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

Optimal. Leaf size=85 \[ \frac{2}{9} a^3 A x^{9/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(
13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(15/2))/15 + (2*b^3*B*x^(17/2))/17

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Rubi [A]  time = 0.110355, 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 \[ \frac{2}{9} a^3 A x^{9/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{15} b^2 x^{15/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{17} b^3 B x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (6*a*b*(A*b + a*B)*x^(
13/2))/13 + (2*b^2*(A*b + 3*a*B)*x^(15/2))/15 + (2*b^3*B*x^(17/2))/17

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Rubi in Sympy [A]  time = 12.3685, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} + \frac{2 a^{2} x^{\frac{11}{2}} \left (3 A b + B a\right )}{11} + \frac{6 a b x^{\frac{13}{2}} \left (A b + B a\right )}{13} + \frac{2 b^{2} x^{\frac{15}{2}} \left (A b + 3 B a\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(9/2)/9 + 2*B*b**3*x**(17/2)/17 + 2*a**2*x**(11/2)*(3*A*b + B*a)/11
+ 6*a*b*x**(13/2)*(A*b + B*a)/13 + 2*b**2*x**(15/2)*(A*b + 3*B*a)/15

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Mathematica [A]  time = 0.0420973, size = 69, normalized size = 0.81 \[ \frac{2 x^{9/2} \left (12155 a^3 A+9945 a^2 x (a B+3 A b)+7293 b^2 x^3 (3 a B+A b)+25245 a b x^2 (a B+A b)+6435 b^3 B x^4\right )}{109395} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(9/2)*(12155*a^3*A + 9945*a^2*(3*A*b + a*B)*x + 25245*a*b*(A*b + a*B)*x^2 +
7293*b^2*(A*b + 3*a*B)*x^3 + 6435*b^3*B*x^4))/109395

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Maple [A]  time = 0.008, size = 76, normalized size = 0.9 \[{\frac{12870\,B{b}^{3}{x}^{4}+14586\,A{b}^{3}{x}^{3}+43758\,B{x}^{3}a{b}^{2}+50490\,aA{b}^{2}{x}^{2}+50490\,B{x}^{2}{a}^{2}b+59670\,{a}^{2}Abx+19890\,{a}^{3}Bx+24310\,{a}^{3}A}{109395}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/109395*x^(9/2)*(6435*B*b^3*x^4+7293*A*b^3*x^3+21879*B*a*b^2*x^3+25245*A*a*b^2*
x^2+25245*B*a^2*b*x^2+29835*A*a^2*b*x+9945*B*a^3*x+12155*A*a^3)

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Maxima [A]  time = 1.37426, size = 99, normalized size = 1.16 \[ \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} + \frac{2}{15} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b^3*x^(17/2) + 2/9*A*a^3*x^(9/2) + 2/15*(3*B*a*b^2 + A*b^3)*x^(15/2) + 6/
13*(B*a^2*b + A*a*b^2)*x^(13/2) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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Fricas [A]  time = 0.207559, size = 105, normalized size = 1.24 \[ \frac{2}{109395} \,{\left (6435 \, B b^{3} x^{8} + 12155 \, A a^{3} x^{4} + 7293 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 25245 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 9945 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/109395*(6435*B*b^3*x^8 + 12155*A*a^3*x^4 + 7293*(3*B*a*b^2 + A*b^3)*x^7 + 2524
5*(B*a^2*b + A*a*b^2)*x^6 + 9945*(B*a^3 + 3*A*a^2*b)*x^5)*sqrt(x)

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Sympy [A]  time = 30.6434, size = 114, normalized size = 1.34 \[ \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{6 A a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 A b^{3} x^{\frac{15}{2}}}{15} + \frac{2 B a^{3} x^{\frac{11}{2}}}{11} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{2 B a b^{2} x^{\frac{15}{2}}}{5} + \frac{2 B b^{3} x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(11/2)/11 + 6*A*a*b**2*x**(13/2)/13 + 2*A*b*
*3*x**(15/2)/15 + 2*B*a**3*x**(11/2)/11 + 6*B*a**2*b*x**(13/2)/13 + 2*B*a*b**2*x
**(15/2)/5 + 2*B*b**3*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.251853, size = 104, normalized size = 1.22 \[ \frac{2}{17} \, B b^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B a b^{2} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b^3*x^(17/2) + 2/5*B*a*b^2*x^(15/2) + 2/15*A*b^3*x^(15/2) + 6/13*B*a^2*b*
x^(13/2) + 6/13*A*a*b^2*x^(13/2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2) +
2/9*A*a^3*x^(9/2)