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

Optimal. Leaf size=180 \[ a^2 A d x+\frac {1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac {1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac {1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac {1}{2} a x^2 (a A e+a B d+2 A b d)+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{7} B c^2 e x^7 \]

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Rubi [A]  time = 0.30, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {771} \begin {gather*} a^2 A d x+\frac {1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac {1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac {1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac {1}{2} a x^2 (a A e+a B d+2 A b d)+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{7} B c^2 e x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*A*d*x + (a*(2*A*b*d + a*B*d + a*A*e)*x^2)/2 + ((a*B*(2*b*d + a*e) + A*(b^2*d + 2*a*c*d + 2*a*b*e))*x^3)/3
+ ((b^2*(B*d + A*e) + 2*a*c*(B*d + A*e) + 2*b*(A*c*d + a*B*e))*x^4)/4 + ((b^2*B*e + 2*b*c*(B*d + A*e) + c*(A*c
*d + 2*a*B*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/7

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 )^2 \, dx &=\int \left (a^2 A d+a (2 A b d+a B d+a A e) x+\left (a B (2 b d+a e)+A \left (b^2 d+2 a c d+2 a b e\right )\right ) x^2+\left (b^2 (B d+A e)+2 a c (B d+A e)+2 b (A c d+a B e)\right ) x^3+\left (b^2 B e+2 b c (B d+A e)+c (A c d+2 a B e)\right ) x^4+c (B c d+2 b B e+A c e) x^5+B c^2 e x^6\right ) \, dx\\ &=a^2 A d x+\frac {1}{2} a (2 A b d+a B d+a A e) x^2+\frac {1}{3} \left (a B (2 b d+a e)+A \left (b^2 d+2 a c d+2 a b e\right )\right ) x^3+\frac {1}{4} \left (b^2 (B d+A e)+2 a c (B d+A e)+2 b (A c d+a B e)\right ) x^4+\frac {1}{5} \left (b^2 B e+2 b c (B d+A e)+c (A c d+2 a B e)\right ) x^5+\frac {1}{6} c (B c d+2 b B e+A c e) x^6+\frac {1}{7} B c^2 e x^7\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 180, normalized size = 1.00 \begin {gather*} a^2 A d x+\frac {1}{5} x^5 \left (c (2 a B e+A c d)+2 b c (A e+B d)+b^2 B e\right )+\frac {1}{4} x^4 \left (2 b (a B e+A c d)+2 a c (A e+B d)+b^2 (A e+B d)\right )+\frac {1}{3} x^3 \left (A \left (2 a b e+2 a c d+b^2 d\right )+a B (a e+2 b d)\right )+\frac {1}{2} a x^2 (a A e+a B d+2 A b d)+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{7} B c^2 e x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*A*d*x + (a*(2*A*b*d + a*B*d + a*A*e)*x^2)/2 + ((a*B*(2*b*d + a*e) + A*(b^2*d + 2*a*c*d + 2*a*b*e))*x^3)/3
+ ((b^2*(B*d + A*e) + 2*a*c*(B*d + A*e) + 2*b*(A*c*d + a*B*e))*x^4)/4 + ((b^2*B*e + 2*b*c*(B*d + A*e) + c*(A*c
*d + 2*a*B*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/7

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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 )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 225, normalized size = 1.25 \begin {gather*} \frac {1}{7} x^{7} e c^{2} B + \frac {1}{6} x^{6} d c^{2} B + \frac {1}{3} x^{6} e c b B + \frac {1}{6} x^{6} e c^{2} A + \frac {2}{5} x^{5} d c b B + \frac {1}{5} x^{5} e b^{2} B + \frac {2}{5} x^{5} e c a B + \frac {1}{5} x^{5} d c^{2} A + \frac {2}{5} x^{5} e c b A + \frac {1}{4} x^{4} d b^{2} B + \frac {1}{2} x^{4} d c a B + \frac {1}{2} x^{4} e b a B + \frac {1}{2} x^{4} d c b A + \frac {1}{4} x^{4} e b^{2} A + \frac {1}{2} x^{4} e c a A + \frac {2}{3} x^{3} d b a B + \frac {1}{3} x^{3} e a^{2} B + \frac {1}{3} x^{3} d b^{2} A + \frac {2}{3} x^{3} d c a A + \frac {2}{3} x^{3} e b a A + \frac {1}{2} x^{2} d a^{2} B + x^{2} d b a A + \frac {1}{2} x^{2} e a^{2} A + x d a^{2} 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)^2,x, algorithm="fricas")

[Out]

