3.49 \(\int x^4 (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{5} a A x^5+\frac{1}{7} b B x^7 \]

[Out]

(a*A*x^5)/5 + ((A*b + a*B)*x^6)/6 + (b*B*x^7)/7

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Rubi [A]  time = 0.0811973, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{5} a A x^5+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x)*(A + B*x),x]

[Out]

(a*A*x^5)/5 + ((A*b + a*B)*x^6)/6 + (b*B*x^7)/7

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Rubi in Sympy [A]  time = 10.0476, size = 29, normalized size = 0.88 \[ \frac{A a x^{5}}{5} + \frac{B b x^{7}}{7} + x^{6} \left (\frac{A b}{6} + \frac{B a}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x+a)*(B*x+A),x)

[Out]

A*a*x**5/5 + B*b*x**7/7 + x**6*(A*b/6 + B*a/6)

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Mathematica [A]  time = 0.00835732, size = 33, normalized size = 1. \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{5} a A x^5+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x)*(A + B*x),x]

[Out]

(a*A*x^5)/5 + ((A*b + a*B)*x^6)/6 + (b*B*x^7)/7

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Maple [A]  time = 0.002, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{5}}{5}}+{\frac{ \left ( Ab+Ba \right ){x}^{6}}{6}}+{\frac{bB{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x+a)*(B*x+A),x)

[Out]

1/5*a*A*x^5+1/6*(A*b+B*a)*x^6+1/7*b*B*x^7

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Maxima [A]  time = 1.34245, size = 36, normalized size = 1.09 \[ \frac{1}{7} \, B b x^{7} + \frac{1}{5} \, A a x^{5} + \frac{1}{6} \,{\left (B a + A b\right )} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)*x^4,x, algorithm="maxima")

[Out]

1/7*B*b*x^7 + 1/5*A*a*x^5 + 1/6*(B*a + A*b)*x^6

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Fricas [A]  time = 0.176897, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} b B + \frac{1}{6} x^{6} a B + \frac{1}{6} x^{6} b A + \frac{1}{5} x^{5} a A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)*x^4,x, algorithm="fricas")

[Out]

1/7*x^7*b*B + 1/6*x^6*a*B + 1/6*x^6*b*A + 1/5*x^5*a*A

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Sympy [A]  time = 0.081733, size = 29, normalized size = 0.88 \[ \frac{A a x^{5}}{5} + \frac{B b x^{7}}{7} + x^{6} \left (\frac{A b}{6} + \frac{B a}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x+a)*(B*x+A),x)

[Out]

A*a*x**5/5 + B*b*x**7/7 + x**6*(A*b/6 + B*a/6)

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GIAC/XCAS [A]  time = 0.237286, size = 39, normalized size = 1.18 \[ \frac{1}{7} \, B b x^{7} + \frac{1}{6} \, B a x^{6} + \frac{1}{6} \, A b x^{6} + \frac{1}{5} \, A a x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)*x^4,x, algorithm="giac")

[Out]

1/7*B*b*x^7 + 1/6*B*a*x^6 + 1/6*A*b*x^6 + 1/5*A*a*x^5