3.1679 \(\int (A+B x) (d+e x) (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac{(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac{B e (a+b x)^7}{7 b^3} \]

[Out]

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

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Rubi [A]  time = 0.137509, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 77} \[ \frac{(a+b x)^6 (-2 a B e+A b e+b B d)}{6 b^3}+\frac{(a+b x)^5 (A b-a B) (b d-a e)}{5 b^3}+\frac{B e (a+b x)^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [B]  time = 0.0579466, size = 172, normalized size = 2.29 \[ \frac{1}{3} a^2 x^3 (2 A b (2 a e+3 b d)+a B (a e+4 b d))+\frac{1}{2} a^3 x^2 (a A e+a B d+4 A b d)+a^4 A d x+\frac{1}{6} b^3 x^6 (4 a B e+A b e+b B d)+\frac{1}{5} b^2 x^5 (A b (4 a e+b d)+2 a B (3 a e+2 b d))+\frac{1}{2} a b x^4 (A b (3 a e+2 b d)+a B (2 a e+3 b d))+\frac{1}{7} b^4 B e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.002, size = 176, normalized size = 2.4 \begin{align*}{\frac{Be{b}^{4}{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){b}^{4}+4\,Bea{b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( Ad{b}^{4}+4\, \left ( Ae+Bd \right ) a{b}^{3}+6\,Be{a}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,Ada{b}^{3}+6\, \left ( Ae+Bd \right ){a}^{2}{b}^{2}+4\,Be{a}^{3}b \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,Ad{a}^{2}{b}^{2}+4\, \left ( Ae+Bd \right ){a}^{3}b+Be{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,Ad{a}^{3}b+ \left ( Ae+Bd \right ){a}^{4} \right ){x}^{2}}{2}}+Ad{a}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.06231, size = 267, normalized size = 3.56 \begin{align*} \frac{1}{7} \, B b^{4} e x^{7} + A a^{4} d x + \frac{1}{6} \,{\left (B b^{4} d +{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left ({\left (4 \, B a b^{3} + A b^{4}\right )} d + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e\right )} x^{5} + \frac{1}{2} \,{\left ({\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d +{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d +{\left (B a^{4} + 4 \, A a^{3} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{4} e +{\left (B a^{4} + 4 \, A a^{3} b\right )} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.36694, size = 504, normalized size = 6.72 \begin{align*} \frac{1}{7} x^{7} e b^{4} B + \frac{1}{6} x^{6} d b^{4} B + \frac{2}{3} x^{6} e b^{3} a B + \frac{1}{6} x^{6} e b^{4} A + \frac{4}{5} x^{5} d b^{3} a B + \frac{6}{5} x^{5} e b^{2} a^{2} B + \frac{1}{5} x^{5} d b^{4} A + \frac{4}{5} x^{5} e b^{3} a A + \frac{3}{2} x^{4} d b^{2} a^{2} B + x^{4} e b a^{3} B + x^{4} d b^{3} a A + \frac{3}{2} x^{4} e b^{2} a^{2} A + \frac{4}{3} x^{3} d b a^{3} B + \frac{1}{3} x^{3} e a^{4} B + 2 x^{3} d b^{2} a^{2} A + \frac{4}{3} x^{3} e b a^{3} A + \frac{1}{2} x^{2} d a^{4} B + 2 x^{2} d b a^{3} A + \frac{1}{2} x^{2} e a^{4} A + x d a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.091704, size = 226, normalized size = 3.01 \begin{align*} A a^{4} d x + \frac{B b^{4} e x^{7}}{7} + x^{6} \left (\frac{A b^{4} e}{6} + \frac{2 B a b^{3} e}{3} + \frac{B b^{4} d}{6}\right ) + x^{5} \left (\frac{4 A a b^{3} e}{5} + \frac{A b^{4} d}{5} + \frac{6 B a^{2} b^{2} e}{5} + \frac{4 B a b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b^{2} e}{2} + A a b^{3} d + B a^{3} b e + \frac{3 B a^{2} b^{2} d}{2}\right ) + x^{3} \left (\frac{4 A a^{3} b e}{3} + 2 A a^{2} b^{2} d + \frac{B a^{4} e}{3} + \frac{4 B a^{3} b d}{3}\right ) + x^{2} \left (\frac{A a^{4} e}{2} + 2 A a^{3} b d + \frac{B a^{4} d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13598, size = 305, normalized size = 4.07 \begin{align*} \frac{1}{7} \, B b^{4} x^{7} e + \frac{1}{6} \, B b^{4} d x^{6} + \frac{2}{3} \, B a b^{3} x^{6} e + \frac{1}{6} \, A b^{4} x^{6} e + \frac{4}{5} \, B a b^{3} d x^{5} + \frac{1}{5} \, A b^{4} d x^{5} + \frac{6}{5} \, B a^{2} b^{2} x^{5} e + \frac{4}{5} \, A a b^{3} x^{5} e + \frac{3}{2} \, B a^{2} b^{2} d x^{4} + A a b^{3} d x^{4} + B a^{3} b x^{4} e + \frac{3}{2} \, A a^{2} b^{2} x^{4} e + \frac{4}{3} \, B a^{3} b d x^{3} + 2 \, A a^{2} b^{2} d x^{3} + \frac{1}{3} \, B a^{4} x^{3} e + \frac{4}{3} \, A a^{3} b x^{3} e + \frac{1}{2} \, B a^{4} d x^{2} + 2 \, A a^{3} b d x^{2} + \frac{1}{2} \, A a^{4} x^{2} e + A a^{4} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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