}}{21} + \frac{2 b^{2} d^{2} x^{\frac{25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c**2*x**(9/2)/9 + 4*a**2*c*d*x**(13/2)/13 + 2*a**2*d**2*x**(17/2)/17 + 4*
a*b*c**2*x**(13/2)/13 + 8*a*b*c*d*x**(17/2)/17 + 4*a*b*d**2*x**(21/2)/21 + 2*b**
2*c**2*x**(17/2)/17 + 4*b**2*c*d*x**(21/2)/21 + 2*b**2*d**2*x**(25/2)/25

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GIAC/XCAS [A]  time = 0.230477, size = 127, normalized size = 1.31 \[ \frac{2}{25} \, b^{2} d^{2} x^{\frac{25}{2}} + \frac{4}{21} \, b^{2} c d x^{\frac{21}{2}} + \frac{4}{21} \, a b d^{2} x^{\frac{21}{2}} + \frac{2}{17} \, b^{2} c^{2} x^{\frac{17}{2}} + \frac{8}{17} \, a b c d x^{\frac{17}{2}} + \frac{2}{17} \, a^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, a b c^{2} x^{\frac{13}{2}} + \frac{4}{13} \, a^{2} c d x^{\frac{13}{2}} + \frac{2}{9} \, a^{2} c^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*b^2*d^2*x^(25/2) + 4/21*b^2*c*d*x^(21/2) + 4/21*a*b*d^2*x^(21/2) + 2/17*b^2
*c^2*x^(17/2) + 8/17*a*b*c*d*x^(17/2) + 2/17*a^2*d^2*x^(17/2) + 4/13*a*b*c^2*x^(
13/2) + 4/13*a^2*c*d*x^(13/2) + 2/9*a^2*c^2*x^(9/2)