3.1000 \(\int x^{5/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx\)

Optimal. Leaf size=182 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{6}{17} c x^{17/2} \left (a B c+A b c+b^2 B\right )+\frac{6}{11} a x^{11/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{15} x^{15/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{13} x^{13/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{21} B c^3 x^{21/2} \]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*(a*b*B + A*(b^2 + a
*c))*x^(11/2))/11 + (2*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(13/2))/13 + (2
*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(15/2))/15 + (6*c*(b^2*B + A*b*c
+ a*B*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(19/2))/19 + (2*B*c^3*x^(21/2))/2
1

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Rubi [A]  time = 0.275547, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{6}{17} c x^{17/2} \left (a B c+A b c+b^2 B\right )+\frac{6}{11} a x^{11/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{15} x^{15/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{13} x^{13/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{21} B c^3 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*(a*b*B + A*(b^2 + a
*c))*x^(11/2))/11 + (2*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(13/2))/13 + (2
*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(15/2))/15 + (6*c*(b^2*B + A*b*c
+ a*B*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(19/2))/19 + (2*B*c^3*x^(21/2))/2
1

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Rubi in Sympy [A]  time = 38.7833, size = 206, normalized size = 1.13 \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 B c^{3} x^{\frac{21}{2}}}{21} + \frac{2 a^{2} x^{\frac{9}{2}} \left (3 A b + B a\right )}{9} + \frac{6 a x^{\frac{11}{2}} \left (A a c + A b^{2} + B a b\right )}{11} + \frac{2 c^{2} x^{\frac{19}{2}} \left (A c + 3 B b\right )}{19} + \frac{6 c x^{\frac{17}{2}} \left (A b c + B a c + B b^{2}\right )}{17} + x^{\frac{15}{2}} \left (\frac{2 A a c^{2}}{5} + \frac{2 A b^{2} c}{5} + \frac{4 B a b c}{5} + \frac{2 B b^{3}}{15}\right ) + x^{\frac{13}{2}} \left (\frac{12 A a b c}{13} + \frac{2 A b^{3}}{13} + \frac{6 B a^{2} c}{13} + \frac{6 B a b^{2}}{13}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(7/2)/7 + 2*B*c**3*x**(21/2)/21 + 2*a**2*x**(9/2)*(3*A*b + B*a)/9 +
6*a*x**(11/2)*(A*a*c + A*b**2 + B*a*b)/11 + 2*c**2*x**(19/2)*(A*c + 3*B*b)/19 +
6*c*x**(17/2)*(A*b*c + B*a*c + B*b**2)/17 + x**(15/2)*(2*A*a*c**2/5 + 2*A*b**2*c
/5 + 4*B*a*b*c/5 + 2*B*b**3/15) + x**(13/2)*(12*A*a*b*c/13 + 2*A*b**3/13 + 6*B*a
**2*c/13 + 6*B*a*b**2/13)

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Mathematica [A]  time = 0.138887, size = 182, normalized size = 1. \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{6}{17} c x^{17/2} \left (a B c+A b c+b^2 B\right )+\frac{6}{11} a x^{11/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac{2}{15} x^{15/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{2}{13} x^{13/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{2}{19} c^2 x^{19/2} (A c+3 b B)+\frac{2}{21} B c^3 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*(a*b*B + A*(b^2 + a
*c))*x^(11/2))/11 + (2*(3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^(13/2))/13 + (2
*(b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)*x^(15/2))/15 + (6*c*(b^2*B + A*b*c
+ a*B*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(19/2))/19 + (2*B*c^3*x^(21/2))/2
1

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Maple [A]  time = 0.009, size = 192, normalized size = 1.1 \[{\frac{1385670\,B{c}^{3}{x}^{7}+1531530\,A{c}^{3}{x}^{6}+4594590\,B{x}^{6}b{c}^{2}+5135130\,A{x}^{5}b{c}^{2}+5135130\,aB{c}^{2}{x}^{5}+5135130\,B{x}^{5}{b}^{2}c+5819814\,aA{c}^{2}{x}^{4}+5819814\,A{x}^{4}{b}^{2}c+11639628\,B{x}^{4}abc+1939938\,B{x}^{4}{b}^{3}[phone]\,A{x}^{3}abc+2238390\,A{b}^{3}{x}^{3}+6715170\,{a}^{2}Bc{x}^{3}+6715170\,B{x}^{3}a{b}^{2}+7936110\,{a}^{2}Ac{x}^{2}+7936110\,A{x}^{2}a{b}^{2}+7936110\,B{x}^{2}{a}^{2}b+9699690\,A{a}^{2}bx+3233230\,{a}^{3}Bx+4157010\,A{a}^{3}}{14549535}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/14549535*x^(7/2)*(692835*B*c^3*x^7+765765*A*c^3*x^6+2297295*B*b*c^2*x^6+256756
5*A*b*c^2*x^5+2567565*B*a*c^2*x^5+2567565*B*b^2*c*x^5+2909907*A*a*c^2*x^4+290990
7*A*b^2*c*x^4+5819814*B*a*b*c*x^4+969969*B*b^3*x^4+6715170*A*a*b*c*x^3+1119195*A
*b^3*x^3+3357585*B*a^2*c*x^3+3357585*B*a*b^2*x^3+3968055*A*a^2*c*x^2+3968055*A*a
*b^2*x^2+3968055*B*a^2*b*x^2+4849845*A*a^2*b*x+1616615*B*a^3*x+2078505*A*a^3)

