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

Optimal. Leaf size=96 \[ a^2 \text {b1} x+\frac {1}{2} x^4 \left (a c \text {c1}+2 b^2 \text {c1}+2 b \text {b1} c\right )+\frac {2}{3} x^3 \left (2 a b \text {c1}+a \text {b1} c+2 b^2 \text {b1}\right )+\frac {1}{2} a x^2 (a \text {c1}+4 b \text {b1})+\frac {1}{5} c x^5 (4 b \text {c1}+\text {b1} c)+\frac {1}{6} c^2 \text {c1} x^6 \]

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Rubi [A]  time = 0.10, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {631} \[ a^2 \text {b1} x+\frac {1}{2} x^4 \left (a c \text {c1}+2 b^2 \text {c1}+2 b \text {b1} c\right )+\frac {2}{3} x^3 \left (2 a b \text {c1}+a \text {b1} c+2 b^2 \text {b1}\right )+\frac {1}{2} a x^2 (a \text {c1}+4 b \text {b1})+\frac {1}{5} c x^5 (4 b \text {c1}+\text {b1} c)+\frac {1}{6} c^2 \text {c1} x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*b1*x + (a*(4*b*b1 + a*c1)*x^2)/2 + (2*(2*b^2*b1 + a*b1*c + 2*a*b*c1)*x^3)/3 + ((2*b*b1*c + 2*b^2*c1 + a*c*
c1)*x^4)/2 + (c*(b1*c + 4*b*c1)*x^5)/5 + (c^2*c1*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 (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx &=\int \left (a^2 \text {b1}+a (4 b \text {b1}+a \text {c1}) x+2 \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^2+2 \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^3+c (\text {b1} c+4 b \text {c1}) x^4+c^2 \text {c1} x^5\right ) \, dx\\ &=a^2 \text {b1} x+\frac {1}{2} a (4 b \text {b1}+a \text {c1}) x^2+\frac {2}{3} \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{2} \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^4+\frac {1}{5} c (\text {b1} c+4 b \text {c1}) x^5+\frac {1}{6} c^2 \text {c1} x^6\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 91, normalized size = 0.95 \[ \frac {1}{30} x \left (15 a^2 (2 \text {b1}+\text {c1} x)+5 a x (4 b (3 \text {b1}+2 \text {c1} x)+c x (4 \text {b1}+3 \text {c1} x))+x^2 \left (10 b^2 (4 \text {b1}+3 \text {c1} x)+6 b c x (5 \text {b1}+4 \text {c1} x)+c^2 x^2 (6 \text {b1}+5 \text {c1} x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(15*a^2*(2*b1 + c1*x) + 5*a*x*(4*b*(3*b1 + 2*c1*x) + c*x*(4*b1 + 3*c1*x)) + x^2*(10*b^2*(4*b1 + 3*c1*x) + 6
*b*c*x*(5*b1 + 4*c1*x) + c^2*x^2*(6*b1 + 5*c1*x))))/30

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IntegrateAlgebraic [A]  time = 0.06, size = 105, normalized size = 1.09 \[ \frac {1}{30} \left (30 a^2 \text {b1} x+15 a^2 \text {c1} x^2+60 a b \text {b1} x^2+40 a b \text {c1} x^3+20 a \text {b1} c x^3+15 a c \text {c1} x^4+40 b^2 \text {b1} x^3+30 b^2 \text {c1} x^4+30 b \text {b1} c x^4+24 b c \text {c1} x^5+6 \text {b1} c^2 x^5+5 c^2 \text {c1} x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(30*a^2*b1*x + 60*a*b*b1*x^2 + 15*a^2*c1*x^2 + 40*b^2*b1*x^3 + 20*a*b1*c*x^3 + 40*a*b*c1*x^3 + 30*b*b1*c*x^4 +
30*b^2*c1*x^4 + 15*a*c*c1*x^4 + 6*b1*c^2*x^5 + 24*b*c*c1*x^5 + 5*c^2*c1*x^6)/30