1/7*x^7*e*c^2*B + 1/6*x^6*d*c^2*B + 1/3*x^6*e*c*b*B + 1/6*x^6*e*c^2*A + 2/5*x^5*d*c*b*B + 1/5*x^5*e*b^2*B + 2/
5*x^5*e*c*a*B + 1/5*x^5*d*c^2*A + 2/5*x^5*e*c*b*A + 1/4*x^4*d*b^2*B + 1/2*x^4*d*c*a*B + 1/2*x^4*e*b*a*B + 1/2*
x^4*d*c*b*A + 1/4*x^4*e*b^2*A + 1/2*x^4*e*c*a*A + 2/3*x^3*d*b*a*B + 1/3*x^3*e*a^2*B + 1/3*x^3*d*b^2*A + 2/3*x^
3*d*c*a*A + 2/3*x^3*e*b*a*A + 1/2*x^2*d*a^2*B + x^2*d*b*a*A + 1/2*x^2*e*a^2*A + x*d*a^2*A

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giac [A]  time = 0.16, size = 237, normalized size = 1.32 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} e + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{3} \, B b c x^{6} e + \frac {1}{6} \, A c^{2} x^{6} e + \frac {2}{5} \, B b c d x^{5} + \frac {1}{5} \, A c^{2} d x^{5} + \frac {1}{5} \, B b^{2} x^{5} e + \frac {2}{5} \, B a c x^{5} e + \frac {2}{5} \, A b c x^{5} e + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, B a c d x^{4} + \frac {1}{2} \, A b c d x^{4} + \frac {1}{2} \, B a b x^{4} e + \frac {1}{4} \, A b^{2} x^{4} e + \frac {1}{2} \, A a c x^{4} e + \frac {2}{3} \, B a b d x^{3} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {2}{3} \, A a c d x^{3} + \frac {1}{3} \, B a^{2} x^{3} e + \frac {2}{3} \, A a b x^{3} e + \frac {1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac {1}{2} \, A a^{2} x^{2} e + A a^{2} 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)^2,x, algorithm="giac")

[Out]

1/7*B*c^2*x^7*e + 1/6*B*c^2*d*x^6 + 1/3*B*b*c*x^6*e + 1/6*A*c^2*x^6*e + 2/5*B*b*c*d*x^5 + 1/5*A*c^2*d*x^5 + 1/
5*B*b^2*x^5*e + 2/5*B*a*c*x^5*e + 2/5*A*b*c*x^5*e + 1/4*B*b^2*d*x^4 + 1/2*B*a*c*d*x^4 + 1/2*A*b*c*d*x^4 + 1/2*
B*a*b*x^4*e + 1/4*A*b^2*x^4*e + 1/2*A*a*c*x^4*e + 2/3*B*a*b*d*x^3 + 1/3*A*b^2*d*x^3 + 2/3*A*a*c*d*x^3 + 1/3*B*
a^2*x^3*e + 2/3*A*a*b*x^3*e + 1/2*B*a^2*d*x^2 + A*a*b*d*x^2 + 1/2*A*a^2*x^2*e + A*a^2*d*x

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maple [A]  time = 0.04, size = 167, normalized size = 0.93 \begin {gather*} \frac {B \,c^{2} e \,x^{7}}{7}+\frac {\left (2 B b c e +\left (A e +B d \right ) c^{2}\right ) x^{6}}{6}+A \,a^{2} d x +\frac {\left (A \,c^{2} d +\left (2 a c +b^{2}\right ) B e +2 \left (A e +B d \right ) b c \right ) x^{5}}{5}+\frac {\left (2 A b c d +2 B a b e +\left (A e +B d \right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (B \,a^{2} e +\left (2 a c +b^{2}\right ) A d +2 \left (A e +B d \right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b d +\left (A e +B d \right ) a^{2}\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)^2,x)

[Out]

1/7*B*c^2*e*x^7+1/6*((A*e+B*d)*c^2+2*B*e*b*c)*x^6+1/5*(A*c^2*d+2*(A*e+B*d)*b*c+B*e*(2*a*c+b^2))*x^5+1/4*(2*A*b
*c*d+(A*e+B*d)*(2*a*c+b^2)+2*B*a*b*e)*x^4+1/3*(A*d*(2*a*c+b^2)+2*(A*e+B*d)*a*b+B*e*a^2)*x^3+1/2*(2*A*d*a*b+(A*
e+B*d)*a^2)*x^2+A*a^2*d*x

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maxima [A]  time = 0.48, size = 184, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{6} \, {\left (B c^{2} d + {\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e\right )} x^{5} + A a^{2} d x + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, 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)^2,x, algorithm="maxima")

[Out]

