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

Optimal. Leaf size=134 \[ -\frac {(d+e x)^4 \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{4 e^4}-\frac {(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {(d+e x)^5 (-A c e-b B e+3 B c d)}{5 e^4}+\frac {B c (d+e x)^6}{6 e^4} \]

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Rubi [A]  time = 0.14, antiderivative size = 133, normalized size of antiderivative = 0.99, 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*} \frac {(d+e x)^4 \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{4 e^4}-\frac {(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {(d+e x)^5 (-A c e-b B e+3 B c d)}{5 e^4}+\frac {B c (d+e x)^6}{6 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 137, normalized size = 1.02 \begin {gather*} \frac {1}{3} x^3 \left (a A e^2+2 a B d e+b d (2 A e+B d)+A c d^2\right )+\frac {1}{4} x^4 \left (B e (a e+2 b d)+A e (b e+2 c d)+B c d^2\right )+\frac {1}{2} d x^2 (2 a A e+a B d+A b d)+a A d^2 x+\frac {1}{5} e x^5 (A c e+b B e+2 B c d)+\frac {1}{6} B c e^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 171, normalized size = 1.28 \begin {gather*} \frac {1}{6} x^{6} e^{2} c B + \frac {2}{5} x^{5} e d c B + \frac {1}{5} x^{5} e^{2} b B + \frac {1}{5} x^{5} e^{2} c A + \frac {1}{4} x^{4} d^{2} c B + \frac {1}{2} x^{4} e d b B + \frac {1}{4} x^{4} e^{2} a B + \frac {1}{2} x^{4} e d c A + \frac {1}{4} x^{4} e^{2} b A + \frac {1}{3} x^{3} d^{2} b B + \frac {2}{3} x^{3} e d a B + \frac {1}{3} x^{3} d^{2} c A + \frac {2}{3} x^{3} e d b A + \frac {1}{3} x^{3} e^{2} a A + \frac {1}{2} x^{2} d^{2} a B + \frac {1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 171, normalized size = 1.28 \begin {gather*} \frac {1}{6} \, B c x^{6} e^{2} + \frac {2}{5} \, B c d x^{5} e + \frac {1}{4} \, B c d^{2} x^{4} + \frac {1}{5} \, B b x^{5} e^{2} + \frac {1}{5} \, A c x^{5} e^{2} + \frac {1}{2} \, B b d x^{4} e + \frac {1}{2} \, A c d x^{4} e + \frac {1}{3} \, B b d^{2} x^{3} + \frac {1}{3} \, A c d^{2} x^{3} + \frac {1}{4} \, B a x^{4} e^{2} + \frac {1}{4} \, A b x^{4} e^{2} + \frac {2}{3} \, B a d x^{3} e + \frac {2}{3} \, A b d x^{3} e + \frac {1}{2} \, B a d^{2} x^{2} + \frac {1}{2} \, A b d^{2} x^{2} + \frac {1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 145, normalized size = 1.08 \begin {gather*} \frac {B c \,e^{2} x^{6}}{6}+A a \,d^{2} x +\frac {\left (B b \,e^{2}+\left (A \,e^{2}+2 B d e \right ) c \right ) x^{5}}{5}+\frac {\left (B a \,e^{2}+\left (A \,e^{2}+2 B d e \right ) b +\left (2 A d e +B \,d^{2}\right ) c \right ) x^{4}}{4}+\frac {\left (A c \,d^{2}+\left (A \,e^{2}+2 B d e \right ) a +\left (2 A d e +B \,d^{2}\right ) b \right ) x^{3}}{3}+\frac {\left (A b \,d^{2}+\left (2 A d e +B \,d^{2}\right ) a \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 132, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, B c e^{2} x^{6} + \frac {1}{5} \, {\left (2 \, B c d e + {\left (B b + A c\right )} e^{2}\right )} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (B c d^{2} + 2 \, {\left (B b + A c\right )} d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a e^{2} + {\left (B b + A c\right )} d^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.32, size = 143, normalized size = 1.07 \begin {gather*} x^3\,\left (\frac {A\,a\,e^2}{3}+\frac {A\,c\,d^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}+\frac {2\,B\,a\,d\,e}{3}\right )+x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,a\,e^2}{4}+\frac {B\,c\,d^2}{4}+\frac {A\,c\,d\,e}{2}+\frac {B\,b\,d\,e}{2}\right )+x^2\,\left (\frac {A\,b\,d^2}{2}+\frac {B\,a\,d^2}{2}+A\,a\,d\,e\right )+x^5\,\left (\frac {A\,c\,e^2}{5}+\frac {B\,b\,e^2}{5}+\frac {2\,B\,c\,d\,e}{5}\right )+A\,a\,d^2\,x+\frac {B\,c\,e^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.09, size = 172, normalized size = 1.28 \begin {gather*} A a d^{2} x + \frac {B c e^{2} x^{6}}{6} + x^{5} \left (\frac {A c e^{2}}{5} + \frac {B b e^{2}}{5} + \frac {2 B c d e}{5}\right ) + x^{4} \left (\frac {A b e^{2}}{4} + \frac {A c d e}{2} + \frac {B a e^{2}}{4} + \frac {B b d e}{2} + \frac {B c d^{2}}{4}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {2 A b d e}{3} + \frac {A c d^{2}}{3} + \frac {2 B a d e}{3} + \frac {B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac {A b d^{2}}{2} + \frac {B a d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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