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

Optimal. Leaf size=113 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{15} x^{15/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{13} x^{13/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

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

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Rubi [A]  time = 0.159665, antiderivative size = 113, 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 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{15} x^{15/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{13} x^{13/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.1304, size = 124, normalized size = 1.1 \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} + \frac{2 a x^{\frac{11}{2}} \left (2 A b + B a\right )}{11} + \frac{2 c x^{\frac{17}{2}} \left (A c + 2 B b\right )}{17} + x^{\frac{15}{2}} \left (\frac{4 A b c}{15} + \frac{4 B a c}{15} + \frac{2 B b^{2}}{15}\right ) + x^{\frac{13}{2}} \left (\frac{4 A a c}{13} + \frac{2 A b^{2}}{13} + \frac{4 B a b}{13}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(9/2)/9 + 2*B*c**2*x**(19/2)/19 + 2*a*x**(11/2)*(2*A*b + B*a)/11 + 2
*c*x**(17/2)*(A*c + 2*B*b)/17 + x**(15/2)*(4*A*b*c/15 + 4*B*a*c/15 + 2*B*b**2/15
) + x**(13/2)*(4*A*a*c/13 + 2*A*b**2/13 + 4*B*a*b/13)

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Mathematica [A]  time = 0.0817525, size = 113, normalized size = 1. \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{15} x^{15/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{13} x^{13/2} \left (2 a A c+2 a b B+A b^2\right )+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 102, normalized size = 0.9 \[{\frac{218790\,B{c}^{2}{x}^{5}+244530\,A{c}^{2}{x}^{4}+489060\,B{x}^{4}bc+554268\,A{x}^{3}bc+554268\,aBc{x}^{3}+277134\,B{b}^{2}{x}^{3}+639540\,aAc{x}^{2}+319770\,A{b}^{2}{x}^{2}+639540\,B{x}^{2}ab+755820\,aAbx+377910\,{a}^{2}Bx+461890\,A{a}^{2}}{2078505}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2078505*x^(9/2)*(109395*B*c^2*x^5+122265*A*c^2*x^4+244530*B*b*c*x^4+277134*A*b
*c*x^3+277134*B*a*c*x^3+138567*B*b^2*x^3+319770*A*a*c*x^2+159885*A*b^2*x^2+31977
0*B*a*b*x^2+377910*A*a*b*x+188955*B*a^2*x+230945*A*a^2)

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Maxima [A]  time = 0.716326, size = 126, normalized size = 1.12 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{17} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.314835, size = 132, normalized size = 1.17 \[ \frac{2}{2078505} \,{\left (109395 \, B c^{2} x^{9} + 122265 \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + 138567 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{7} + 230945 \, A a^{2} x^{4} + 159885 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{6} + 188955 \,{\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2078505*(109395*B*c^2*x^9 + 122265*(2*B*b*c + A*c^2)*x^8 + 138567*(B*b^2 + 2*(
B*a + A*b)*c)*x^7 + 230945*A*a^2*x^4 + 159885*(2*B*a*b + A*b^2 + 2*A*a*c)*x^6 +
188955*(B*a^2 + 2*A*a*b)*x^5)*sqrt(x)

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Sympy [A]  time = 45.5581, size = 162, normalized size = 1.43 \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{11}{2}}}{11} + \frac{4 A a c x^{\frac{13}{2}}}{13} + \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A b c x^{\frac{15}{2}}}{15} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a b x^{\frac{13}{2}}}{13} + \frac{4 B a c x^{\frac{15}{2}}}{15} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} + \frac{4 B b c x^{\frac{17}{2}}}{17} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(11/2)/11 + 4*A*a*c*x**(13/2)/13 + 2*A*b**2*x**
(13/2)/13 + 4*A*b*c*x**(15/2)/15 + 2*A*c**2*x**(17/2)/17 + 2*B*a**2*x**(11/2)/11
+ 4*B*a*b*x**(13/2)/13 + 4*B*a*c*x**(15/2)/15 + 2*B*b**2*x**(15/2)/15 + 4*B*b*c
*x**(17/2)/17 + 2*B*c**2*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.272648, size = 139, normalized size = 1.23 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{4}{17} \, B b c x^{\frac{17}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, B a c x^{\frac{15}{2}} + \frac{4}{15} \, A b c x^{\frac{15}{2}} + \frac{4}{13} \, B a b x^{\frac{13}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A a c x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*B*c^2*x^(19/2) + 4/17*B*b*c*x^(17/2) + 2/17*A*c^2*x^(17/2) + 2/15*B*b^2*x^(
15/2) + 4/15*B*a*c*x^(15/2) + 4/15*A*b*c*x^(15/2) + 4/13*B*a*b*x^(13/2) + 2/13*A
*b^2*x^(13/2) + 4/13*A*a*c*x^(13/2) + 2/11*B*a^2*x^(11/2) + 4/11*A*a*b*x^(11/2)
+ 2/9*A*a^2*x^(9/2)