3.9.57 \(\int (A+B x) (a+b x+c x^2)^2 \, dx\) [857]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {645} \begin {gather*} a^2 A x+\frac {1}{4} x^4 \left (2 a B c+2 A b c+b^2 B\right )+\frac {1}{3} x^3 \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {1}{2} a x^2 (a B+2 A b)+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{6} B c^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + b*x + c*x^2)^2,x]

[Out]

a^2*A*x + (a*(2*A*b + a*B)*x^2)/2 + ((2*a*b*B + A*(b^2 + 2*a*c))*x^3)/3 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^4)/4
+ (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^6)/6

Rule 645

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

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Mathematica [A]
time = 0.01, size = 96, normalized size = 1.00 \begin {gather*} a^2 A x+\frac {1}{2} a (2 A b+a B) x^2+\frac {1}{3} \left (A b^2+2 a b B+2 a A c\right ) x^3+\frac {1}{4} \left (b^2 B+2 A b c+2 a B c\right ) x^4+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{6} B c^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*A*x + (a*(2*A*b + a*B)*x^2)/2 + ((A*b^2 + 2*a*b*B + 2*a*A*c)*x^3)/3 + ((b^2*B + 2*A*b*c + 2*a*B*c)*x^4)/4
+ (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^6)/6

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Maple [A]
time = 3.16, size = 91, normalized size = 0.95

method	result	size
norman	\(\frac {B \,c^{2} x^{6}}{6}+\left (\frac {1}{5} A \,c^{2}+\frac {2}{5} b B c \right ) x^{5}+\left (\frac {1}{2} A b c +\frac {1}{2} a B c +\frac {1}{4} b^{2} B \right ) x^{4}+\left (\frac {2}{3} A a c +\frac {1}{3} b^{2} A +\frac {2}{3} a b B \right ) x^{3}+\left (a b A +\frac {1}{2} a^{2} B \right ) x^{2}+a^{2} A x\)	\(90\)
default	\(\frac {B \,c^{2} x^{6}}{6}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{5}}{5}+\frac {\left (2 A b c +B \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (2 a b B +A \left (2 a c +b^{2}\right )\right ) x^{3}}{3}+\frac {\left (2 a b A +a^{2} B \right ) x^{2}}{2}+a^{2} A x\)	\(91\)
gosper	\(\frac {1}{6} B \,c^{2} x^{6}+\frac {1}{5} A \,c^{2} x^{5}+\frac {2}{5} x^{5} b B c +\frac {1}{2} x^{4} A b c +\frac {1}{2} B a c \,x^{4}+\frac {1}{4} b^{2} B \,x^{4}+\frac {2}{3} a A c \,x^{3}+\frac {1}{3} A \,b^{2} x^{3}+\frac {2}{3} B a b \,x^{3}+a A b \,x^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\)	\(100\)
risch	\(\frac {1}{6} B \,c^{2} x^{6}+\frac {1}{5} A \,c^{2} x^{5}+\frac {2}{5} x^{5} b B c +\frac {1}{2} x^{4} A b c +\frac {1}{2} B a c \,x^{4}+\frac {1}{4} b^{2} B \,x^{4}+\frac {2}{3} a A c \,x^{3}+\frac {1}{3} A \,b^{2} x^{3}+\frac {2}{3} B a b \,x^{3}+a A b \,x^{2}+\frac {1}{2} B \,a^{2} x^{2}+a^{2} A x\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 90, normalized size = 0.94 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.13, size = 90, normalized size = 0.94 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.02, size = 100, normalized size = 1.04 \begin {gather*} A a^{2} x + \frac {B c^{2} x^{6}}{6} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {B a c}{2} + \frac {B b^{2}}{4}\right ) + x^{3} \cdot \left (\frac {2 A a c}{3} + \frac {A b^{2}}{3} + \frac {2 B a b}{3}\right ) + x^{2} \left (A a b + \frac {B a^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**2,x)

[Out]

A*a**2*x + B*c**2*x**6/6 + x**5*(A*c**2/5 + 2*B*b*c/5) + x**4*(A*b*c/2 + B*a*c/2 + B*b**2/4) + x**3*(2*A*a*c/3
+ A*b**2/3 + 2*B*a*b/3) + x**2*(A*a*b + B*a**2/2)

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Giac [A]
time = 1.00, size = 99, normalized size = 1.03 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, B a c x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {2}{3} \, B a b x^{3} + \frac {1}{3} \, A b^{2} x^{3} + \frac {2}{3} \, A a c x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a b x^{2} + A a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c^2*x^6 + 2/5*B*b*c*x^5 + 1/5*A*c^2*x^5 + 1/4*B*b^2*x^4 + 1/2*B*a*c*x^4 + 1/2*A*b*c*x^4 + 2/3*B*a*b*x^3
+ 1/3*A*b^2*x^3 + 2/3*A*a*c*x^3 + 1/2*B*a^2*x^2 + A*a*b*x^2 + A*a^2*x

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Mupad [B]
time = 0.03, size = 89, normalized size = 0.93 \begin {gather*} x^2\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+x^3\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}+\frac {2\,A\,a\,c}{3}\right )+x^4\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}+\frac {B\,a\,c}{2}\right )+\frac {B\,c^2\,x^6}{6}+A\,a^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x + c*x^2)^2,x)

[Out]

x^2*((B*a^2)/2 + A*a*b) + x^5*((A*c^2)/5 + (2*B*b*c)/5) + x^3*((A*b^2)/3 + (2*A*a*c)/3 + (2*B*a*b)/3) + x^4*((
B*b^2)/4 + (A*b*c)/2 + (B*a*c)/2) + (B*c^2*x^6)/6 + A*a^2*x

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