3.9.75 \(\int (a+b x) (A+B x) (d+e x) \, dx\)

Optimal. Leaf size=56 \[ \frac {1}{3} x^3 (a B e+A b e+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} b B e x^4 \]

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Rubi [A]  time = 0.04, antiderivative size = 56, 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 = {77} \begin {gather*} \frac {1}{3} x^3 (a B e+A b e+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} b B e x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(A + B*x)*(d + e*x),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 53, normalized size = 0.95 \begin {gather*} \frac {1}{12} x \left (4 x^2 (a B e+A b e+b B d)+6 x (a A e+a B d+A b d)+12 a A d+3 b B e x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(A + B*x)*(d + e*x),x]

[Out]

(x*(12*a*A*d + 6*(A*b*d + a*B*d + a*A*e)*x + 4*(b*B*d + A*b*e + a*B*e)*x^2 + 3*b*B*e*x^3))/12

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(A + B*x)*(d + e*x),x]

[Out]

IntegrateAlgebraic[(a + b*x)*(A + B*x)*(d + e*x), x]

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fricas [A]  time = 0.59, size = 62, normalized size = 1.11 \begin {gather*} \frac {1}{4} x^{4} e b B + \frac {1}{3} x^{3} d b B + \frac {1}{3} x^{3} e a B + \frac {1}{3} x^{3} e b A + \frac {1}{2} x^{2} d a B + \frac {1}{2} x^{2} d b A + \frac {1}{2} x^{2} e a A + x d a A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.13, size = 66, normalized size = 1.18 \begin {gather*} \frac {1}{4} \, B b x^{4} e + \frac {1}{3} \, B b d x^{3} + \frac {1}{3} \, B a x^{3} e + \frac {1}{3} \, A b x^{3} e + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A b d x^{2} + \frac {1}{2} \, A a x^{2} e + A a d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 53, normalized size = 0.95 \begin {gather*} \frac {B b e \,x^{4}}{4}+A a d x +\frac {\left (B b d +\left (A b +B a \right ) e \right ) x^{3}}{3}+\frac {\left (A a e +\left (A b +B a \right ) d \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)*(e*x+d),x)

[Out]

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

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maxima [A]  time = 0.50, size = 52, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, B b e x^{4} + A a d x + \frac {1}{3} \, {\left (B b d + {\left (B a + A b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a e + {\left (B a + A b\right )} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 54, normalized size = 0.96 \begin {gather*} \frac {B\,b\,e\,x^4}{4}+\left (\frac {A\,b\,e}{3}+\frac {B\,a\,e}{3}+\frac {B\,b\,d}{3}\right )\,x^3+\left (\frac {A\,a\,e}{2}+\frac {A\,b\,d}{2}+\frac {B\,a\,d}{2}\right )\,x^2+A\,a\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)*(d + e*x),x)

[Out]

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

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sympy [A]  time = 0.07, size = 63, normalized size = 1.12 \begin {gather*} A a d x + \frac {B b e x^{4}}{4} + x^{3} \left (\frac {A b e}{3} + \frac {B a e}{3} + \frac {B b d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {A b d}{2} + \frac {B a d}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d),x)

[Out]

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

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