3.5.81 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{10}} \, dx\)

Optimal. Leaf size=72 \[ -\frac {b (a+b x)^7 (2 A b-9 a B)}{504 a^3 x^7}+\frac {(a+b x)^7 (2 A b-9 a B)}{72 a^2 x^8}-\frac {A (a+b x)^7}{9 a x^9} \]

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {27, 78, 45, 37} \begin {gather*} -\frac {b (a+b x)^7 (2 A b-9 a B)}{504 a^3 x^7}+\frac {(a+b x)^7 (2 A b-9 a B)}{72 a^2 x^8}-\frac {A (a+b x)^7}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^10,x]

[Out]

-(A*(a + b*x)^7)/(9*a*x^9) + ((2*A*b - 9*a*B)*(a + b*x)^7)/(72*a^2*x^8) - (b*(2*A*b - 9*a*B)*(a + b*x)^7)/(504
*a^3*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 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]

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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]))))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^{10}} \, dx\\ &=-\frac {A (a+b x)^7}{9 a x^9}+\frac {(-2 A b+9 a B) \int \frac {(a+b x)^6}{x^9} \, dx}{9 a}\\ &=-\frac {A (a+b x)^7}{9 a x^9}+\frac {(2 A b-9 a B) (a+b x)^7}{72 a^2 x^8}+\frac {(b (2 A b-9 a B)) \int \frac {(a+b x)^6}{x^8} \, dx}{72 a^2}\\ &=-\frac {A (a+b x)^7}{9 a x^9}+\frac {(2 A b-9 a B) (a+b x)^7}{72 a^2 x^8}-\frac {b (2 A b-9 a B) (a+b x)^7}{504 a^3 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 126, normalized size = 1.75 \begin {gather*} -\frac {7 a^6 (8 A+9 B x)+54 a^5 b x (7 A+8 B x)+180 a^4 b^2 x^2 (6 A+7 B x)+336 a^3 b^3 x^3 (5 A+6 B x)+378 a^2 b^4 x^4 (4 A+5 B x)+252 a b^5 x^5 (3 A+4 B x)+84 b^6 x^6 (2 A+3 B x)}{504 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^10,x]

[Out]

-1/504*(84*b^6*x^6*(2*A + 3*B*x) + 252*a*b^5*x^5*(3*A + 4*B*x) + 378*a^2*b^4*x^4*(4*A + 5*B*x) + 336*a^3*b^3*x
^3*(5*A + 6*B*x) + 180*a^4*b^2*x^2*(6*A + 7*B*x) + 54*a^5*b*x*(7*A + 8*B*x) + 7*a^6*(8*A + 9*B*x))/x^9

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^10,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^10, x]

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fricas [B]  time = 0.39, size = 147, normalized size = 2.04 \begin {gather*} -\frac {252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{504 \, x^{9}} \end {gather*}

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^10,x, algorithm="fricas")

[Out]

-1/504*(252*B*b^6*x^7 + 56*A*a^6 + 168*(6*B*a*b^5 + A*b^6)*x^6 + 378*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 504*(4*B*
a^3*b^3 + 3*A*a^2*b^4)*x^4 + 420*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 216*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 63*(B*a
^6 + 6*A*a^5*b)*x)/x^9

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giac [B]  time = 0.18, size = 147, normalized size = 2.04 \begin {gather*} -\frac {252 \, B b^{6} x^{7} + 1008 \, B a b^{5} x^{6} + 168 \, A b^{6} x^{6} + 1890 \, B a^{2} b^{4} x^{5} + 756 \, A a b^{5} x^{5} + 2016 \, B a^{3} b^{3} x^{4} + 1512 \, A a^{2} b^{4} x^{4} + 1260 \, B a^{4} b^{2} x^{3} + 1680 \, A a^{3} b^{3} x^{3} + 432 \, B a^{5} b x^{2} + 1080 \, A a^{4} b^{2} x^{2} + 63 \, B a^{6} x + 378 \, A a^{5} b x + 56 \, A a^{6}}{504 \, x^{9}} \end {gather*}

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^10,x, algorithm="giac")

[Out]

-1/504*(252*B*b^6*x^7 + 1008*B*a*b^5*x^6 + 168*A*b^6*x^6 + 1890*B*a^2*b^4*x^5 + 756*A*a*b^5*x^5 + 2016*B*a^3*b
^3*x^4 + 1512*A*a^2*b^4*x^4 + 1260*B*a^4*b^2*x^3 + 1680*A*a^3*b^3*x^3 + 432*B*a^5*b*x^2 + 1080*A*a^4*b^2*x^2 +
63*B*a^6*x + 378*A*a^5*b*x + 56*A*a^6)/x^9

