3.409 \(\int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx\)

Optimal. Leaf size=139 \[ \frac{2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2} \]

[Out]

(2*a^2*c^3*x^(5/2))/5 + (2*a*c^2*(2*b*c + 3*a*d)*x^(9/2))/9 + (2*c*(b^2*c^2 + 6*
a*b*c*d + 3*a^2*d^2)*x^(13/2))/13 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(17
/2))/17 + (2*b*d^2*(3*b*c + 2*a*d)*x^(21/2))/21 + (2*b^2*d^3*x^(25/2))/25

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Rubi [A]  time = 0.175668, antiderivative size = 139, 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 \[ \frac{2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*a^2*c^3*x^(5/2))/5 + (2*a*c^2*(2*b*c + 3*a*d)*x^(9/2))/9 + (2*c*(b^2*c^2 + 6*
a*b*c*d + 3*a^2*d^2)*x^(13/2))/13 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(17
/2))/17 + (2*b*d^2*(3*b*c + 2*a*d)*x^(21/2))/21 + (2*b^2*d^3*x^(25/2))/25

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Rubi in Sympy [A]  time = 29.827, size = 144, normalized size = 1.04 \[ \frac{2 a^{2} c^{3} x^{\frac{5}{2}}}{5} + \frac{2 a c^{2} x^{\frac{9}{2}} \left (3 a d + 2 b c\right )}{9} + \frac{2 b^{2} d^{3} x^{\frac{25}{2}}}{25} + \frac{2 b d^{2} x^{\frac{21}{2}} \left (2 a d + 3 b c\right )}{21} + \frac{2 c x^{\frac{13}{2}} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{13} + \frac{2 d x^{\frac{17}{2}} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c**3*x**(5/2)/5 + 2*a*c**2*x**(9/2)*(3*a*d + 2*b*c)/9 + 2*b**2*d**3*x**(2
5/2)/25 + 2*b*d**2*x**(21/2)*(2*a*d + 3*b*c)/21 + 2*c*x**(13/2)*(3*a**2*d**2 + 6
*a*b*c*d + b**2*c**2)/13 + 2*d*x**(17/2)*(a**2*d**2 + 6*a*b*c*d + 3*b**2*c**2)/1
7

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Mathematica [A]  time = 0.0609849, size = 139, normalized size = 1. \[ \frac{2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*c^3*x^(5/2))/5 + (2*a*c^2*(2*b*c + 3*a*d)*x^(9/2))/9 + (2*c*(b^2*c^2 + 6*
a*b*c*d + 3*a^2*d^2)*x^(13/2))/13 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(17
/2))/17 + (2*b*d^2*(3*b*c + 2*a*d)*x^(21/2))/21 + (2*b^2*d^3*x^(25/2))/25

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Maple [A]  time = 0.011, size = 138, normalized size = 1. \[{\frac{27846\,{b}^{2}{d}^{3}{x}^{10}+66300\,{x}^{8}ab{d}^{3}+99450\,{x}^{8}{b}^{2}c{d}^{2}+40950\,{x}^{6}{a}^{2}{d}^{3}+245700\,{x}^{6}abc{d}^{2}+122850\,{x}^{6}{b}^{2}{c}^{2}d+160650\,{x}^{4}{a}^{2}c{d}^{2}+321300\,{x}^{4}ab{c}^{2}d+53550\,{x}^{4}{b}^{2}{c}^{3}+232050\,{x}^{2}{a}^{2}{c}^{2}d+154700\,{x}^{2}ab{c}^{3}+139230\,{a}^{2}{c}^{3}}{348075}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(b*x^2+a)^2*(d*x^2+c)^3,x)

[Out]

2/348075*x^(5/2)*(13923*b^2*d^3*x^10+33150*a*b*d^3*x^8+49725*b^2*c*d^2*x^8+20475
*a^2*d^3*x^6+122850*a*b*c*d^2*x^6+61425*b^2*c^2*d*x^6+80325*a^2*c*d^2*x^4+160650
*a*b*c^2*d*x^4+26775*b^2*c^3*x^4+116025*a^2*c^2*d*x^2+77350*a*b*c^3*x^2+69615*a^
2*c^3)

