3.9.99 \(\int (a+b x)^3 (A+B x) (d+e x)^2 \, dx\)

Optimal. Leaf size=118 \[ \frac {e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac {(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^2}{4 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4} \]

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Rubi [A]  time = 0.16, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac {(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac {(a+b x)^4 (A b-a B) (b d-a e)^2}{4 b^4}+\frac {B e^2 (a+b x)^7}{7 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^2,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 224, normalized size = 1.90 \begin {gather*} a^3 A d^2 x+\frac {1}{4} x^4 \left (A b \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a B \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )\right )+\frac {1}{3} a x^3 \left (A \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+a B d (2 a e+3 b d)\right )+\frac {1}{5} b x^5 \left (3 a^2 B e^2+3 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+\frac {1}{2} a^2 d x^2 (2 a A e+a B d+3 A b d)+\frac {1}{6} b^2 e x^6 (3 a B e+A b e+2 b B d)+\frac {1}{7} b^3 B e^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^2,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^3 (A+B x) (d+e x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^3*(A + B*x)*(d + e*x)^2,x]

[Out]

IntegrateAlgebraic[(a + b*x)^3*(A + B*x)*(d + e*x)^2, x]

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fricas [B]  time = 0.92, size = 287, normalized size = 2.43 \begin {gather*} \frac {1}{7} x^{7} e^{2} b^{3} B + \frac {1}{3} x^{6} e d b^{3} B + \frac {1}{2} x^{6} e^{2} b^{2} a B + \frac {1}{6} x^{6} e^{2} b^{3} A + \frac {1}{5} x^{5} d^{2} b^{3} B + \frac {6}{5} x^{5} e d b^{2} a B + \frac {3}{5} x^{5} e^{2} b a^{2} B + \frac {2}{5} x^{5} e d b^{3} A + \frac {3}{5} x^{5} e^{2} b^{2} a A + \frac {3}{4} x^{4} d^{2} b^{2} a B + \frac {3}{2} x^{4} e d b a^{2} B + \frac {1}{4} x^{4} e^{2} a^{3} B + \frac {1}{4} x^{4} d^{2} b^{3} A + \frac {3}{2} x^{4} e d b^{2} a A + \frac {3}{4} x^{4} e^{2} b a^{2} A + x^{3} d^{2} b a^{2} B + \frac {2}{3} x^{3} e d a^{3} B + x^{3} d^{2} b^{2} a A + 2 x^{3} e d b a^{2} A + \frac {1}{3} x^{3} e^{2} a^{3} A + \frac {1}{2} x^{2} d^{2} a^{3} B + \frac {3}{2} x^{2} d^{2} b a^{2} A + x^{2} e d a^{3} A + x d^{2} a^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.26, size = 287, normalized size = 2.43 \begin {gather*} \frac {1}{7} \, B b^{3} x^{7} e^{2} + \frac {1}{3} \, B b^{3} d x^{6} e + \frac {1}{5} \, B b^{3} d^{2} x^{5} + \frac {1}{2} \, B a b^{2} x^{6} e^{2} + \frac {1}{6} \, A b^{3} x^{6} e^{2} + \frac {6}{5} \, B a b^{2} d x^{5} e + \frac {2}{5} \, A b^{3} d x^{5} e + \frac {3}{4} \, B a b^{2} d^{2} x^{4} + \frac {1}{4} \, A b^{3} d^{2} x^{4} + \frac {3}{5} \, B a^{2} b x^{5} e^{2} + \frac {3}{5} \, A a b^{2} x^{5} e^{2} + \frac {3}{2} \, B a^{2} b d x^{4} e + \frac {3}{2} \, A a b^{2} d x^{4} e + B a^{2} b d^{2} x^{3} + A a b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{3} x^{4} e^{2} + \frac {3}{4} \, A a^{2} b x^{4} e^{2} + \frac {2}{3} \, B a^{3} d x^{3} e + 2 \, A a^{2} b d x^{3} e + \frac {1}{2} \, B a^{3} d^{2} x^{2} + \frac {3}{2} \, A a^{2} b d^{2} x^{2} + \frac {1}{3} \, A a^{3} x^{3} e^{2} + A a^{3} d x^{2} e + A a^{3} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^2,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.00, size = 244, normalized size = 2.07 \begin {gather*} \frac {B \,b^{3} e^{2} x^{7}}{7}+A \,a^{3} d^{2} x +\frac {\left (2 B \,b^{3} d e +\left (b^{3} A +3 a \,b^{2} B \right ) e^{2}\right ) x^{6}}{6}+\frac {\left (B \,b^{3} d^{2}+2 \left (b^{3} A +3 a \,b^{2} B \right ) d e +\left (3 a \,b^{2} A +3 a^{2} b B \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) d^{2}+2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) d e +\left (3 A \,a^{2} b +B \,a^{3}\right ) e^{2}\right ) x^{4}}{4}+\frac {\left (A \,a^{3} e^{2}+\left (3 a \,b^{2} A +3 a^{2} b B \right ) d^{2}+2 \left (3 A \,a^{2} b +B \,a^{3}\right ) d e \right ) x^{3}}{3}+\frac {\left (2 A \,a^{3} d e +\left (3 A \,a^{2} b +B \,a^{3}\right ) d^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)*(e*x+d)^2,x)