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maple [A]  time = 0.06, size = 128, normalized size = 1.78 \begin {gather*} -\frac {B \,b^{6}}{2 x^{2}}-\frac {\left (A b +6 B a \right ) b^{5}}{3 x^{3}}-\frac {3 \left (2 A b +5 B a \right ) a \,b^{4}}{4 x^{4}}-\frac {\left (3 A b +4 B a \right ) a^{2} b^{3}}{x^{5}}-\frac {5 \left (4 A b +3 B a \right ) a^{3} b^{2}}{6 x^{6}}-\frac {A \,a^{6}}{9 x^{9}}-\frac {3 \left (5 A b +2 B a \right ) a^{4} b}{7 x^{7}}-\frac {\left (6 A b +B a \right ) a^{5}}{8 x^{8}} \end {gather*}

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^10,x)

[Out]

-a^2*b^3*(3*A*b+4*B*a)/x^5-3/4*a*b^4*(2*A*b+5*B*a)/x^4-1/3*b^5*(A*b+6*B*a)/x^3-1/8*a^5*(6*A*b+B*a)/x^8-1/2*B*b
^6/x^2-1/9*A*a^6/x^9-3/7*a^4*b*(5*A*b+2*B*a)/x^7-5/6*a^3*b^2*(4*A*b+3*B*a)/x^6

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maxima [B]  time = 0.56, size = 147, normalized size = 2.04 \begin {gather*} -\frac {252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{504 \, x^{9}} \end {gather*}

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^10,x, algorithm="maxima")

[Out]

-1/504*(252*B*b^6*x^7 + 56*A*a^6 + 168*(6*B*a*b^5 + A*b^6)*x^6 + 378*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 504*(4*B*
a^3*b^3 + 3*A*a^2*b^4)*x^4 + 420*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 216*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 63*(B*a
^6 + 6*A*a^5*b)*x)/x^9

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mupad [B]  time = 0.07, size = 143, normalized size = 1.99 \begin {gather*} -\frac {x\,\left (\frac {B\,a^6}{8}+\frac {3\,A\,b\,a^5}{4}\right )+\frac {A\,a^6}{9}+x^5\,\left (\frac {15\,B\,a^2\,b^4}{4}+\frac {3\,A\,a\,b^5}{2}\right )+x^2\,\left (\frac {6\,B\,a^5\,b}{7}+\frac {15\,A\,a^4\,b^2}{7}\right )+x^6\,\left (\frac {A\,b^6}{3}+2\,B\,a\,b^5\right )+x^4\,\left (4\,B\,a^3\,b^3+3\,A\,a^2\,b^4\right )+x^3\,\left (\frac {5\,B\,a^4\,b^2}{2}+\frac {10\,A\,a^3\,b^3}{3}\right )+\frac {B\,b^6\,x^7}{2}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^10,x)

[Out]

-(x*((B*a^6)/8 + (3*A*a^5*b)/4) + (A*a^6)/9 + x^5*((15*B*a^2*b^4)/4 + (3*A*a*b^5)/2) + x^2*((15*A*a^4*b^2)/7 +
(6*B*a^5*b)/7) + x^6*((A*b^6)/3 + 2*B*a*b^5) + x^4*(3*A*a^2*b^4 + 4*B*a^3*b^3) + x^3*((10*A*a^3*b^3)/3 + (5*B
*a^4*b^2)/2) + (B*b^6*x^7)/2)/x^9

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sympy [B]  time = 6.88, size = 158, normalized size = 2.19 \begin {gather*} \frac {- 56 A a^{6} - 252 B b^{6} x^{7} + x^{6} \left (- 168 A b^{6} - 1008 B a b^{5}\right ) + x^{5} \left (- 756 A a b^{5} - 1890 B a^{2} b^{4}\right ) + x^{4} \left (- 1512 A a^{2} b^{4} - 2016 B a^{3} b^{3}\right ) + x^{3} \left (- 1680 A a^{3} b^{3} - 1260 B a^{4} b^{2}\right ) + x^{2} \left (- 1080 A a^{4} b^{2} - 432 B a^{5} b\right ) + x \left (- 378 A a^{5} b - 63 B a^{6}\right )}{504 x^{9}} \end {gather*}

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**10,x)

[Out]

(-56*A*a**6 - 252*B*b**6*x**7 + x**6*(-168*A*b**6 - 1008*B*a*b**5) + x**5*(-756*A*a*b**5 - 1890*B*a**2*b**4) +
x**4*(-1512*A*a**2*b**4 - 2016*B*a**3*b**3) + x**3*(-1680*A*a**3*b**3 - 1260*B*a**4*b**2) + x**2*(-1080*A*a**
4*b**2 - 432*B*a**5*b) + x*(-378*A*a**5*b - 63*B*a**6))/(504*x**9)

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