3.11.24 \(\int (a+b x)^2 (A+B x) (d+e x)^2 \, dx\) [1024]

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

[Out]

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

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Rubi [A]
time = 0.09, 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 = {78} \begin {gather*} \frac {e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac {(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac {(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.04, size = 157, normalized size = 1.33 \begin {gather*} a^2 A d^2 x+\frac {1}{2} a d (a B d+2 A (b d+a e)) x^2+\frac {1}{3} \left (2 a B d (b d+a e)+A \left (b^2 d^2+4 a b d e+a^2 e^2\right )\right ) x^3+\frac {1}{4} \left (a^2 B e^2+2 a b e (2 B d+A e)+b^2 d (B d+2 A e)\right ) x^4+\frac {1}{5} b e (2 b B d+A b e+2 a B e) x^5+\frac {1}{6} b^2 B e^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.08, size = 169, normalized size = 1.43

method	result	size
default	\(\frac {b^{2} B \,e^{2} x^{6}}{6}+\frac {\left (\left (b^{2} A +2 a b B \right ) e^{2}+2 b^{2} B d e \right ) x^{5}}{5}+\frac {\left (\left (2 a b A +a^{2} B \right ) e^{2}+2 \left (b^{2} A +2 a b B \right ) d e +b^{2} B \,d^{2}\right ) x^{4}}{4}+\frac {\left (a^{2} A \,e^{2}+2 \left (2 a b A +a^{2} B \right ) d e +\left (b^{2} A +2 a b B \right ) d^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{2} A d e +\left (2 a b A +a^{2} B \right ) d^{2}\right ) x^{2}}{2}+a^{2} A \,d^{2} x\)	\(169\)
norman	\(\frac {b^{2} B \,e^{2} x^{6}}{6}+\left (\frac {1}{5} A \,b^{2} e^{2}+\frac {2}{5} B a b \,e^{2}+\frac {2}{5} b^{2} B d e \right ) x^{5}+\left (\frac {1}{2} A a b \,e^{2}+\frac {1}{2} A \,b^{2} d e +\frac {1}{4} B \,a^{2} e^{2}+B a b d e +\frac {1}{4} b^{2} B \,d^{2}\right ) x^{4}+\left (\frac {1}{3} a^{2} A \,e^{2}+\frac {4}{3} A a b d e +\frac {1}{3} A \,b^{2} d^{2}+\frac {2}{3} B \,a^{2} d e +\frac {2}{3} B a b \,d^{2}\right ) x^{3}+\left (a^{2} A d e +A a b \,d^{2}+\frac {1}{2} B \,a^{2} d^{2}\right ) x^{2}+a^{2} A \,d^{2} x\)	\(172\)
gosper	\(\frac {1}{6} b^{2} B \,e^{2} x^{6}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} b^{2} B d e +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{4} x^{4} B \,a^{2} e^{2}+x^{4} B a b d e +\frac {1}{4} x^{4} b^{2} B \,d^{2}+\frac {1}{3} x^{3} a^{2} A \,e^{2}+\frac {4}{3} x^{3} A a b d e +\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} a^{2} A d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{2}+a^{2} A \,d^{2} x\)	\(200\)
risch	\(\frac {1}{6} b^{2} B \,e^{2} x^{6}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} b^{2} B d e +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {1}{4} x^{4} B \,a^{2} e^{2}+x^{4} B a b d e +\frac {1}{4} x^{4} b^{2} B \,d^{2}+\frac {1}{3} x^{3} a^{2} A \,e^{2}+\frac {4}{3} x^{3} A a b d e +\frac {1}{3} x^{3} A \,b^{2} d^{2}+\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} a^{2} A d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{2} B \,a^{2} d^{2}+a^{2} A \,d^{2} x\)	\(200\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 172, normalized size = 1.46 \begin {gather*} \frac {1}{6} \, B b^{2} x^{6} e^{2} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (2 \, B b^{2} d e + 2 \, B a b e^{2} + A b^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{2} + B a^{2} e^{2} + 2 \, A a b e^{2} + 2 \, {\left (2 \, B a b e + A b^{2} e\right )} d\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{2} + 2 \, {\left (B a^{2} e + 2 \, A a b e\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, 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)^2*(B*x+A)*(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 174, normalized size = 1.47 \begin {gather*} \frac {1}{4} \, B b^{2} d^{2} x^{4} + A a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, B a b + A b^{2}\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} x^{2} + \frac {1}{60} \, {\left (10 \, B b^{2} x^{6} + 20 \, A a^{2} x^{3} + 12 \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} e^{2} + \frac {1}{30} \, {\left (12 \, B b^{2} d x^{5} + 30 \, A a^{2} d x^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} d x^{4} + 20 \, {\left (B a^{2} + 2 \, A a b\right )} d 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)^2,x, algorithm="fricas")

[Out]

1/4*B*b^2*d^2*x^4 + A*a^2*d^2*x + 1/3*(2*B*a*b + A*b^2)*d^2*x^3 + 1/2*(B*a^2 + 2*A*a*b)*d^2*x^2 + 1/60*(10*B*b
^2*x^6 + 20*A*a^2*x^3 + 12*(2*B*a*b + A*b^2)*x^5 + 15*(B*a^2 + 2*A*a*b)*x^4)*e^2 + 1/30*(12*B*b^2*d*x^5 + 30*A
*a^2*d*x^2 + 15*(2*B*a*b + A*b^2)*d*x^4 + 20*(B*a^2 + 2*A*a*b)*d*x^3)*e

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Sympy [A]
time = 0.02, size = 202, normalized size = 1.71 \begin {gather*} A a^{2} d^{2} x + \frac {B b^{2} e^{2} x^{6}}{6} + x^{5} \left (\frac {A b^{2} e^{2}}{5} + \frac {2 B a b e^{2}}{5} + \frac {2 B b^{2} d e}{5}\right ) + x^{4} \left (\frac {A a b e^{2}}{2} + \frac {A b^{2} d e}{2} + \frac {B a^{2} e^{2}}{4} + B a b d e + \frac {B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {4 A a b d e}{3} + \frac {A b^{2} d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac {B a^{2} d^{2}}{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)**2,x)

[Out]

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

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Giac [A]
time = 2.34, size = 199, normalized size = 1.69 \begin {gather*} \frac {1}{6} \, B b^{2} x^{6} e^{2} + \frac {2}{5} \, B b^{2} d x^{5} e + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {2}{5} \, B a b x^{5} e^{2} + \frac {1}{5} \, A b^{2} x^{5} e^{2} + B a b d x^{4} e + \frac {1}{2} \, A b^{2} d x^{4} e + \frac {2}{3} \, B a b d^{2} x^{3} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{2} x^{4} e^{2} + \frac {1}{2} \, A a b x^{4} e^{2} + \frac {2}{3} \, B a^{2} d x^{3} e + \frac {4}{3} \, A a b d x^{3} e + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + \frac {1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} d^{2} 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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.07, size = 157, normalized size = 1.33 \begin {gather*} x^3\,\left (\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {4\,A\,a\,b\,d\,e}{3}+\frac {A\,b^2\,d^2}{3}\right )+x^4\,\left (\frac {B\,a^2\,e^2}{4}+B\,a\,b\,d\,e+\frac {A\,a\,b\,e^2}{2}+\frac {B\,b^2\,d^2}{4}+\frac {A\,b^2\,d\,e}{2}\right )+\frac {a\,d\,x^2\,\left (2\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e\,x^5\,\left (A\,b\,e+2\,B\,a\,e+2\,B\,b\,d\right )}{5}+A\,a^2\,d^2\,x+\frac {B\,b^2\,e^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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