3.1680 \(\int (A+B x) (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^5 (A b-a B)}{5 b^2}+\frac{B (a+b x)^6}{6 b^2} \]

[Out]

((A*b - a*B)*(a + b*x)^5)/(5*b^2) + (B*(a + b*x)^6)/(6*b^2)

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Rubi [A]  time = 0.0162216, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac{(a+b x)^5 (A b-a B)}{5 b^2}+\frac{B (a+b x)^6}{6 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(a + b*x)^5)/(5*b^2) + (B*(a + b*x)^6)/(6*b^2)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [B]  time = 0.0169338, size = 84, normalized size = 2.21 \[ \frac{1}{30} x \left (15 a^2 b^2 x^2 (4 A+3 B x)+20 a^3 b x (3 A+2 B x)+15 a^4 (2 A+B x)+6 a b^3 x^3 (5 A+4 B x)+b^4 x^4 (6 A+5 B x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(15*a^4*(2*A + B*x) + 20*a^3*b*x*(3*A + 2*B*x) + 15*a^2*b^2*x^2*(4*A + 3*B*x) + 6*a*b^3*x^3*(5*A + 4*B*x) +
b^4*x^4*(6*A + 5*B*x)))/30

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Maple [B]  time = 0.001, size = 97, normalized size = 2.6 \begin{align*}{\frac{{b}^{4}B{x}^{6}}{6}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,{b}^{2}B{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{b}^{2}{a}^{2}+4\,{a}^{3}bB \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{2}}{2}}+{a}^{4}Ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.03214, size = 130, normalized size = 3.42 \begin{align*} \frac{1}{6} \, B b^{4} x^{6} + A a^{4} x + \frac{1}{5} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{5} + \frac{1}{2} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.34725, size = 217, normalized size = 5.71 \begin{align*} \frac{1}{6} x^{6} b^{4} B + \frac{4}{5} x^{5} b^{3} a B + \frac{1}{5} x^{5} b^{4} A + \frac{3}{2} x^{4} b^{2} a^{2} B + x^{4} b^{3} a A + \frac{4}{3} x^{3} b a^{3} B + 2 x^{3} b^{2} a^{2} A + \frac{1}{2} x^{2} a^{4} B + 2 x^{2} b a^{3} A + x a^{4} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.084416, size = 100, normalized size = 2.63 \begin{align*} A a^{4} x + \frac{B b^{4} x^{6}}{6} + x^{5} \left (\frac{A b^{4}}{5} + \frac{4 B a b^{3}}{5}\right ) + x^{4} \left (A a b^{3} + \frac{3 B a^{2} b^{2}}{2}\right ) + x^{3} \left (2 A a^{2} b^{2} + \frac{4 B a^{3} b}{3}\right ) + x^{2} \left (2 A a^{3} b + \frac{B a^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.12674, size = 131, normalized size = 3.45 \begin{align*} \frac{1}{6} \, B b^{4} x^{6} + \frac{4}{5} \, B a b^{3} x^{5} + \frac{1}{5} \, A b^{4} x^{5} + \frac{3}{2} \, B a^{2} b^{2} x^{4} + A a b^{3} x^{4} + \frac{4}{3} \, B a^{3} b x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac{1}{2} \, B a^{4} x^{2} + 2 \, A a^{3} b x^{2} + A a^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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