3.408 \(\int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx\)

Optimal. Leaf size=139 \[ \frac{2}{19} d x^{19/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{15} c x^{15/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^3 x^{7/2}+\frac{2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac{2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac{2}{27} b^2 d^3 x^{27/2} \]

[Out]

(2*a^2*c^3*x^(7/2))/7 + (2*a*c^2*(2*b*c + 3*a*d)*x^(11/2))/11 + (2*c*(b^2*c^2 +
6*a*b*c*d + 3*a^2*d^2)*x^(15/2))/15 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(
19/2))/19 + (2*b*d^2*(3*b*c + 2*a*d)*x^(23/2))/23 + (2*b^2*d^3*x^(27/2))/27

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Rubi [A]  time = 0.177869, 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 \[ \frac{2}{19} d x^{19/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{15} c x^{15/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^3 x^{7/2}+\frac{2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac{2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac{2}{27} b^2 d^3 x^{27/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*a^2*c^3*x^(7/2))/7 + (2*a*c^2*(2*b*c + 3*a*d)*x^(11/2))/11 + (2*c*(b^2*c^2 +
6*a*b*c*d + 3*a^2*d^2)*x^(15/2))/15 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(
19/2))/19 + (2*b*d^2*(3*b*c + 2*a*d)*x^(23/2))/23 + (2*b^2*d^3*x^(27/2))/27

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Rubi in Sympy [A]  time = 29.4869, size = 144, normalized size = 1.04 \[ \frac{2 a^{2} c^{3} x^{\frac{7}{2}}}{7} + \frac{2 a c^{2} x^{\frac{11}{2}} \left (3 a d + 2 b c\right )}{11} + \frac{2 b^{2} d^{3} x^{\frac{27}{2}}}{27} + \frac{2 b d^{2} x^{\frac{23}{2}} \left (2 a d + 3 b c\right )}{23} + \frac{2 c x^{\frac{15}{2}} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{15} + \frac{2 d x^{\frac{19}{2}} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

2*a**2*c**3*x**(7/2)/7 + 2*a*c**2*x**(11/2)*(3*a*d + 2*b*c)/11 + 2*b**2*d**3*x**
(27/2)/27 + 2*b*d**2*x**(23/2)*(2*a*d + 3*b*c)/23 + 2*c*x**(15/2)*(3*a**2*d**2 +
6*a*b*c*d + b**2*c**2)/15 + 2*d*x**(19/2)*(a**2*d**2 + 6*a*b*c*d + 3*b**2*c**2)
/19

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Mathematica [A]  time = 0.064363, size = 139, normalized size = 1. \[ \frac{2}{19} d x^{19/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{15} c x^{15/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^3 x^{7/2}+\frac{2}{11} a c^2 x^{11/2} (3 a d+2 b c)+\frac{2}{23} b d^2 x^{23/2} (2 a d+3 b c)+\frac{2}{27} b^2 d^3 x^{27/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3,x]

[Out]

(2*a^2*c^3*x^(7/2))/7 + (2*a*c^2*(2*b*c + 3*a*d)*x^(11/2))/11 + (2*c*(b^2*c^2 +
6*a*b*c*d + 3*a^2*d^2)*x^(15/2))/15 + (2*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^(
19/2))/19 + (2*b*d^2*(3*b*c + 2*a*d)*x^(23/2))/23 + (2*b^2*d^3*x^(27/2))/27

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Maple [A]  time = 0.01, size = 138, normalized size = 1. \[{\frac{336490\,{b}^{2}{d}^{3}{x}^{10}+790020\,{x}^{8}ab{d}^{3}+1185030\,{x}^{8}{b}^{2}c{d}^{2}+478170\,{x}^{6}{a}^{2}{d}^{3}+2869020\,{x}^{6}abc{d}^{2}+1434510\,{x}^{6}{b}^{2}{c}^{2}d+1817046\,{x}^{4}{a}^{2}c{d}^{2}+3634092\,{x}^{4}ab{c}^{2}d+605682\,{x}^{4}{b}^{2}{c}^{3}+2477790\,{x}^{2}{a}^{2}{c}^{2}d+1651860\,{x}^{2}ab{c}^{3}+1297890\,{a}^{2}{c}^{3}}{4542615}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^3,x)

[Out]

2/4542615*x^(7/2)*(168245*b^2*d^3*x^10+395010*a*b*d^3*x^8+592515*b^2*c*d^2*x^8+2
39085*a^2*d^3*x^6+1434510*a*b*c*d^2*x^6+717255*b^2*c^2*d*x^6+908523*a^2*c*d^2*x^
4+1817046*a*b*c^2*d*x^4+302841*b^2*c^3*x^4+1238895*a^2*c^2*d*x^2+825930*a*b*c^3*
x^2+648945*a^2*c^3)