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fricas [A]  time = 1.08, size = 98, normalized size = 1.02 \[ \frac {1}{6} x^{6} c_{1} c^{2} + \frac {1}{5} x^{5} c^{2} b_{1} + \frac {4}{5} x^{5} c_{1} c b + x^{4} c b_{1} b + x^{4} c_{1} b^{2} + \frac {1}{2} x^{4} c_{1} c a + \frac {4}{3} x^{3} b_{1} b^{2} + \frac {2}{3} x^{3} c b_{1} a + \frac {4}{3} x^{3} c_{1} b a + 2 x^{2} b_{1} b a + \frac {1}{2} x^{2} c_{1} a^{2} + x b_{1} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*c1*c^2 + 1/5*x^5*c^2*b1 + 4/5*x^5*c1*c*b + x^4*c*b1*b + x^4*c1*b^2 + 1/2*x^4*c1*c*a + 4/3*x^3*b1*b^2 +
2/3*x^3*c*b1*a + 4/3*x^3*c1*b*a + 2*x^2*b1*b*a + 1/2*x^2*c1*a^2 + x*b1*a^2

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giac [A]  time = 1.11, size = 98, normalized size = 1.02 \[ \frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, b_{1} c^{2} x^{5} + \frac {4}{5} \, b c c_{1} x^{5} + b b_{1} c x^{4} + b^{2} c_{1} x^{4} + \frac {1}{2} \, a c c_{1} x^{4} + \frac {4}{3} \, b^{2} b_{1} x^{3} + \frac {2}{3} \, a b_{1} c x^{3} + \frac {4}{3} \, a b c_{1} x^{3} + 2 \, a b b_{1} x^{2} + \frac {1}{2} \, a^{2} c_{1} x^{2} + a^{2} b_{1} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*c1*x^6 + 1/5*b1*c^2*x^5 + 4/5*b*c*c1*x^5 + b*b1*c*x^4 + b^2*c1*x^4 + 1/2*a*c*c1*x^4 + 4/3*b^2*b1*x^3 +
2/3*a*b1*c*x^3 + 4/3*a*b*c1*x^3 + 2*a*b*b1*x^2 + 1/2*a^2*c1*x^2 + a^2*b1*x

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maple [A]  time = 0.32, size = 89, normalized size = 0.93

method	result	size
norman	\(\frac {c^{2} \mathit {c1} \,x^{6}}{6}+\left (\frac {4}{5} \mathit {c1} b c +\frac {1}{5} \mathit {b1} \,c^{2}\right ) x^{5}+\left (\frac {1}{2} a c \mathit {c1} +b^{2} \mathit {c1} +b \mathit {b1} c \right ) x^{4}+\left (\frac {4}{3} a b \mathit {c1} +\frac {2}{3} a \mathit {b1} c +\frac {4}{3} b^{2} \mathit {b1} \right ) x^{3}+\left (\frac {1}{2} \mathit {c1} \,a^{2}+2 \mathit {b1} a b \right ) x^{2}+a^{2} \mathit {b1} x\)	\(89\)
default	\(\frac {c^{2} \mathit {c1} \,x^{6}}{6}+\frac {\left (4 \mathit {c1} b c +\mathit {b1} \,c^{2}\right ) x^{5}}{5}+\frac {\left (4 b \mathit {b1} c +\mathit {c1} \left (2 a c +4 b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\mathit {b1} \left (2 a c +4 b^{2}\right )+4 a b \mathit {c1} \right ) x^{3}}{3}+\frac {\left (\mathit {c1} \,a^{2}+4 \mathit {b1} a b \right ) x^{2}}{2}+a^{2} \mathit {b1} x\)	\(95\)
gosper	\(\frac {1}{6} c^{2} \mathit {c1} \,x^{6}+\frac {4}{5} x^{5} \mathit {c1} b c +\frac {1}{5} x^{5} \mathit {b1} \,c^{2}+\frac {1}{2} x^{4} a c \mathit {c1} +x^{4} b^{2} \mathit {c1} +x^{4} b \mathit {b1} c +\frac {4}{3} x^{3} a b \mathit {c1} +\frac {2}{3} x^{3} a \mathit {b1} c +\frac {4}{3} x^{3} b^{2} \mathit {b1} +\frac {1}{2} x^{2} \mathit {c1} \,a^{2}+2 x^{2} \mathit {b1} a b +a^{2} \mathit {b1} x\)	\(99\)
risch	\(\frac {1}{6} c^{2} \mathit {c1} \,x^{6}+\frac {4}{5} x^{5} \mathit {c1} b c +\frac {1}{5} x^{5} \mathit {b1} \,c^{2}+\frac {1}{2} x^{4} a c \mathit {c1} +x^{4} b^{2} \mathit {c1} +x^{4} b \mathit {b1} c +\frac {4}{3} x^{3} a b \mathit {c1} +\frac {2}{3} x^{3} a \mathit {b1} c +\frac {4}{3} x^{3} b^{2} \mathit {b1} +\frac {1}{2} x^{2} \mathit {c1} \,a^{2}+2 x^{2} \mathit {b1} a b +a^{2} \mathit {b1} x\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*c1*x^6+(4/5*c1*b*c+1/5*b1*c^2)*x^5+(1/2*a*c*c1+b^2*c1+b*b1*c)*x^4+(4/3*a*b*c1+2/3*a*b1*c+4/3*b^2*b1)*x
^3+(1/2*c1*a^2+2*b1*a*b)*x^2+a^2*b1*x

