3.265 \(\int x^3 (A+B x) \left (a+c x^2\right )^3 \, dx\)

Optimal. Leaf size=93 \[ \frac{1}{4} a^3 A x^4+\frac{1}{5} a^3 B x^5+\frac{1}{2} a^2 A c x^6+\frac{3}{7} a^2 B c x^7+\frac{3}{8} a A c^2 x^8+\frac{1}{3} a B c^2 x^9+\frac{1}{10} A c^3 x^{10}+\frac{1}{11} B c^3 x^{11} \]

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*A*c*x^6)/2 + (3*a^2*B*c*x^7)/7 + (3*a*A*c^2
*x^8)/8 + (a*B*c^2*x^9)/3 + (A*c^3*x^10)/10 + (B*c^3*x^11)/11

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Rubi [A]  time = 0.207684, antiderivative size = 93, 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 \[ \frac{1}{4} a^3 A x^4+\frac{1}{5} a^3 B x^5+\frac{1}{2} a^2 A c x^6+\frac{3}{7} a^2 B c x^7+\frac{3}{8} a A c^2 x^8+\frac{1}{3} a B c^2 x^9+\frac{1}{10} A c^3 x^{10}+\frac{1}{11} B c^3 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*A*c*x^6)/2 + (3*a^2*B*c*x^7)/7 + (3*a*A*c^2
*x^8)/8 + (a*B*c^2*x^9)/3 + (A*c^3*x^10)/10 + (B*c^3*x^11)/11

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Rubi in Sympy [A]  time = 15.9174, size = 90, normalized size = 0.97 \[ \frac{A a^{3} x^{4}}{4} + \frac{A a^{2} c x^{6}}{2} + \frac{3 A a c^{2} x^{8}}{8} + \frac{A c^{3} x^{10}}{10} + \frac{B a^{3} x^{5}}{5} + \frac{3 B a^{2} c x^{7}}{7} + \frac{B a c^{2} x^{9}}{3} + \frac{B c^{3} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**3*x**4/4 + A*a**2*c*x**6/2 + 3*A*a*c**2*x**8/8 + A*c**3*x**10/10 + B*a**3*x
**5/5 + 3*B*a**2*c*x**7/7 + B*a*c**2*x**9/3 + B*c**3*x**11/11

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Mathematica [A]  time = 0.00578753, size = 93, normalized size = 1. \[ \frac{1}{4} a^3 A x^4+\frac{1}{5} a^3 B x^5+\frac{1}{2} a^2 A c x^6+\frac{3}{7} a^2 B c x^7+\frac{3}{8} a A c^2 x^8+\frac{1}{3} a B c^2 x^9+\frac{1}{10} A c^3 x^{10}+\frac{1}{11} B c^3 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^4)/4 + (a^3*B*x^5)/5 + (a^2*A*c*x^6)/2 + (3*a^2*B*c*x^7)/7 + (3*a*A*c^2
*x^8)/8 + (a*B*c^2*x^9)/3 + (A*c^3*x^10)/10 + (B*c^3*x^11)/11

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Maple [A]  time = 0.001, size = 78, normalized size = 0.8 \[{\frac{{a}^{3}A{x}^{4}}{4}}+{\frac{{a}^{3}B{x}^{5}}{5}}+{\frac{{a}^{2}Ac{x}^{6}}{2}}+{\frac{3\,{a}^{2}Bc{x}^{7}}{7}}+{\frac{3\,aA{c}^{2}{x}^{8}}{8}}+{\frac{aB{c}^{2}{x}^{9}}{3}}+{\frac{A{c}^{3}{x}^{10}}{10}}+{\frac{B{c}^{3}{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+a)^3,x)

[Out]

1/4*a^3*A*x^4+1/5*a^3*B*x^5+1/2*a^2*A*c*x^6+3/7*a^2*B*c*x^7+3/8*a*A*c^2*x^8+1/3*
a*B*c^2*x^9+1/10*A*c^3*x^10+1/11*B*c^3*x^11

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Maxima [A]  time = 0.691677, size = 104, normalized size = 1.12 \[ \frac{1}{11} \, B c^{3} x^{11} + \frac{1}{10} \, A c^{3} x^{10} + \frac{1}{3} \, B a c^{2} x^{9} + \frac{3}{8} \, A a c^{2} x^{8} + \frac{3}{7} \, B a^{2} c x^{7} + \frac{1}{2} \, A a^{2} c x^{6} + \frac{1}{5} \, B a^{3} x^{5} + \frac{1}{4} \, A a^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*B*c^3*x^11 + 1/10*A*c^3*x^10 + 1/3*B*a*c^2*x^9 + 3/8*A*a*c^2*x^8 + 3/7*B*a^
2*c*x^7 + 1/2*A*a^2*c*x^6 + 1/5*B*a^3*x^5 + 1/4*A*a^3*x^4

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Fricas [A]  time = 0.261666, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} c^{3} B + \frac{1}{10} x^{10} c^{3} A + \frac{1}{3} x^{9} c^{2} a B + \frac{3}{8} x^{8} c^{2} a A + \frac{3}{7} x^{7} c a^{2} B + \frac{1}{2} x^{6} c a^{2} A + \frac{1}{5} x^{5} a^{3} B + \frac{1}{4} x^{4} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*x^11*c^3*B + 1/10*x^10*c^3*A + 1/3*x^9*c^2*a*B + 3/8*x^8*c^2*a*A + 3/7*x^7*
c*a^2*B + 1/2*x^6*c*a^2*A + 1/5*x^5*a^3*B + 1/4*x^4*a^3*A

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Sympy [A]  time = 0.126461, size = 90, normalized size = 0.97 \[ \frac{A a^{3} x^{4}}{4} + \frac{A a^{2} c x^{6}}{2} + \frac{3 A a c^{2} x^{8}}{8} + \frac{A c^{3} x^{10}}{10} + \frac{B a^{3} x^{5}}{5} + \frac{3 B a^{2} c x^{7}}{7} + \frac{B a c^{2} x^{9}}{3} + \frac{B c^{3} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**3*x**4/4 + A*a**2*c*x**6/2 + 3*A*a*c**2*x**8/8 + A*c**3*x**10/10 + B*a**3*x
**5/5 + 3*B*a**2*c*x**7/7 + B*a*c**2*x**9/3 + B*c**3*x**11/11

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GIAC/XCAS [A]  time = 0.266828, size = 104, normalized size = 1.12 \[ \frac{1}{11} \, B c^{3} x^{11} + \frac{1}{10} \, A c^{3} x^{10} + \frac{1}{3} \, B a c^{2} x^{9} + \frac{3}{8} \, A a c^{2} x^{8} + \frac{3}{7} \, B a^{2} c x^{7} + \frac{1}{2} \, A a^{2} c x^{6} + \frac{1}{5} \, B a^{3} x^{5} + \frac{1}{4} \, A a^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*B*c^3*x^11 + 1/10*A*c^3*x^10 + 1/3*B*a*c^2*x^9 + 3/8*A*a*c^2*x^8 + 3/7*B*a^
2*c*x^7 + 1/2*A*a^2*c*x^6 + 1/5*B*a^3*x^5 + 1/4*A*a^3*x^4