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Maxima [A]  time = 0.717484, size = 224, normalized size = 1.23 \[ \frac{2}{21} \, B c^{3} x^{\frac{21}{2}} + \frac{2}{19} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{19}{2}} + \frac{6}{17} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} + \frac{2}{13} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac{13}{2}} + \frac{6}{11} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*B*c^3*x^(21/2) + 2/19*(3*B*b*c^2 + A*c^3)*x^(19/2) + 6/17*(B*b^2*c + (B*a +
A*b)*c^2)*x^(17/2) + 2/15*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^(15/2)
+ 2/7*A*a^3*x^(7/2) + 2/13*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^(13/2)
+ 6/11*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^(11/2) + 2/9*(B*a^3 + 3*A*a^2*b)*x^(9/2)

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Fricas [A]  time = 0.278042, size = 231, normalized size = 1.27 \[ \frac{2}{14549535} \,{\left (692835 \, B c^{3} x^{10} + 765765 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 2567565 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{8} + 969969 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{7} + 2078505 \, A a^{3} x^{3} + 1119195 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{6} + 3968055 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{5} + 1616615 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/14549535*(692835*B*c^3*x^10 + 765765*(3*B*b*c^2 + A*c^3)*x^9 + 2567565*(B*b^2*
c + (B*a + A*b)*c^2)*x^8 + 969969*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^
7 + 2078505*A*a^3*x^3 + 1119195*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^6
+ 3968055*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^5 + 1616615*(B*a^3 + 3*A*a^2*b)*x^4)*s
qrt(x)

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Sympy [A]  time = 51.418, size = 294, normalized size = 1.62 \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a^{2} c x^{\frac{11}{2}}}{11} + \frac{6 A a b^{2} x^{\frac{11}{2}}}{11} + \frac{12 A a b c x^{\frac{13}{2}}}{13} + \frac{2 A a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 A b^{3} x^{\frac{13}{2}}}{13} + \frac{2 A b^{2} c x^{\frac{15}{2}}}{5} + \frac{6 A b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 A c^{3} x^{\frac{19}{2}}}{19} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{11}{2}}}{11} + \frac{6 B a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 B a b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B a b c x^{\frac{15}{2}}}{5} + \frac{6 B a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} + \frac{6 B b^{2} c x^{\frac{17}{2}}}{17} + \frac{6 B b c^{2} x^{\frac{19}{2}}}{19} + \frac{2 B c^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**3*x**(7/2)/7 + 2*A*a**2*b*x**(9/2)/3 + 6*A*a**2*c*x**(11/2)/11 + 6*A*a*b*
*2*x**(11/2)/11 + 12*A*a*b*c*x**(13/2)/13 + 2*A*a*c**2*x**(15/2)/5 + 2*A*b**3*x*
*(13/2)/13 + 2*A*b**2*c*x**(15/2)/5 + 6*A*b*c**2*x**(17/2)/17 + 2*A*c**3*x**(19/
2)/19 + 2*B*a**3*x**(9/2)/9 + 6*B*a**2*b*x**(11/2)/11 + 6*B*a**2*c*x**(13/2)/13
+ 6*B*a*b**2*x**(13/2)/13 + 4*B*a*b*c*x**(15/2)/5 + 6*B*a*c**2*x**(17/2)/17 + 2*
B*b**3*x**(15/2)/15 + 6*B*b**2*c*x**(17/2)/17 + 6*B*b*c**2*x**(19/2)/19 + 2*B*c*
*3*x**(21/2)/21

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GIAC/XCAS [A]  time = 0.275195, size = 261, normalized size = 1.43 \[ \frac{2}{21} \, B c^{3} x^{\frac{21}{2}} + \frac{6}{19} \, B b c^{2} x^{\frac{19}{2}} + \frac{2}{19} \, A c^{3} x^{\frac{19}{2}} + \frac{6}{17} \, B b^{2} c x^{\frac{17}{2}} + \frac{6}{17} \, B a c^{2} x^{\frac{17}{2}} + \frac{6}{17} \, A b c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{4}{5} \, B a b c x^{\frac{15}{2}} + \frac{2}{5} \, A b^{2} c x^{\frac{15}{2}} + \frac{2}{5} \, A a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, B a b^{2} x^{\frac{13}{2}} + \frac{2}{13} \, A b^{3} x^{\frac{13}{2}} + \frac{6}{13} \, B a^{2} c x^{\frac{13}{2}} + \frac{12}{13} \, A a b c x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} b x^{\frac{11}{2}} + \frac{6}{11} \, A a b^{2} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*B*c^3*x^(21/2) + 6/19*B*b*c^2*x^(19/2) + 2/19*A*c^3*x^(19/2) + 6/17*B*b^2*c
*x^(17/2) + 6/17*B*a*c^2*x^(17/2) + 6/17*A*b*c^2*x^(17/2) + 2/15*B*b^3*x^(15/2)
+ 4/5*B*a*b*c*x^(15/2) + 2/5*A*b^2*c*x^(15/2) + 2/5*A*a*c^2*x^(15/2) + 6/13*B*a*
b^2*x^(13/2) + 2/13*A*b^3*x^(13/2) + 6/13*B*a^2*c*x^(13/2) + 12/13*A*a*b*c*x^(13
/2) + 6/11*B*a^2*b*x^(11/2) + 6/11*A*a*b^2*x^(11/2) + 6/11*A*a^2*c*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)