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

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

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {76} \begin {gather*} \frac {1}{6} x^6 (a B+A b)+\frac {1}{5} a A x^5+\frac {1}{7} b B x^7 \end {gather*}

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int x^4 (a+b x) (A+B x) \, dx &=\int \left (a A x^4+(A b+a B) x^5+b B x^6\right ) \, dx\\ &=\frac {1}{5} a A x^5+\frac {1}{6} (A b+a B) x^6+\frac {1}{7} b B x^7\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {1}{6} x^6 (a B+A b)+\frac {1}{5} a A x^5+\frac {1}{7} b B x^7 \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 (a+b x) (A+B x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.11, size = 29, normalized size = 0.88 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)*(B*x+A),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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giac [A]  time = 1.22, size = 29, normalized size = 0.88 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)*(B*x+A),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

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maple [A]  time = 0.00, size = 28, normalized size = 0.85 \begin {gather*} \frac {B b \,x^{7}}{7}+\frac {A a \,x^{5}}{5}+\frac {\left (A b +B a \right ) x^{6}}{6} \end {gather*}

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 = 1.10, size = 27, normalized size = 0.82 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)*(B*x+A),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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mupad [B]  time = 0.05, size = 28, normalized size = 0.85 \begin {gather*} \frac {B\,b\,x^7}{7}+\left (\frac {A\,b}{6}+\frac {B\,a}{6}\right )\,x^6+\frac {A\,a\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(A + B*x)*(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} \frac {A a x^{5}}{5} + \frac {B b x^{7}}{7} + x^{6} \left (\frac {A b}{6} + \frac {B a}{6}\right ) \end {gather*}

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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