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

Optimal. Leaf size=109 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^3 B x^{9/2}+\frac{6}{11} a^2 A c x^{11/2}+\frac{6}{13} a^2 B c x^{13/2}+\frac{2}{5} a A c^2 x^{15/2}+\frac{6}{17} a B c^2 x^{17/2}+\frac{2}{19} A c^3 x^{19/2}+\frac{2}{21} B c^3 x^{21/2} \]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^3*B*x^(9/2))/9 + (6*a^2*A*c*x^(11/2))/11 + (6*a^2*B*c
*x^(13/2))/13 + (2*a*A*c^2*x^(15/2))/5 + (6*a*B*c^2*x^(17/2))/17 + (2*A*c^3*x^(1
9/2))/19 + (2*B*c^3*x^(21/2))/21

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Rubi [A]  time = 0.0987637, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{9} a^3 B x^{9/2}+\frac{6}{11} a^2 A c x^{11/2}+\frac{6}{13} a^2 B c x^{13/2}+\frac{2}{5} a A c^2 x^{15/2}+\frac{6}{17} a B c^2 x^{17/2}+\frac{2}{19} A c^3 x^{19/2}+\frac{2}{21} B c^3 x^{21/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^3*B*x^(9/2))/9 + (6*a^2*A*c*x^(11/2))/11 + (6*a^2*B*c
*x^(13/2))/13 + (2*a*A*c^2*x^(15/2))/5 + (6*a*B*c^2*x^(17/2))/17 + (2*A*c^3*x^(1
9/2))/19 + (2*B*c^3*x^(21/2))/21

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Rubi in Sympy [A]  time = 11.4099, size = 114, normalized size = 1.05 \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{6 A a^{2} c x^{\frac{11}{2}}}{11} + \frac{2 A a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 A c^{3} x^{\frac{19}{2}}}{19} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 B a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B c^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0483907, size = 83, normalized size = 0.76 \[ \frac{2}{63} a^3 x^{7/2} (9 A+7 B x)+\frac{6}{143} a^2 c x^{11/2} (13 A+11 B x)+\frac{2}{85} a c^2 x^{15/2} (17 A+15 B x)+\frac{2}{399} c^3 x^{19/2} (21 A+19 B x) \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*x^(7/2)*(9*A + 7*B*x))/63 + (6*a^2*c*x^(11/2)*(13*A + 11*B*x))/143 + (2*a
*c^2*x^(15/2)*(17*A + 15*B*x))/85 + (2*c^3*x^(19/2)*(21*A + 19*B*x))/399

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Maple [A]  time = 0.009, size = 78, normalized size = 0.7 \[{\frac{1385670\,B{c}^{3}{x}^{7}+1531530\,A{c}^{3}{x}^{6}+5135130\,aB{c}^{2}{x}^{5}+5819814\,aA{c}^{2}{x}^{4}+6715170\,{a}^{2}Bc{x}^{3}+7936110\,{a}^{2}Ac{x}^{2}+3233230\,{a}^{3}Bx+4157010\,A{a}^{3}}{14549535}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/14549535*x^(7/2)*(692835*B*c^3*x^7+765765*A*c^3*x^6+2567565*B*a*c^2*x^5+290990
7*A*a*c^2*x^4+3357585*B*a^2*c*x^3+3968055*A*a^2*c*x^2+1616615*B*a^3*x+2078505*A*
a^3)

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Maxima [A]  time = 0.685499, size = 104, normalized size = 0.95 \[ \frac{2}{21} \, B c^{3} x^{\frac{21}{2}} + \frac{2}{19} \, A c^{3} x^{\frac{19}{2}} + \frac{6}{17} \, B a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, A a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} c x^{\frac{13}{2}} + \frac{6}{11} \, A a^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.281493, size = 111, normalized size = 1.02 \[ \frac{2}{14549535} \,{\left (692835 \, B c^{3} x^{10} + 765765 \, A c^{3} x^{9} + 2567565 \, B a c^{2} x^{8} + 2909907 \, A a c^{2} x^{7} + 3357585 \, B a^{2} c x^{6} + 3968055 \, A a^{2} c x^{5} + 1616615 \, B a^{3} x^{4} + 2078505 \, A a^{3} x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/14549535*(692835*B*c^3*x^10 + 765765*A*c^3*x^9 + 2567565*B*a*c^2*x^8 + 2909907
*A*a*c^2*x^7 + 3357585*B*a^2*c*x^6 + 3968055*A*a^2*c*x^5 + 1616615*B*a^3*x^4 + 2
078505*A*a^3*x^3)*sqrt(x)

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Sympy [A]  time = 42.1443, size = 114, normalized size = 1.05 \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{6 A a^{2} c x^{\frac{11}{2}}}{11} + \frac{2 A a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 A c^{3} x^{\frac{19}{2}}}{19} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} c x^{\frac{13}{2}}}{13} + \frac{6 B a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B c^{3} x^{\frac{21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.275733, size = 104, normalized size = 0.95 \[ \frac{2}{21} \, B c^{3} x^{\frac{21}{2}} + \frac{2}{19} \, A c^{3} x^{\frac{19}{2}} + \frac{6}{17} \, B a c^{2} x^{\frac{17}{2}} + \frac{2}{5} \, A a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, B a^{2} c x^{\frac{13}{2}} + \frac{6}{11} \, A a^{2} c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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