3.21.68 \(\int (A+B x) (d+e x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=80 \[ \frac {1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} x^4 (A c e+b B e+B c d)+\frac {1}{5} B c e x^5 \]

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Rubi [A]  time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {771} \begin {gather*} \frac {1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} x^4 (A c e+b B e+B c d)+\frac {1}{5} B c e x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)*(a + b*x + c*x^2),x]

[Out]

a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + A*b*e + a*B*e)*x^3)/3 + ((B*c*d + b*B*e + A*c*e)
*x^4)/4 + (B*c*e*x^5)/5

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (A+B x) (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A d+(A b d+a B d+a A e) x+(b B d+A c d+A b e+a B e) x^2+(B c d+b B e+A c e) x^3+B c e x^4\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 c d+A b e+a B e) x^3+\frac {1}{4} (B c d+b B e+A c e) x^4+\frac {1}{5} B c e x^5\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 80, normalized size = 1.00 \begin {gather*} \frac {1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac {1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac {1}{4} x^4 (A c e+b B e+B c d)+\frac {1}{5} B c e x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*A*d*x + ((A*b*d + a*B*d + a*A*e)*x^2)/2 + ((b*B*d + A*c*d + A*b*e + a*B*e)*x^3)/3 + ((B*c*d + b*B*e + A*c*e)
*x^4)/4 + (B*c*e*x^5)/5

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 94, normalized size = 1.18 \begin {gather*} \frac {1}{5} x^{5} e c B + \frac {1}{4} x^{4} d c B + \frac {1}{4} x^{4} e b B + \frac {1}{4} x^{4} e c A + \frac {1}{3} x^{3} d b B + \frac {1}{3} x^{3} e a B + \frac {1}{3} x^{3} d c A + \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)*(e*x+d)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/5*x^5*e*c*B + 1/4*x^4*d*c*B + 1/4*x^4*e*b*B + 1/4*x^4*e*c*A + 1/3*x^3*d*b*B + 1/3*x^3*e*a*B + 1/3*x^3*d*c*A
+ 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 = 0.15, size = 100, normalized size = 1.25 \begin {gather*} \frac {1}{5} \, B c x^{5} e + \frac {1}{4} \, B c d x^{4} + \frac {1}{4} \, B b x^{4} e + \frac {1}{4} \, A c x^{4} e + \frac {1}{3} \, B b d x^{3} + \frac {1}{3} \, A c 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)*(e*x+d)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/5*B*c*x^5*e + 1/4*B*c*d*x^4 + 1/4*B*b*x^4*e + 1/4*A*c*x^4*e + 1/3*B*b*d*x^3 + 1/3*A*c*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 = 0.04, size = 76, normalized size = 0.95 \begin {gather*} \frac {B c e \,x^{5}}{5}+A a d x +\frac {\left (B b e +\left (A e +B d \right ) c \right ) x^{4}}{4}+\frac {\left (A c d +B a e +\left (A e +B d \right ) b \right ) x^{3}}{3}+\frac {\left (A b d +\left (A e +B d \right ) a \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*B*c*e*x^5+1/4*((A*e+B*d)*c+B*b*e)*x^4+1/3*(A*c*d+(A*e+B*d)*b+a*B*e)*x^3+1/2*(A*b*d+a*(A*e+B*d))*x^2+A*a*d*
x

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maxima [A]  time = 0.46, size = 76, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, B c e x^{5} + \frac {1}{4} \, {\left (B c d + {\left (B b + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left ({\left (B b + A c\right )} 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)*(e*x+d)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/5*B*c*e*x^5 + 1/4*(B*c*d + (B*b + A*c)*e)*x^4 + A*a*d*x + 1/3*((B*b + A*c)*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 = 0.03, size = 79, normalized size = 0.99 \begin {gather*} \frac {B\,c\,e\,x^5}{5}+\left (\frac {A\,c\,e}{4}+\frac {B\,b\,e}{4}+\frac {B\,c\,d}{4}\right )\,x^4+\left (\frac {A\,b\,e}{3}+\frac {A\,c\,d}{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)*(d + e*x)*(a + b*x + c*x^2),x)

[Out]

x^3*((A*b*e)/3 + (A*c*d)/3 + (B*a*e)/3 + (B*b*d)/3) + x^2*((A*a*e)/2 + (A*b*d)/2 + (B*a*d)/2) + x^4*((A*c*e)/4
+ (B*b*e)/4 + (B*c*d)/4) + (B*c*e*x^5)/5 + A*a*d*x

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sympy [A]  time = 0.08, size = 94, normalized size = 1.18 \begin {gather*} A a d x + \frac {B c e x^{5}}{5} + x^{4} \left (\frac {A c e}{4} + \frac {B b e}{4} + \frac {B c d}{4}\right ) + x^{3} \left (\frac {A b e}{3} + \frac {A c d}{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)*(e*x+d)*(c*x**2+b*x+a),x)

[Out]

A*a*d*x + B*c*e*x**5/5 + x**4*(A*c*e/4 + B*b*e/4 + B*c*d/4) + x**3*(A*b*e/3 + A*c*d/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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