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

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

[Out]

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

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Rubi [A]  time = 0.09958, 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 \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{7} a^3 B x^{7/2}+\frac{2}{3} a^2 A c x^{9/2}+\frac{6}{11} a^2 B c x^{11/2}+\frac{6}{13} a A c^2 x^{13/2}+\frac{2}{5} a B c^2 x^{15/2}+\frac{2}{17} A c^3 x^{17/2}+\frac{2}{19} B c^3 x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.4138, size = 114, normalized size = 1.05 \[ \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 A a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 A a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 A c^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{7}{2}}}{7} + \frac{6 B a^{2} c x^{\frac{11}{2}}}{11} + \frac{2 B a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 B c^{3} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0444604, size = 83, normalized size = 0.76 \[ \frac{2}{35} a^3 x^{5/2} (7 A+5 B x)+\frac{2}{33} a^2 c x^{9/2} (11 A+9 B x)+\frac{2}{65} a c^2 x^{13/2} (15 A+13 B x)+\frac{2}{323} c^3 x^{17/2} (19 A+17 B x) \]

Antiderivative was successfully verified.

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

[Out]

(2*a^3*x^(5/2)*(7*A + 5*B*x))/35 + (2*a^2*c*x^(9/2)*(11*A + 9*B*x))/33 + (2*a*c^
2*x^(13/2)*(15*A + 13*B*x))/65 + (2*c^3*x^(17/2)*(19*A + 17*B*x))/323

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Maple [A]  time = 0.008, size = 78, normalized size = 0.7 \[{\frac{510510\,B{c}^{3}{x}^{7}+570570\,A{c}^{3}{x}^{6}+1939938\,aB{c}^{2}{x}^{5}+2238390\,aA{c}^{2}{x}^{4}+2645370\,{a}^{2}Bc{x}^{3}+3233230\,{a}^{2}Ac{x}^{2}+1385670\,{a}^{3}Bx+1939938\,A{a}^{3}}{4849845}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/4849845*x^(5/2)*(255255*B*c^3*x^7+285285*A*c^3*x^6+969969*B*a*c^2*x^5+1119195*
A*a*c^2*x^4+1322685*B*a^2*c*x^3+1616615*A*a^2*c*x^2+692835*B*a^3*x+969969*A*a^3)

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Maxima [A]  time = 0.682856, size = 104, normalized size = 0.95 \[ \frac{2}{19} \, B c^{3} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, A a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} c x^{\frac{11}{2}} + \frac{2}{3} \, A a^{2} c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.264313, size = 111, normalized size = 1.02 \[ \frac{2}{4849845} \,{\left (255255 \, B c^{3} x^{9} + 285285 \, A c^{3} x^{8} + 969969 \, B a c^{2} x^{7} + 1119195 \, A a c^{2} x^{6} + 1322685 \, B a^{2} c x^{5} + 1616615 \, A a^{2} c x^{4} + 692835 \, B a^{3} x^{3} + 969969 \, A a^{3} x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/4849845*(255255*B*c^3*x^9 + 285285*A*c^3*x^8 + 969969*B*a*c^2*x^7 + 1119195*A*
a*c^2*x^6 + 1322685*B*a^2*c*x^5 + 1616615*A*a^2*c*x^4 + 692835*B*a^3*x^3 + 96996
9*A*a^3*x^2)*sqrt(x)

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Sympy [A]  time = 20.7443, size = 114, normalized size = 1.05 \[ \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 A a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 A a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 A c^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{7}{2}}}{7} + \frac{6 B a^{2} c x^{\frac{11}{2}}}{11} + \frac{2 B a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 B c^{3} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.267492, size = 104, normalized size = 0.95 \[ \frac{2}{19} \, B c^{3} x^{\frac{19}{2}} + \frac{2}{17} \, A c^{3} x^{\frac{17}{2}} + \frac{2}{5} \, B a c^{2} x^{\frac{15}{2}} + \frac{6}{13} \, A a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, B a^{2} c x^{\frac{11}{2}} + \frac{2}{3} \, A a^{2} c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{3} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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