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Maxima [A]  time = 1.36623, size = 171, normalized size = 1.23 \[ \frac{2}{27} \, b^{2} d^{3} x^{\frac{27}{2}} + \frac{2}{23} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{19}{2}} + \frac{2}{7} \, a^{2} c^{3} x^{\frac{7}{2}} + \frac{2}{15} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^3*x^(5/2),x, algorithm="maxima")

[Out]

2/27*b^2*d^3*x^(27/2) + 2/23*(3*b^2*c*d^2 + 2*a*b*d^3)*x^(23/2) + 2/19*(3*b^2*c^
2*d + 6*a*b*c*d^2 + a^2*d^3)*x^(19/2) + 2/7*a^2*c^3*x^(7/2) + 2/15*(b^2*c^3 + 6*
a*b*c^2*d + 3*a^2*c*d^2)*x^(15/2) + 2/11*(2*a*b*c^3 + 3*a^2*c^2*d)*x^(11/2)

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Fricas [A]  time = 0.218149, size = 178, normalized size = 1.28 \[ \frac{2}{4542615} \,{\left (168245 \, b^{2} d^{3} x^{13} + 197505 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{11} + 239085 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{9} + 648945 \, a^{2} c^{3} x^{3} + 302841 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + 412965 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^3*x^(5/2),x, algorithm="fricas")

[Out]

2/4542615*(168245*b^2*d^3*x^13 + 197505*(3*b^2*c*d^2 + 2*a*b*d^3)*x^11 + 239085*
(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^9 + 648945*a^2*c^3*x^3 + 302841*(b^2*c^3
+ 6*a*b*c^2*d + 3*a^2*c*d^2)*x^7 + 412965*(2*a*b*c^3 + 3*a^2*c^2*d)*x^5)*sqrt(x
)

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Sympy [A]  time = 125.663, size = 192, normalized size = 1.38 \[ \frac{2 a^{2} c^{3} x^{\frac{7}{2}}}{7} + \frac{6 a^{2} c^{2} d x^{\frac{11}{2}}}{11} + \frac{2 a^{2} c d^{2} x^{\frac{15}{2}}}{5} + \frac{2 a^{2} d^{3} x^{\frac{19}{2}}}{19} + \frac{4 a b c^{3} x^{\frac{11}{2}}}{11} + \frac{4 a b c^{2} d x^{\frac{15}{2}}}{5} + \frac{12 a b c d^{2} x^{\frac{19}{2}}}{19} + \frac{4 a b d^{3} x^{\frac{23}{2}}}{23} + \frac{2 b^{2} c^{3} x^{\frac{15}{2}}}{15} + \frac{6 b^{2} c^{2} d x^{\frac{19}{2}}}{19} + \frac{6 b^{2} c d^{2} x^{\frac{23}{2}}}{23} + \frac{2 b^{2} d^{3} x^{\frac{27}{2}}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

2*a**2*c**3*x**(7/2)/7 + 6*a**2*c**2*d*x**(11/2)/11 + 2*a**2*c*d**2*x**(15/2)/5
+ 2*a**2*d**3*x**(19/2)/19 + 4*a*b*c**3*x**(11/2)/11 + 4*a*b*c**2*d*x**(15/2)/5
+ 12*a*b*c*d**2*x**(19/2)/19 + 4*a*b*d**3*x**(23/2)/23 + 2*b**2*c**3*x**(15/2)/1
5 + 6*b**2*c**2*d*x**(19/2)/19 + 6*b**2*c*d**2*x**(23/2)/23 + 2*b**2*d**3*x**(27
/2)/27

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GIAC/XCAS [A]  time = 0.229959, size = 182, normalized size = 1.31 \[ \frac{2}{27} \, b^{2} d^{3} x^{\frac{27}{2}} + \frac{6}{23} \, b^{2} c d^{2} x^{\frac{23}{2}} + \frac{4}{23} \, a b d^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c^{2} d x^{\frac{19}{2}} + \frac{12}{19} \, a b c d^{2} x^{\frac{19}{2}} + \frac{2}{19} \, a^{2} d^{3} x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} c^{3} x^{\frac{15}{2}} + \frac{4}{5} \, a b c^{2} d x^{\frac{15}{2}} + \frac{2}{5} \, a^{2} c d^{2} x^{\frac{15}{2}} + \frac{4}{11} \, a b c^{3} x^{\frac{11}{2}} + \frac{6}{11} \, a^{2} c^{2} d x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} c^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^3*x^(5/2),x, algorithm="giac")

[Out]

2/27*b^2*d^3*x^(27/2) + 6/23*b^2*c*d^2*x^(23/2) + 4/23*a*b*d^3*x^(23/2) + 6/19*b
^2*c^2*d*x^(19/2) + 12/19*a*b*c*d^2*x^(19/2) + 2/19*a^2*d^3*x^(19/2) + 2/15*b^2*
c^3*x^(15/2) + 4/5*a*b*c^2*d*x^(15/2) + 2/5*a^2*c*d^2*x^(15/2) + 4/11*a*b*c^3*x^
(11/2) + 6/11*a^2*c^2*d*x^(11/2) + 2/7*a^2*c^3*x^(7/2)