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

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{21} b x^{21/2} (2 a B+A b)+\frac{2}{15} a x^{15/2} (a B+2 A b)+\frac{2}{27} b^2 B x^{27/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(15/2))/15 + (2*b*(A*b + 2*a*B)*x^(21
/2))/21 + (2*b^2*B*x^(27/2))/27

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Rubi [A]  time = 0.104011, 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 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{21} b x^{21/2} (2 a B+A b)+\frac{2}{15} a x^{15/2} (a B+2 A b)+\frac{2}{27} b^2 B x^{27/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(15/2))/15 + (2*b*(A*b + 2*a*B)*x^(21
/2))/21 + (2*b^2*B*x^(27/2))/27

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Rubi in Sympy [A]  time = 10.7559, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{27}{2}}}{27} + \frac{2 a x^{\frac{15}{2}} \left (2 A b + B a\right )}{15} + \frac{2 b x^{\frac{21}{2}} \left (A b + 2 B a\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(9/2)/9 + 2*B*b**2*x**(27/2)/27 + 2*a*x**(15/2)*(2*A*b + B*a)/15 + 2
*b*x**(21/2)*(A*b + 2*B*a)/21

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Mathematica [A]  time = 0.0367779, size = 53, normalized size = 0.84 \[ \frac{2}{945} x^{9/2} \left (105 a^2 A+45 b x^6 (2 a B+A b)+63 a x^3 (a B+2 A b)+35 b^2 B x^9\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(105*a^2*A + 63*a*(2*A*b + a*B)*x^3 + 45*b*(A*b + 2*a*B)*x^6 + 35*b^2
*B*x^9))/945

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Maple [A]  time = 0.007, size = 56, normalized size = 0.9 \[{\frac{70\,B{x}^{9}{b}^{2}+90\,A{b}^{2}{x}^{6}+180\,B{x}^{6}ab+252\,aAb{x}^{3}+126\,B{x}^{3}{a}^{2}+210\,A{a}^{2}}{945}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/945*x^(9/2)*(35*B*b^2*x^9+45*A*b^2*x^6+90*B*a*b*x^6+126*A*a*b*x^3+63*B*a^2*x^3
+105*A*a^2)

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Maxima [A]  time = 1.61315, size = 69, normalized size = 1.1 \[ \frac{2}{27} \, B b^{2} x^{\frac{27}{2}} + \frac{2}{21} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{21}{2}} + \frac{2}{15} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*B*b^2*x^(27/2) + 2/21*(2*B*a*b + A*b^2)*x^(21/2) + 2/15*(B*a^2 + 2*A*a*b)*x
^(15/2) + 2/9*A*a^2*x^(9/2)

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Fricas [A]  time = 0.234867, size = 76, normalized size = 1.21 \[ \frac{2}{945} \,{\left (35 \, B b^{2} x^{13} + 45 \,{\left (2 \, B a b + A b^{2}\right )} x^{10} + 63 \,{\left (B a^{2} + 2 \, A a b\right )} x^{7} + 105 \, A a^{2} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/945*(35*B*b^2*x^13 + 45*(2*B*a*b + A*b^2)*x^10 + 63*(B*a^2 + 2*A*a*b)*x^7 + 10
5*A*a^2*x^4)*sqrt(x)

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Sympy [A]  time = 167.797, size = 80, normalized size = 1.27 \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{15}{2}}}{15} + \frac{2 A b^{2} x^{\frac{21}{2}}}{21} + \frac{2 B a^{2} x^{\frac{15}{2}}}{15} + \frac{4 B a b x^{\frac{21}{2}}}{21} + \frac{2 B b^{2} x^{\frac{27}{2}}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(15/2)/15 + 2*A*b**2*x**(21/2)/21 + 2*B*a**2*x*
*(15/2)/15 + 4*B*a*b*x**(21/2)/21 + 2*B*b**2*x**(27/2)/27

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GIAC/XCAS [A]  time = 0.215062, size = 72, normalized size = 1.14 \[ \frac{2}{27} \, B b^{2} x^{\frac{27}{2}} + \frac{4}{21} \, B a b x^{\frac{21}{2}} + \frac{2}{21} \, A b^{2} x^{\frac{21}{2}} + \frac{2}{15} \, B a^{2} x^{\frac{15}{2}} + \frac{4}{15} \, A a b x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*B*b^2*x^(27/2) + 4/21*B*a*b*x^(21/2) + 2/21*A*b^2*x^(21/2) + 2/15*B*a^2*x^(
15/2) + 4/15*A*a*b*x^(15/2) + 2/9*A*a^2*x^(9/2)