[Out]

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

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maxima [B]  time = 0.63, size = 239, normalized size = 2.03 \begin {gather*} \frac {1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac {1}{6} \, {\left (2 \, B b^{3} d e + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B b^{3} d^{2} + 2 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{3} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{3} d e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.07, size = 231, normalized size = 1.96 \begin {gather*} x^4\,\left (\frac {B\,a^3\,e^2}{4}+\frac {3\,B\,a^2\,b\,d\,e}{2}+\frac {3\,A\,a^2\,b\,e^2}{4}+\frac {3\,B\,a\,b^2\,d^2}{4}+\frac {3\,A\,a\,b^2\,d\,e}{2}+\frac {A\,b^3\,d^2}{4}\right )+x^3\,\left (\frac {2\,B\,a^3\,d\,e}{3}+\frac {A\,a^3\,e^2}{3}+B\,a^2\,b\,d^2+2\,A\,a^2\,b\,d\,e+A\,a\,b^2\,d^2\right )+x^5\,\left (\frac {3\,B\,a^2\,b\,e^2}{5}+\frac {6\,B\,a\,b^2\,d\,e}{5}+\frac {3\,A\,a\,b^2\,e^2}{5}+\frac {B\,b^3\,d^2}{5}+\frac {2\,A\,b^3\,d\,e}{5}\right )+A\,a^3\,d^2\,x+\frac {a^2\,d\,x^2\,\left (2\,A\,a\,e+3\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b^2\,e\,x^6\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{6}+\frac {B\,b^3\,e^2\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^3*(d + e*x)^2,x)

[Out]

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

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sympy [B]  time = 0.11, size = 296, normalized size = 2.51 \begin {gather*} A a^{3} d^{2} x + \frac {B b^{3} e^{2} x^{7}}{7} + x^{6} \left (\frac {A b^{3} e^{2}}{6} + \frac {B a b^{2} e^{2}}{2} + \frac {B b^{3} d e}{3}\right ) + x^{5} \left (\frac {3 A a b^{2} e^{2}}{5} + \frac {2 A b^{3} d e}{5} + \frac {3 B a^{2} b e^{2}}{5} + \frac {6 B a b^{2} d e}{5} + \frac {B b^{3} d^{2}}{5}\right ) + x^{4} \left (\frac {3 A a^{2} b e^{2}}{4} + \frac {3 A a b^{2} d e}{2} + \frac {A b^{3} d^{2}}{4} + \frac {B a^{3} e^{2}}{4} + \frac {3 B a^{2} b d e}{2} + \frac {3 B a b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{3} e^{2}}{3} + 2 A a^{2} b d e + A a b^{2} d^{2} + \frac {2 B a^{3} d e}{3} + B a^{2} b d^{2}\right ) + x^{2} \left (A a^{3} d e + \frac {3 A a^{2} b d^{2}}{2} + \frac {B a^{3} d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**2,x)

[Out]

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

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