3.306 \(\int x^{5/2} (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(11
/2))/11 + (2*b^2*(A*b + 3*a*B)*x^(13/2))/13 + (2*b^3*B*x^(15/2))/15

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Rubi [A]  time = 0.108089, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{13} b^2 x^{13/2} (3 a B+A b)+\frac{6}{11} a b x^{11/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(11
/2))/11 + (2*b^2*(A*b + 3*a*B)*x^(13/2))/13 + (2*b^3*B*x^(15/2))/15

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Rubi in Sympy [A]  time = 12.44, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} + \frac{2 a^{2} x^{\frac{9}{2}} \left (3 A b + B a\right )}{9} + \frac{6 a b x^{\frac{11}{2}} \left (A b + B a\right )}{11} + \frac{2 b^{2} x^{\frac{13}{2}} \left (A b + 3 B a\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(7/2)/7 + 2*B*b**3*x**(15/2)/15 + 2*a**2*x**(9/2)*(3*A*b + B*a)/9 +
6*a*b*x**(11/2)*(A*b + B*a)/11 + 2*b**2*x**(13/2)*(A*b + 3*B*a)/13

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Mathematica [A]  time = 0.0407242, size = 71, normalized size = 0.84 \[ \frac{2 x^{7/2} \left (715 a^3 (9 A+7 B x)+1365 a^2 b x (11 A+9 B x)+945 a b^2 x^2 (13 A+11 B x)+231 b^3 x^3 (15 A+13 B x)\right )}{45045} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(7/2)*(715*a^3*(9*A + 7*B*x) + 1365*a^2*b*x*(11*A + 9*B*x) + 945*a*b^2*x^2*
(13*A + 11*B*x) + 231*b^3*x^3*(15*A + 13*B*x)))/45045

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Maple [A]  time = 0.007, size = 76, normalized size = 0.9 \[{\frac{6006\,B{b}^{3}{x}^{4}+6930\,A{b}^{3}{x}^{3}+20790\,B{x}^{3}a{b}^{2}+24570\,aA{b}^{2}{x}^{2}+24570\,B{x}^{2}{a}^{2}b+30030\,{a}^{2}Abx+10010\,{a}^{3}Bx+12870\,{a}^{3}A}{45045}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*x^(7/2)*(3003*B*b^3*x^4+3465*A*b^3*x^3+10395*B*a*b^2*x^3+12285*A*a*b^2*x
^2+12285*B*a^2*b*x^2+15015*A*a^2*b*x+5005*B*a^3*x+6435*A*a^3)

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Maxima [A]  time = 1.39021, size = 99, normalized size = 1.16 \[ \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*B*b^3*x^(15/2) + 2/7*A*a^3*x^(7/2) + 2/13*(3*B*a*b^2 + A*b^3)*x^(13/2) + 6/
11*(B*a^2*b + A*a*b^2)*x^(11/2) + 2/9*(B*a^3 + 3*A*a^2*b)*x^(9/2)

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Fricas [A]  time = 0.205881, size = 105, normalized size = 1.24 \[ \frac{2}{45045} \,{\left (3003 \, B b^{3} x^{7} + 6435 \, A a^{3} x^{3} + 3465 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 12285 \,{\left (B a^{2} b + A a b^{2}\right )} x^{5} + 5005 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*B*b^3*x^7 + 6435*A*a^3*x^3 + 3465*(3*B*a*b^2 + A*b^3)*x^6 + 12285*
(B*a^2*b + A*a*b^2)*x^5 + 5005*(B*a^3 + 3*A*a^2*b)*x^4)*sqrt(x)

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Sympy [A]  time = 8.01276, size = 114, normalized size = 1.34 \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 A b^{3} x^{\frac{13}{2}}}{13} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 B a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(7/2)/7 + 2*A*a**2*b*x**(9/2)/3 + 6*A*a*b**2*x**(11/2)/11 + 2*A*b**3
*x**(13/2)/13 + 2*B*a**3*x**(9/2)/9 + 6*B*a**2*b*x**(11/2)/11 + 6*B*a*b**2*x**(1
3/2)/13 + 2*B*b**3*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.259042, size = 104, normalized size = 1.22 \[ \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a b^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*B*b^3*x^(15/2) + 6/13*B*a*b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 6/11*B*a^2*b
*x^(11/2) + 6/11*A*a*b^2*x^(11/2) + 2/9*B*a^3*x^(9/2) + 2/3*A*a^2*b*x^(9/2) + 2/
7*A*a^3*x^(7/2)