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maxima [A]  time = 0.55, size = 91, normalized size = 0.95 \[ \frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, {\left (b_{1} c^{2} + 4 \, b c c_{1}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, b b_{1} c + {\left (2 \, b^{2} + a c\right )} c_{1}\right )} x^{4} + a^{2} b_{1} x + \frac {2}{3} \, {\left (2 \, b^{2} b_{1} + a b_{1} c + 2 \, a b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b b_{1} + a^{2} c_{1}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*c1*x^6 + 1/5*(b1*c^2 + 4*b*c*c1)*x^5 + 1/2*(2*b*b1*c + (2*b^2 + a*c)*c1)*x^4 + a^2*b1*x + 2/3*(2*b^2*b
1 + a*b1*c + 2*a*b*c1)*x^3 + 1/2*(4*a*b*b1 + a^2*c1)*x^2

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mupad [B]  time = 0.19, size = 88, normalized size = 0.92 \[ x^3\,\left (\frac {4\,b_{1}\,b^2}{3}+\frac {4\,a\,c_{1}\,b}{3}+\frac {2\,a\,b_{1}\,c}{3}\right )+x^4\,\left (c_{1}\,b^2+b_{1}\,c\,b+\frac {a\,c\,c_{1}}{2}\right )+x^2\,\left (\frac {c_{1}\,a^2}{2}+2\,b\,b_{1}\,a\right )+x^5\,\left (\frac {b_{1}\,c^2}{5}+\frac {4\,b\,c_{1}\,c}{5}\right )+\frac {c^2\,c_{1}\,x^6}{6}+a^2\,b_{1}\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((4*b^2*b1)/3 + (4*a*b*c1)/3 + (2*a*b1*c)/3) + x^4*(b^2*c1 + (a*c*c1)/2 + b*b1*c) + x^2*((a^2*c1)/2 + 2*a*
b*b1) + x^5*((b1*c^2)/5 + (4*b*c*c1)/5) + (c^2*c1*x^6)/6 + a^2*b1*x

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sympy [A]  time = 0.09, size = 100, normalized size = 1.04 \[ a^{2} b_{1} x + \frac {c^{2} c_{1} x^{6}}{6} + x^{5} \left (\frac {4 b c c_{1}}{5} + \frac {b_{1} c^{2}}{5}\right ) + x^{4} \left (\frac {a c c_{1}}{2} + b^{2} c_{1} + b b_{1} c\right ) + x^{3} \left (\frac {4 a b c_{1}}{3} + \frac {2 a b_{1} c}{3} + \frac {4 b^{2} b_{1}}{3}\right ) + x^{2} \left (\frac {a^{2} c_{1}}{2} + 2 a b b_{1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b1*x + c**2*c1*x**6/6 + x**5*(4*b*c*c1/5 + b1*c**2/5) + x**4*(a*c*c1/2 + b**2*c1 + b*b1*c) + x**3*(4*a*b*
c1/3 + 2*a*b1*c/3 + 4*b**2*b1/3) + x**2*(a**2*c1/2 + 2*a*b*b1)

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