3.189 \(\int (\text {b1}+\text {c1} x) (a+2 b x+c x^2) \, dx\)

Optimal. Leaf size=44 \[ \frac {1}{2} x^2 (a \text {c1}+2 b \text {b1})+a \text {b1} x+\frac {1}{3} x^3 (2 b \text {c1}+\text {b1} c)+\frac {1}{4} c \text {c1} x^4 \]

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Rubi [A]  time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \[ \frac {1}{2} x^2 (a \text {c1}+2 b \text {b1})+a \text {b1} x+\frac {1}{3} x^3 (2 b \text {c1}+\text {b1} c)+\frac {1}{4} c \text {c1} x^4 \]

Antiderivative was successfully verified.

[In]

Int[(b1 + c1*x)*(a + 2*b*x + c*x^2),x]

[Out]

a*b1*x + ((2*b*b1 + a*c1)*x^2)/2 + ((b1*c + 2*b*c1)*x^3)/3 + (c*c1*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 (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx &=\int \left (a \text {b1}+(2 b \text {b1}+a \text {c1}) x+(\text {b1} c+2 b \text {c1}) x^2+c \text {c1} x^3\right ) \, dx\\ &=a \text {b1} x+\frac {1}{2} (2 b \text {b1}+a \text {c1}) x^2+\frac {1}{3} (\text {b1} c+2 b \text {c1}) x^3+\frac {1}{4} c \text {c1} x^4\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.93 \[ \frac {1}{12} x (6 a (2 \text {b1}+\text {c1} x)+x (4 b (3 \text {b1}+2 \text {c1} x)+c x (4 \text {b1}+3 \text {c1} x))) \]

Antiderivative was successfully verified.

[In]

Integrate[(b1 + c1*x)*(a + 2*b*x + c*x^2),x]

[Out]

(x*(6*a*(2*b1 + c1*x) + x*(4*b*(3*b1 + 2*c1*x) + c*x*(4*b1 + 3*c1*x))))/12

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IntegrateAlgebraic [A]  time = 0.03, size = 41, normalized size = 0.93 \[ \frac {1}{12} x \left (12 a \text {b1}+6 a \text {c1} x+12 b \text {b1} x+8 b \text {c1} x^2+4 \text {b1} c x^2+3 c \text {c1} x^3\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b1 + c1*x)*(a + 2*b*x + c*x^2),x]

[Out]

(x*(12*a*b1 + 12*b*b1*x + 6*a*c1*x + 4*b1*c*x^2 + 8*b*c1*x^2 + 3*c*c1*x^3))/12

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fricas [A]  time = 1.01, size = 39, normalized size = 0.89 \[ \frac {1}{4} x^{4} c_{1} c + \frac {1}{3} x^{3} c b_{1} + \frac {2}{3} x^{3} c_{1} b + x^{2} b_{1} b + \frac {1}{2} x^{2} c_{1} a + x b_{1} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a),x, algorithm="fricas")

[Out]

1/4*x^4*c1*c + 1/3*x^3*c*b1 + 2/3*x^3*c1*b + x^2*b1*b + 1/2*x^2*c1*a + x*b1*a

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giac [A]  time = 0.97, size = 39, normalized size = 0.89 \[ \frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, b_{1} c x^{3} + \frac {2}{3} \, b c_{1} x^{3} + b b_{1} x^{2} + \frac {1}{2} \, a c_{1} x^{2} + a b_{1} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a),x, algorithm="giac")

[Out]

1/4*c*c1*x^4 + 1/3*b1*c*x^3 + 2/3*b*c1*x^3 + b*b1*x^2 + 1/2*a*c1*x^2 + a*b1*x

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maple [A]  time = 0.02, size = 38, normalized size = 0.86

method	result	size
norman	\(\frac {c \mathit {c1} \,x^{4}}{4}+\left (\frac {2 b \mathit {c1}}{3}+\frac {\mathit {b1} c}{3}\right ) x^{3}+\left (\frac {a \mathit {c1}}{2}+b \mathit {b1} \right ) x^{2}+a \mathit {b1} x\)	\(38\)
default	\(a \mathit {b1} x +\frac {\left (a \mathit {c1} +2 b \mathit {b1} \right ) x^{2}}{2}+\frac {\left (2 b \mathit {c1} +\mathit {b1} c \right ) x^{3}}{3}+\frac {c \mathit {c1} \,x^{4}}{4}\)	\(39\)
gosper	\(\frac {1}{4} c \mathit {c1} \,x^{4}+\frac {2}{3} x^{3} b \mathit {c1} +\frac {1}{3} x^{3} \mathit {b1} c +\frac {1}{2} x^{2} a \mathit {c1} +x^{2} b \mathit {b1} +a \mathit {b1} x\)	\(40\)
risch	\(\frac {1}{4} c \mathit {c1} \,x^{4}+\frac {2}{3} x^{3} b \mathit {c1} +\frac {1}{3} x^{3} \mathit {b1} c +\frac {1}{2} x^{2} a \mathit {c1} +x^{2} b \mathit {b1} +a \mathit {b1} x\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c1*x+b1)*(c*x^2+2*b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/4*c*c1*x^4+(2/3*b*c1+1/3*b1*c)*x^3+(1/2*a*c1+b*b1)*x^2+a*b1*x

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maxima [A]  time = 0.59, size = 38, normalized size = 0.86 \[ \frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, {\left (b_{1} c + 2 \, b c_{1}\right )} x^{3} + a b_{1} x + \frac {1}{2} \, {\left (2 \, b b_{1} + a c_{1}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x^2+2*b*x+a),x, algorithm="maxima")

[Out]

1/4*c*c1*x^4 + 1/3*(b1*c + 2*b*c1)*x^3 + a*b1*x + 1/2*(2*b*b1 + a*c1)*x^2

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mupad [B]  time = 0.05, size = 37, normalized size = 0.84 \[ \frac {c\,c_{1}\,x^4}{4}+\left (\frac {2\,b\,c_{1}}{3}+\frac {b_{1}\,c}{3}\right )\,x^3+\left (\frac {a\,c_{1}}{2}+b\,b_{1}\right )\,x^2+a\,b_{1}\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b1 + c1*x)*(a + 2*b*x + c*x^2),x)

[Out]

x^2*((a*c1)/2 + b*b1) + x^3*((2*b*c1)/3 + (b1*c)/3) + a*b1*x + (c*c1*x^4)/4

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sympy [A]  time = 0.07, size = 39, normalized size = 0.89 \[ a b_{1} x + \frac {c c_{1} x^{4}}{4} + x^{3} \left (\frac {2 b c_{1}}{3} + \frac {b_{1} c}{3}\right ) + x^{2} \left (\frac {a c_{1}}{2} + b b_{1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c1*x+b1)*(c*x**2+2*b*x+a),x)

[Out]

a*b1*x + c*c1*x**4/4 + x**3*(2*b*c1/3 + b1*c/3) + x**2*(a*c1/2 + b*b1)

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