3.11.25 \(\int (a+b x)^2 (A+B x) (d+e x) \, dx\) [1025]

Optimal. Leaf size=75 \[ \frac {(A b-a B) (b d-a e) (a+b x)^3}{3 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^4}{4 b^3}+\frac {B e (a+b x)^5}{5 b^3} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 75, 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 = {78} \begin {gather*} \frac {(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac {(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {B e (a+b x)^5}{5 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

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

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Mathematica [A]
time = 0.02, size = 96, normalized size = 1.28 \begin {gather*} 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 d+2 a b B d+2 a A b e+a^2 B e\right ) x^3+\frac {1}{4} b (b B d+A b e+2 a B e) x^4+\frac {1}{5} b^2 B e x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 101, normalized size = 1.35

method	result	size
norman	\(\frac {b^{2} B e \,x^{5}}{5}+\left (\frac {1}{4} A \,b^{2} e +\frac {1}{2} B a b e +\frac {1}{4} b^{2} B d \right ) x^{4}+\left (\frac {2}{3} A a b e +\frac {1}{3} A \,b^{2} d +\frac {1}{3} B \,a^{2} e +\frac {2}{3} B a b d \right ) x^{3}+\left (\frac {1}{2} a^{2} A e +A a b d +\frac {1}{2} B \,a^{2} d \right ) x^{2}+a^{2} A d x\)	\(99\)
default	\(\frac {b^{2} B e \,x^{5}}{5}+\frac {\left (\left (b^{2} A +2 a b B \right ) e +b^{2} B d \right ) x^{4}}{4}+\frac {\left (\left (2 a b A +a^{2} B \right ) e +\left (b^{2} A +2 a b B \right ) d \right ) x^{3}}{3}+\frac {\left (a^{2} A e +\left (2 a b A +a^{2} B \right ) d \right ) x^{2}}{2}+a^{2} A d x\)	\(101\)
gosper	\(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\)	\(114\)
risch	\(\frac {1}{5} b^{2} B e \,x^{5}+\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} B a b e +\frac {1}{4} x^{4} b^{2} B d +\frac {2}{3} x^{3} A a b e +\frac {1}{3} x^{3} A \,b^{2} d +\frac {1}{3} x^{3} B \,a^{2} e +\frac {2}{3} x^{3} B a b d +\frac {1}{2} x^{2} a^{2} A e +x^{2} A a b d +\frac {1}{2} x^{2} B \,a^{2} d +a^{2} A d x\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 104, normalized size = 1.39 \begin {gather*} \frac {1}{5} \, B b^{2} x^{5} e + A a^{2} d x + \frac {1}{4} \, {\left (B b^{2} d + 2 \, B a b e + A b^{2} e\right )} x^{4} + \frac {1}{3} \, {\left (B a^{2} e + 2 \, A a b e + {\left (2 \, B a b + A b^{2}\right )} d\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)^2*(B*x+A)*(e*x+d),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 107, normalized size = 1.43 \begin {gather*} \frac {1}{4} \, B b^{2} d x^{4} + A a^{2} d x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} d x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} d x^{2} + \frac {1}{60} \, {\left (12 \, B b^{2} x^{5} + 30 \, A a^{2} x^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 20 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*b^2*d*x^4 + A*a^2*d*x + 1/3*(2*B*a*b + A*b^2)*d*x^3 + 1/2*(B*a^2 + 2*A*a*b)*d*x^2 + 1/60*(12*B*b^2*x^5 +
30*A*a^2*x^2 + 15*(2*B*a*b + A*b^2)*x^4 + 20*(B*a^2 + 2*A*a*b)*x^3)*e

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Sympy [A]
time = 0.02, size = 116, normalized size = 1.55 \begin {gather*} A a^{2} d x + \frac {B b^{2} e x^{5}}{5} + x^{4} \left (\frac {A b^{2} e}{4} + \frac {B a b e}{2} + \frac {B b^{2} d}{4}\right ) + x^{3} \cdot \left (\frac {2 A a b e}{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)**2*(B*x+A)*(e*x+d),x)

[Out]

A*a**2*d*x + B*b**2*e*x**5/5 + x**4*(A*b**2*e/4 + B*a*b*e/2 + B*b**2*d/4) + x**3*(2*A*a*b*e/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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Giac [A]
time = 1.91, size = 119, normalized size = 1.59 \begin {gather*} \frac {1}{5} \, B b^{2} x^{5} e + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, B a b x^{4} e + \frac {1}{4} \, A b^{2} x^{4} e + \frac {2}{3} \, B a b d x^{3} + \frac {1}{3} \, A b^{2} 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)^2*(B*x+A)*(e*x+d),x, algorithm="giac")

[Out]

1/5*B*b^2*x^5*e + 1/4*B*b^2*d*x^4 + 1/2*B*a*b*x^4*e + 1/4*A*b^2*x^4*e + 2/3*B*a*b*d*x^3 + 1/3*A*b^2*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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Mupad [B]
time = 0.05, size = 98, normalized size = 1.31 \begin {gather*} x^3\,\left (\frac {A\,b^2\,d}{3}+\frac {B\,a^2\,e}{3}+\frac {2\,A\,a\,b\,e}{3}+\frac {2\,B\,a\,b\,d}{3}\right )+x^2\,\left (\frac {A\,a^2\,e}{2}+\frac {B\,a^2\,d}{2}+A\,a\,b\,d\right )+x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {B\,a\,b\,e}{2}\right )+A\,a^2\,d\,x+\frac {B\,b^2\,e\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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