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

Optimal. Leaf size=61 \[ \frac {(a+b x)^6 (A b-2 a B)}{6 b^3}-\frac {a (a+b x)^5 (A b-a B)}{5 b^3}+\frac {B (a+b x)^7}{7 b^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 76} \begin {gather*} \frac {(a+b x)^6 (A b-2 a B)}{6 b^3}-\frac {a (a+b x)^5 (A b-a B)}{5 b^3}+\frac {B (a+b x)^7}{7 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(A*b - a*B)*(a + b*x)^5)/(5*b^3) + ((A*b - 2*a*B)*(a + b*x)^6)/(6*b^3) + (B*(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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 88, normalized size = 1.44 \begin {gather*} \frac {1}{210} x^2 \left (35 a^4 (3 A+2 B x)+70 a^3 b x (4 A+3 B x)+63 a^2 b^2 x^2 (5 A+4 B x)+28 a b^3 x^3 (6 A+5 B x)+5 b^4 x^4 (7 A+6 B x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(35*a^4*(3*A + 2*B*x) + 70*a^3*b*x*(4*A + 3*B*x) + 63*a^2*b^2*x^2*(5*A + 4*B*x) + 28*a*b^3*x^3*(6*A + 5*B
*x) + 5*b^4*x^4*(7*A + 6*B*x)))/210

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 100, normalized size = 1.64 \begin {gather*} \frac {1}{7} x^{7} b^{4} B + \frac {2}{3} x^{6} b^{3} a B + \frac {1}{6} x^{6} b^{4} A + \frac {6}{5} x^{5} b^{2} a^{2} B + \frac {4}{5} x^{5} b^{3} a A + x^{4} b a^{3} B + \frac {3}{2} x^{4} b^{2} a^{2} A + \frac {1}{3} x^{3} a^{4} B + \frac {4}{3} x^{3} b a^{3} A + \frac {1}{2} x^{2} a^{4} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 100, normalized size = 1.64 \begin {gather*} \frac {1}{7} \, B b^{4} x^{7} + \frac {2}{3} \, B a b^{3} x^{6} + \frac {1}{6} \, A b^{4} x^{6} + \frac {6}{5} \, B a^{2} b^{2} x^{5} + \frac {4}{5} \, A a b^{3} x^{5} + B a^{3} b x^{4} + \frac {3}{2} \, A a^{2} b^{2} x^{4} + \frac {1}{3} \, B a^{4} x^{3} + \frac {4}{3} \, A a^{3} b x^{3} + \frac {1}{2} \, A a^{4} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 100, normalized size = 1.64 \begin {gather*} \frac {B \,b^{4} x^{7}}{7}+\frac {A \,a^{4} x^{2}}{2}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{6}}{6}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) x^{5}}{5}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) x^{4}}{4}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.46, size = 99, normalized size = 1.62 \begin {gather*} \frac {1}{7} \, B b^{4} x^{7} + \frac {1}{2} \, A a^{4} x^{2} + \frac {1}{6} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{6} + \frac {2}{5} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 91, normalized size = 1.49 \begin {gather*} x^3\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,b\,a^3}{3}\right )+x^6\,\left (\frac {A\,b^4}{6}+\frac {2\,B\,a\,b^3}{3}\right )+\frac {A\,a^4\,x^2}{2}+\frac {B\,b^4\,x^7}{7}+\frac {a^2\,b\,x^4\,\left (3\,A\,b+2\,B\,a\right )}{2}+\frac {2\,a\,b^2\,x^5\,\left (2\,A\,b+3\,B\,a\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.09, size = 107, normalized size = 1.75 \begin {gather*} \frac {A a^{4} x^{2}}{2} + \frac {B b^{4} x^{7}}{7} + x^{6} \left (\frac {A b^{4}}{6} + \frac {2 B a b^{3}}{3}\right ) + x^{5} \left (\frac {4 A a b^{3}}{5} + \frac {6 B a^{2} b^{2}}{5}\right ) + x^{4} \left (\frac {3 A a^{2} b^{2}}{2} + B a^{3} b\right ) + x^{3} \left (\frac {4 A a^{3} b}{3} + \frac {B a^{4}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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