3.553 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^9} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^7 (A b-8 a B)}{56 a^2 x^7}-\frac{A (a+b x)^7}{8 a x^8} \]

[Out]

-(A*(a + b*x)^7)/(8*a*x^8) + ((A*b - 8*a*B)*(a + b*x)^7)/(56*a^2*x^7)

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Rubi [A]  time = 0.012539, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 78, 37} \[ \frac{(a+b x)^7 (A b-8 a B)}{56 a^2 x^7}-\frac{A (a+b x)^7}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^9,x]

[Out]

-(A*(a + b*x)^7)/(8*a*x^8) + ((A*b - 8*a*B)*(a + b*x)^7)/(56*a^2*x^7)

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^9} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^9} \, dx\\ &=-\frac{A (a+b x)^7}{8 a x^8}+\frac{(-A b+8 a B) \int \frac{(a+b x)^6}{x^8} \, dx}{8 a}\\ &=-\frac{A (a+b x)^7}{8 a x^8}+\frac{(A b-8 a B) (a+b x)^7}{56 a^2 x^7}\\ \end{align*}

Mathematica [B]  time = 0.0332997, size = 123, normalized size = 2.8 \[ -\frac{28 a^4 b^2 x^2 (5 A+6 B x)+56 a^3 b^3 x^3 (4 A+5 B x)+70 a^2 b^4 x^4 (3 A+4 B x)+8 a^5 b x (6 A+7 B x)+a^6 (7 A+8 B x)+56 a b^5 x^5 (2 A+3 B x)+28 b^6 x^6 (A+2 B x)}{56 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^9,x]

[Out]

-(28*b^6*x^6*(A + 2*B*x) + 56*a*b^5*x^5*(2*A + 3*B*x) + 70*a^2*b^4*x^4*(3*A + 4*B*x) + 56*a^3*b^3*x^3*(4*A + 5
*B*x) + 28*a^4*b^2*x^2*(5*A + 6*B*x) + 8*a^5*b*x*(6*A + 7*B*x) + a^6*(7*A + 8*B*x))/(56*x^8)

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Maple [B]  time = 0.007, size = 128, normalized size = 2.9 \begin{align*} -{\frac{a{b}^{4} \left ( 2\,Ab+5\,aB \right ) }{{x}^{3}}}-{\frac{{b}^{5} \left ( Ab+6\,aB \right ) }{2\,{x}^{2}}}-{\frac{B{b}^{6}}{x}}-{\frac{{a}^{5} \left ( 6\,Ab+aB \right ) }{7\,{x}^{7}}}-{\frac{{a}^{4}b \left ( 5\,Ab+2\,aB \right ) }{2\,{x}^{6}}}-{\frac{{a}^{3}{b}^{2} \left ( 4\,Ab+3\,aB \right ) }{{x}^{5}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( 3\,Ab+4\,aB \right ) }{4\,{x}^{4}}}-{\frac{A{a}^{6}}{8\,{x}^{8}}} \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)^3/x^9,x)

[Out]

-a*b^4*(2*A*b+5*B*a)/x^3-1/2*b^5*(A*b+6*B*a)/x^2-B*b^6/x-1/7*a^5*(6*A*b+B*a)/x^7-1/2*a^4*b*(5*A*b+2*B*a)/x^6-a
^3*b^2*(4*A*b+3*B*a)/x^5-5/4*a^2*b^3*(3*A*b+4*B*a)/x^4-1/8*A*a^6/x^8

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Maxima [B]  time = 0.993295, size = 198, normalized size = 4.5 \begin{align*} -\frac{56 \, B b^{6} x^{7} + 7 \, A a^{6} + 28 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 56 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 70 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 56 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 28 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 8 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{56 \, x^{8}} \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)^3/x^9,x, algorithm="maxima")

[Out]

-1/56*(56*B*b^6*x^7 + 7*A*a^6 + 28*(6*B*a*b^5 + A*b^6)*x^6 + 56*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 70*(4*B*a^3*b^
3 + 3*A*a^2*b^4)*x^4 + 56*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 28*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 8*(B*a^6 + 6*A*
a^5*b)*x)/x^8

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Fricas [B]  time = 1.36571, size = 317, normalized size = 7.2 \begin{align*} -\frac{56 \, B b^{6} x^{7} + 7 \, A a^{6} + 28 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 56 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 70 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 56 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 28 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 8 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{56 \, x^{8}} \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)^3/x^9,x, algorithm="fricas")

[Out]

-1/56*(56*B*b^6*x^7 + 7*A*a^6 + 28*(6*B*a*b^5 + A*b^6)*x^6 + 56*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 70*(4*B*a^3*b^
3 + 3*A*a^2*b^4)*x^4 + 56*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 28*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 8*(B*a^6 + 6*A*
a^5*b)*x)/x^8

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Sympy [B]  time = 8.10809, size = 150, normalized size = 3.41 \begin{align*} - \frac{7 A a^{6} + 56 B b^{6} x^{7} + x^{6} \left (28 A b^{6} + 168 B a b^{5}\right ) + x^{5} \left (112 A a b^{5} + 280 B a^{2} b^{4}\right ) + x^{4} \left (210 A a^{2} b^{4} + 280 B a^{3} b^{3}\right ) + x^{3} \left (224 A a^{3} b^{3} + 168 B a^{4} b^{2}\right ) + x^{2} \left (140 A a^{4} b^{2} + 56 B a^{5} b\right ) + x \left (48 A a^{5} b + 8 B a^{6}\right )}{56 x^{8}} \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)**3/x**9,x)

[Out]

-(7*A*a**6 + 56*B*b**6*x**7 + x**6*(28*A*b**6 + 168*B*a*b**5) + x**5*(112*A*a*b**5 + 280*B*a**2*b**4) + x**4*(
210*A*a**2*b**4 + 280*B*a**3*b**3) + x**3*(224*A*a**3*b**3 + 168*B*a**4*b**2) + x**2*(140*A*a**4*b**2 + 56*B*a
**5*b) + x*(48*A*a**5*b + 8*B*a**6))/(56*x**8)

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Giac [B]  time = 1.16749, size = 198, normalized size = 4.5 \begin{align*} -\frac{56 \, B b^{6} x^{7} + 168 \, B a b^{5} x^{6} + 28 \, A b^{6} x^{6} + 280 \, B a^{2} b^{4} x^{5} + 112 \, A a b^{5} x^{5} + 280 \, B a^{3} b^{3} x^{4} + 210 \, A a^{2} b^{4} x^{4} + 168 \, B a^{4} b^{2} x^{3} + 224 \, A a^{3} b^{3} x^{3} + 56 \, B a^{5} b x^{2} + 140 \, A a^{4} b^{2} x^{2} + 8 \, B a^{6} x + 48 \, A a^{5} b x + 7 \, A a^{6}}{56 \, x^{8}} \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)^3/x^9,x, algorithm="giac")

[Out]

-1/56*(56*B*b^6*x^7 + 168*B*a*b^5*x^6 + 28*A*b^6*x^6 + 280*B*a^2*b^4*x^5 + 112*A*a*b^5*x^5 + 280*B*a^3*b^3*x^4
+ 210*A*a^2*b^4*x^4 + 168*B*a^4*b^2*x^3 + 224*A*a^3*b^3*x^3 + 56*B*a^5*b*x^2 + 140*A*a^4*b^2*x^2 + 8*B*a^6*x
+ 48*A*a^5*b*x + 7*A*a^6)/x^8