3.8.71 \(\int (A+B x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=42 \[ \frac {1}{2} x^2 (a B+A b)+a A x+\frac {1}{3} x^3 (A c+b B)+\frac {1}{4} B c x^4 \]

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Rubi [A]  time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {631} \begin {gather*} \frac {1}{2} x^2 (a B+A b)+a A x+\frac {1}{3} x^3 (A c+b B)+\frac {1}{4} B c x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + b*x + c*x^2),x]

[Out]

a*A*x + ((A*b + a*B)*x^2)/2 + ((b*B + A*c)*x^3)/3 + (B*c*x^4)/4

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

\begin {align*} \int (A+B x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A+(A b+a B) x+(b B+A c) x^2+B c x^3\right ) \, dx\\ &=a A x+\frac {1}{2} (A b+a B) x^2+\frac {1}{3} (b B+A c) x^3+\frac {1}{4} B c x^4\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 (a B+A b)+a A x+\frac {1}{3} x^3 (A c+b B)+\frac {1}{4} B c x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*A*x + ((A*b + a*B)*x^2)/2 + ((b*B + A*c)*x^3)/3 + (B*c*x^4)/4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(a + b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[(A + B*x)*(a + b*x + c*x^2), x]

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fricas [A]  time = 0.35, size = 40, normalized size = 0.95 \begin {gather*} \frac {1}{4} x^{4} c B + \frac {1}{3} x^{3} b B + \frac {1}{3} x^{3} c A + \frac {1}{2} x^{2} a B + \frac {1}{2} x^{2} b A + x a A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/4*x^4*c*B + 1/3*x^3*b*B + 1/3*x^3*c*A + 1/2*x^2*a*B + 1/2*x^2*b*A + x*a*A

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giac [A]  time = 0.15, size = 40, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, B b x^{3} + \frac {1}{3} \, A c x^{3} + \frac {1}{2} \, B a x^{2} + \frac {1}{2} \, A b x^{2} + A a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/4*B*c*x^4 + 1/3*B*b*x^3 + 1/3*A*c*x^3 + 1/2*B*a*x^2 + 1/2*A*b*x^2 + A*a*x

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maple [A]  time = 0.06, size = 37, normalized size = 0.88 \begin {gather*} \frac {B c \,x^{4}}{4}+A a x +\frac {\left (A c +b B \right ) x^{3}}{3}+\frac {\left (A b +B a \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a),x)

[Out]

A*a*x+1/2*(A*b+B*a)*x^2+1/3*(A*c+B*b)*x^3+1/4*B*c*x^4

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maxima [A]  time = 0.56, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, B c x^{4} + \frac {1}{3} \, {\left (B b + A c\right )} x^{3} + A a x + \frac {1}{2} \, {\left (B a + A b\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/4*B*c*x^4 + 1/3*(B*b + A*c)*x^3 + A*a*x + 1/2*(B*a + A*b)*x^2

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mupad [B]  time = 0.04, size = 38, normalized size = 0.90 \begin {gather*} \frac {B\,c\,x^4}{4}+\left (\frac {A\,c}{3}+\frac {B\,b}{3}\right )\,x^3+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x^2+A\,a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x + c*x^2),x)

[Out]

x^2*((A*b)/2 + (B*a)/2) + x^3*((A*c)/3 + (B*b)/3) + A*a*x + (B*c*x^4)/4

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sympy [A]  time = 0.07, size = 39, normalized size = 0.93 \begin {gather*} A a x + \frac {B c x^{4}}{4} + x^{3} \left (\frac {A c}{3} + \frac {B b}{3}\right ) + x^{2} \left (\frac {A b}{2} + \frac {B a}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a),x)

[Out]

A*a*x + B*c*x**4/4 + x**3*(A*c/3 + B*b/3) + x**2*(A*b/2 + B*a/2)

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