3.18.90 \(\int (d+e x) (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=96 \[ a^2 d x+\frac {1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{2} a x^2 (a e+2 b d)+\frac {1}{5} c x^5 (2 b e+c d)+\frac {1}{6} c^2 e x^6 \]

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Rubi [A]  time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {631} \begin {gather*} a^2 d x+\frac {1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{2} a x^2 (a e+2 b d)+\frac {1}{5} c x^5 (2 b e+c d)+\frac {1}{6} c^2 e x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^2,x]

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4
+ (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

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 (d+e x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 d+a (2 b d+a e) x+\left (b^2 d+2 a c d+2 a b e\right ) x^2+\left (2 b c d+b^2 e+2 a c e\right ) x^3+c (c d+2 b e) x^4+c^2 e x^5\right ) \, dx\\ &=a^2 d x+\frac {1}{2} a (2 b d+a e) x^2+\frac {1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 96, normalized size = 1.00 \begin {gather*} a^2 d x+\frac {1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{2} a x^2 (a e+2 b d)+\frac {1}{5} c x^5 (2 b e+c d)+\frac {1}{6} c^2 e x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4
+ (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 99, normalized size = 1.03 \begin {gather*} \frac {1}{6} x^{6} e c^{2} + \frac {1}{5} x^{5} d c^{2} + \frac {2}{5} x^{5} e c b + \frac {1}{2} x^{4} d c b + \frac {1}{4} x^{4} e b^{2} + \frac {1}{2} x^{4} e c a + \frac {1}{3} x^{3} d b^{2} + \frac {2}{3} x^{3} d c a + \frac {2}{3} x^{3} e b a + x^{2} d b a + \frac {1}{2} x^{2} e a^{2} + x d a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*e*c^2 + 1/5*x^5*d*c^2 + 2/5*x^5*e*c*b + 1/2*x^4*d*c*b + 1/4*x^4*e*b^2 + 1/2*x^4*e*c*a + 1/3*x^3*d*b^2
+ 2/3*x^3*d*c*a + 2/3*x^3*e*b*a + x^2*d*b*a + 1/2*x^2*e*a^2 + x*d*a^2

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giac [A]  time = 0.15, size = 105, normalized size = 1.09 \begin {gather*} \frac {1}{6} \, c^{2} x^{6} e + \frac {1}{5} \, c^{2} d x^{5} + \frac {2}{5} \, b c x^{5} e + \frac {1}{2} \, b c d x^{4} + \frac {1}{4} \, b^{2} x^{4} e + \frac {1}{2} \, a c x^{4} e + \frac {1}{3} \, b^{2} d x^{3} + \frac {2}{3} \, a c d x^{3} + \frac {2}{3} \, a b x^{3} e + a b d x^{2} + \frac {1}{2} \, a^{2} x^{2} e + a^{2} d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*x^6*e + 1/5*c^2*d*x^5 + 2/5*b*c*x^5*e + 1/2*b*c*d*x^4 + 1/4*b^2*x^4*e + 1/2*a*c*x^4*e + 1/3*b^2*d*x^3
+ 2/3*a*c*d*x^3 + 2/3*a*b*x^3*e + a*b*d*x^2 + 1/2*a^2*x^2*e + a^2*d*x

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maple [A]  time = 0.05, size = 91, normalized size = 0.95 \begin {gather*} \frac {c^{2} e \,x^{6}}{6}+\frac {\left (2 b c e +c^{2} d \right ) x^{5}}{5}+a^{2} d x +\frac {\left (2 b c d +\left (2 a c +b^{2}\right ) e \right ) x^{4}}{4}+\frac {\left (2 a b e +\left (2 a c +b^{2}\right ) d \right ) x^{3}}{3}+\frac {\left (e \,a^{2}+2 a b d \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*e*x^6+1/5*(2*b*c*e+c^2*d)*x^5+1/4*(2*b*c*d+e*(2*a*c+b^2))*x^4+1/3*((2*a*c+b^2)*d+2*b*e*a)*x^3+1/2*(a^2
*e+2*a*b*d)*x^2+a^2*d*x

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maxima [A]  time = 0.99, size = 90, normalized size = 0.94 \begin {gather*} \frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*e*x^6 + 1/5*(c^2*d + 2*b*c*e)*x^5 + 1/4*(2*b*c*d + (b^2 + 2*a*c)*e)*x^4 + a^2*d*x + 1/3*(2*a*b*e + (b^
2 + 2*a*c)*d)*x^3 + 1/2*(2*a*b*d + a^2*e)*x^2

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mupad [B]  time = 0.64, size = 89, normalized size = 0.93 \begin {gather*} x^3\,\left (\frac {d\,b^2}{3}+\frac {2\,a\,e\,b}{3}+\frac {2\,a\,c\,d}{3}\right )+x^4\,\left (\frac {e\,b^2}{4}+\frac {c\,d\,b}{2}+\frac {a\,c\,e}{2}\right )+x^2\,\left (\frac {e\,a^2}{2}+b\,d\,a\right )+x^5\,\left (\frac {d\,c^2}{5}+\frac {2\,b\,e\,c}{5}\right )+\frac {c^2\,e\,x^6}{6}+a^2\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)*(a + b*x + c*x^2)^2,x)

[Out]

x^3*((b^2*d)/3 + (2*a*b*e)/3 + (2*a*c*d)/3) + x^4*((b^2*e)/4 + (a*c*e)/2 + (b*c*d)/2) + x^2*((a^2*e)/2 + a*b*d
) + x^5*((c^2*d)/5 + (2*b*c*e)/5) + (c^2*e*x^6)/6 + a^2*d*x

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sympy [A]  time = 0.09, size = 100, normalized size = 1.04 \begin {gather*} a^{2} d x + \frac {c^{2} e x^{6}}{6} + x^{5} \left (\frac {2 b c e}{5} + \frac {c^{2} d}{5}\right ) + x^{4} \left (\frac {a c e}{2} + \frac {b^{2} e}{4} + \frac {b c d}{2}\right ) + x^{3} \left (\frac {2 a b e}{3} + \frac {2 a c d}{3} + \frac {b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{2} e}{2} + a b d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**2+b*x+a)**2,x)

[Out]

a**2*d*x + c**2*e*x**6/6 + x**5*(2*b*c*e/5 + c**2*d/5) + x**4*(a*c*e/2 + b**2*e/4 + b*c*d/2) + x**3*(2*a*b*e/3
+ 2*a*c*d/3 + b**2*d/3) + x**2*(a**2*e/2 + a*b*d)

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