3.535 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^8} \, dx\)

Optimal. Leaf size=99 \[ -\frac{a^3 (a B+4 A b)}{6 x^6}-\frac{2 a^2 b (2 a B+3 A b)}{5 x^5}-\frac{a^4 A}{7 x^7}-\frac{a b^2 (3 a B+2 A b)}{2 x^4}-\frac{b^3 (4 a B+A b)}{3 x^3}-\frac{b^4 B}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0476249, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{a^3 (a B+4 A b)}{6 x^6}-\frac{2 a^2 b (2 a B+3 A b)}{5 x^5}-\frac{a^4 A}{7 x^7}-\frac{a b^2 (3 a B+2 A b)}{2 x^4}-\frac{b^3 (4 a B+A b)}{3 x^3}-\frac{b^4 B}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^8,x]

[Out]

-(a^4*A)/(7*x^7) - (a^3*(4*A*b + a*B))/(6*x^6) - (2*a^2*b*(3*A*b + 2*a*B))/(5*x^5) - (a*b^2*(2*A*b + 3*a*B))/(
2*x^4) - (b^3*(A*b + 4*a*B))/(3*x^3) - (b^4*B)/(2*x^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 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 \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^8} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{x^8} \, dx\\ &=\int \left (\frac{a^4 A}{x^8}+\frac{a^3 (4 A b+a B)}{x^7}+\frac{2 a^2 b (3 A b+2 a B)}{x^6}+\frac{2 a b^2 (2 A b+3 a B)}{x^5}+\frac{b^3 (A b+4 a B)}{x^4}+\frac{b^4 B}{x^3}\right ) \, dx\\ &=-\frac{a^4 A}{7 x^7}-\frac{a^3 (4 A b+a B)}{6 x^6}-\frac{2 a^2 b (3 A b+2 a B)}{5 x^5}-\frac{a b^2 (2 A b+3 a B)}{2 x^4}-\frac{b^3 (A b+4 a B)}{3 x^3}-\frac{b^4 B}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0242833, size = 88, normalized size = 0.89 \[ -\frac{63 a^2 b^2 x^2 (4 A+5 B x)+28 a^3 b x (5 A+6 B x)+5 a^4 (6 A+7 B x)+70 a b^3 x^3 (3 A+4 B x)+35 b^4 x^4 (2 A+3 B x)}{210 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^8,x]

[Out]

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

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Maple [A]  time = 0.006, size = 88, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{4}}{7\,{x}^{7}}}-{\frac{{a}^{3} \left ( 4\,Ab+aB \right ) }{6\,{x}^{6}}}-{\frac{2\,{a}^{2}b \left ( 3\,Ab+2\,aB \right ) }{5\,{x}^{5}}}-{\frac{a{b}^{2} \left ( 2\,Ab+3\,aB \right ) }{2\,{x}^{4}}}-{\frac{{b}^{3} \left ( Ab+4\,aB \right ) }{3\,{x}^{3}}}-{\frac{{b}^{4}B}{2\,{x}^{2}}} \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^8,x)

[Out]

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

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Maxima [A]  time = 0.999069, size = 134, normalized size = 1.35 \begin{align*} -\frac{105 \, B b^{4} x^{5} + 30 \, A a^{4} + 70 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 105 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 84 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{210 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-1/210*(105*B*b^4*x^5 + 30*A*a^4 + 70*(4*B*a*b^3 + A*b^4)*x^4 + 105*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 84*(2*B*a^
3*b + 3*A*a^2*b^2)*x^2 + 35*(B*a^4 + 4*A*a^3*b)*x)/x^7

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Fricas [A]  time = 1.2929, size = 224, normalized size = 2.26 \begin{align*} -\frac{105 \, B b^{4} x^{5} + 30 \, A a^{4} + 70 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 105 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 84 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{210 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/210*(105*B*b^4*x^5 + 30*A*a^4 + 70*(4*B*a*b^3 + A*b^4)*x^4 + 105*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 84*(2*B*a^
3*b + 3*A*a^2*b^2)*x^2 + 35*(B*a^4 + 4*A*a^3*b)*x)/x^7

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Sympy [A]  time = 3.83808, size = 102, normalized size = 1.03 \begin{align*} - \frac{30 A a^{4} + 105 B b^{4} x^{5} + x^{4} \left (70 A b^{4} + 280 B a b^{3}\right ) + x^{3} \left (210 A a b^{3} + 315 B a^{2} b^{2}\right ) + x^{2} \left (252 A a^{2} b^{2} + 168 B a^{3} b\right ) + x \left (140 A a^{3} b + 35 B a^{4}\right )}{210 x^{7}} \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**8,x)

[Out]

-(30*A*a**4 + 105*B*b**4*x**5 + x**4*(70*A*b**4 + 280*B*a*b**3) + x**3*(210*A*a*b**3 + 315*B*a**2*b**2) + x**2
*(252*A*a**2*b**2 + 168*B*a**3*b) + x*(140*A*a**3*b + 35*B*a**4))/(210*x**7)

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Giac [A]  time = 1.14141, size = 134, normalized size = 1.35 \begin{align*} -\frac{105 \, B b^{4} x^{5} + 280 \, B a b^{3} x^{4} + 70 \, A b^{4} x^{4} + 315 \, B a^{2} b^{2} x^{3} + 210 \, A a b^{3} x^{3} + 168 \, B a^{3} b x^{2} + 252 \, A a^{2} b^{2} x^{2} + 35 \, B a^{4} x + 140 \, A a^{3} b x + 30 \, A a^{4}}{210 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

-1/210*(105*B*b^4*x^5 + 280*B*a*b^3*x^4 + 70*A*b^4*x^4 + 315*B*a^2*b^2*x^3 + 210*A*a*b^3*x^3 + 168*B*a^3*b*x^2
+ 252*A*a^2*b^2*x^2 + 35*B*a^4*x + 140*A*a^3*b*x + 30*A*a^4)/x^7