1/7*B*c^2*e*x^7 + 1/6*(B*c^2*d + (2*B*b*c + A*c^2)*e)*x^6 + 1/5*((2*B*b*c + A*c^2)*d + (B*b^2 + 2*(B*a + A*b)*
c)*e)*x^5 + A*a^2*d*x + 1/4*((B*b^2 + 2*(B*a + A*b)*c)*d + (2*B*a*b + A*b^2 + 2*A*a*c)*e)*x^4 + 1/3*((2*B*a*b
+ A*b^2 + 2*A*a*c)*d + (B*a^2 + 2*A*a*b)*e)*x^3 + 1/2*(A*a^2*e + (B*a^2 + 2*A*a*b)*d)*x^2

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mupad [B]  time = 2.33, size = 184, normalized size = 1.02 \begin {gather*} x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {A\,a\,c\,e}{2}+\frac {A\,b\,c\,d}{2}+\frac {B\,a\,b\,e}{2}+\frac {B\,a\,c\,d}{2}\right )+x^3\,\left (\frac {A\,b^2\,d}{3}+\frac {B\,a^2\,e}{3}+\frac {2\,A\,a\,b\,e}{3}+\frac {2\,A\,a\,c\,d}{3}+\frac {2\,B\,a\,b\,d}{3}\right )+x^5\,\left (\frac {A\,c^2\,d}{5}+\frac {B\,b^2\,e}{5}+\frac {2\,A\,b\,c\,e}{5}+\frac {2\,B\,a\,c\,e}{5}+\frac {2\,B\,b\,c\,d}{5}\right )+x^2\,\left (\frac {A\,a^2\,e}{2}+\frac {B\,a^2\,d}{2}+A\,a\,b\,d\right )+x^6\,\left (\frac {A\,c^2\,e}{6}+\frac {B\,c^2\,d}{6}+\frac {B\,b\,c\,e}{3}\right )+A\,a^2\,d\,x+\frac {B\,c^2\,e\,x^7}{7} \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)^2,x)

[Out]

x^4*((A*b^2*e)/4 + (B*b^2*d)/4 + (A*a*c*e)/2 + (A*b*c*d)/2 + (B*a*b*e)/2 + (B*a*c*d)/2) + x^3*((A*b^2*d)/3 + (
B*a^2*e)/3 + (2*A*a*b*e)/3 + (2*A*a*c*d)/3 + (2*B*a*b*d)/3) + x^5*((A*c^2*d)/5 + (B*b^2*e)/5 + (2*A*b*c*e)/5 +
(2*B*a*c*e)/5 + (2*B*b*c*d)/5) + x^2*((A*a^2*e)/2 + (B*a^2*d)/2 + A*a*b*d) + x^6*((A*c^2*e)/6 + (B*c^2*d)/6 +
(B*b*c*e)/3) + A*a^2*d*x + (B*c^2*e*x^7)/7

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sympy [A]  time = 0.10, size = 231, normalized size = 1.28 \begin {gather*} A a^{2} d x + \frac {B c^{2} e x^{7}}{7} + x^{6} \left (\frac {A c^{2} e}{6} + \frac {B b c e}{3} + \frac {B c^{2} d}{6}\right ) + x^{5} \left (\frac {2 A b c e}{5} + \frac {A c^{2} d}{5} + \frac {2 B a c e}{5} + \frac {B b^{2} e}{5} + \frac {2 B b c d}{5}\right ) + x^{4} \left (\frac {A a c e}{2} + \frac {A b^{2} e}{4} + \frac {A b c d}{2} + \frac {B a b e}{2} + \frac {B a c d}{2} + \frac {B b^{2} d}{4}\right ) + x^{3} \left (\frac {2 A a b e}{3} + \frac {2 A a c d}{3} + \frac {A b^{2} d}{3} + \frac {B a^{2} e}{3} + \frac {2 B a b d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + A a b d + \frac {B a^{2} 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)**2,x)

[Out]

A*a**2*d*x + B*c**2*e*x**7/7 + x**6*(A*c**2*e/6 + B*b*c*e/3 + B*c**2*d/6) + x**5*(2*A*b*c*e/5 + A*c**2*d/5 + 2
*B*a*c*e/5 + B*b**2*e/5 + 2*B*b*c*d/5) + x**4*(A*a*c*e/2 + A*b**2*e/4 + A*b*c*d/2 + B*a*b*e/2 + B*a*c*d/2 + B*
b**2*d/4) + x**3*(2*A*a*b*e/3 + 2*A*a*c*d/3 + A*b**2*d/3 + B*a**2*e/3 + 2*B*a*b*d/3) + x**2*(A*a**2*e/2 + A*a*
b*d + B*a**2*d/2)

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