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Maxima [A]  time = 1.3291, size = 171, normalized size = 1.23 \[ \frac{2}{25} \, b^{2} d^{3} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{17}{2}} + \frac{2}{5} \, a^{2} c^{3} x^{\frac{5}{2}} + \frac{2}{13} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*b^2*d^3*x^(25/2) + 2/21*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(21/2) + 2/17*(3*b^2*c^
2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(17/2) + 2/5*a^2*c^3*x^(5/2) + 2/13*(b^2*c^3 + 6*
a*b*c^2*d + 3*a^2*c*d^2)*x^(13/2) + 2/9*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(9/2)

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Fricas [A]  time = 0.221278, size = 178, normalized size = 1.28 \[ \frac{2}{348075} \,{\left (13923 \, b^{2} d^{3} x^{12} + 16575 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + 20475 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + 69615 \, a^{2} c^{3} x^{2} + 26775 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + 38675 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/348075*(13923*b^2*d^3*x^12 + 16575*(3*b^2*c*d^2 + 2*a*b*d^3)*x^10 + 20475*(3*b
^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^8 + 69615*a^2*c^3*x^2 + 26775*(b^2*c^3 + 6*a
*b*c^2*d + 3*a^2*c*d^2)*x^6 + 38675*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4)*sqrt(x)

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Sympy [A]  time = 73.2787, size = 192, normalized size = 1.38 \[ \frac{2 a^{2} c^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} c^{2} d x^{\frac{9}{2}}}{3} + \frac{6 a^{2} c d^{2} x^{\frac{13}{2}}}{13} + \frac{2 a^{2} d^{3} x^{\frac{17}{2}}}{17} + \frac{4 a b c^{3} x^{\frac{9}{2}}}{9} + \frac{12 a b c^{2} d x^{\frac{13}{2}}}{13} + \frac{12 a b c d^{2} x^{\frac{17}{2}}}{17} + \frac{4 a b d^{3} x^{\frac{21}{2}}}{21} + \frac{2 b^{2} c^{3} x^{\frac{13}{2}}}{13} + \frac{6 b^{2} c^{2} d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c d^{2} x^{\frac{21}{2}}}{7} + \frac{2 b^{2} d^{3} x^{\frac{25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*c**3*x**(5/2)/5 + 2*a**2*c**2*d*x**(9/2)/3 + 6*a**2*c*d**2*x**(13/2)/13 +
2*a**2*d**3*x**(17/2)/17 + 4*a*b*c**3*x**(9/2)/9 + 12*a*b*c**2*d*x**(13/2)/13 +
12*a*b*c*d**2*x**(17/2)/17 + 4*a*b*d**3*x**(21/2)/21 + 2*b**2*c**3*x**(13/2)/13
+ 6*b**2*c**2*d*x**(17/2)/17 + 2*b**2*c*d**2*x**(21/2)/7 + 2*b**2*d**3*x**(25/2
)/25

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GIAC/XCAS [A]  time = 0.229972, size = 182, normalized size = 1.31 \[ \frac{2}{25} \, b^{2} d^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c d^{2} x^{\frac{21}{2}} + \frac{4}{21} \, a b d^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c^{2} d x^{\frac{17}{2}} + \frac{12}{17} \, a b c d^{2} x^{\frac{17}{2}} + \frac{2}{17} \, a^{2} d^{3} x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c^{3} x^{\frac{13}{2}} + \frac{12}{13} \, a b c^{2} d x^{\frac{13}{2}} + \frac{6}{13} \, a^{2} c d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b c^{3} x^{\frac{9}{2}} + \frac{2}{3} \, a^{2} c^{2} d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c^{3} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/25*b^2*d^3*x^(25/2) + 2/7*b^2*c*d^2*x^(21/2) + 4/21*a*b*d^3*x^(21/2) + 6/17*b^
2*c^2*d*x^(17/2) + 12/17*a*b*c*d^2*x^(17/2) + 2/17*a^2*d^3*x^(17/2) + 2/13*b^2*c
^3*x^(13/2) + 12/13*a*b*c^2*d*x^(13/2) + 6/13*a^2*c*d^2*x^(13/2) + 4/9*a*b*c^3*x
^(9/2) + 2/3*a^2*c^2*d*x^(9/2) + 2/5*a^2*c^3*x^(5/2)