3.2.18 \(\int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=44 \[ \frac {(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac {A (a+b x)^6}{7 a x^7} \]

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 37} \begin {gather*} \frac {(a+b x)^6 (A b-7 a B)}{42 a^2 x^6}-\frac {A (a+b x)^6}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

-(A*(a + b*x)^6)/(7*a*x^7) + ((A*b - 7*a*B)*(a + b*x)^6)/(42*a^2*x^6)

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 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)^5 (A+B x)}{x^8} \, dx &=-\frac {A (a+b x)^6}{7 a x^7}+\frac {(-A b+7 a B) \int \frac {(a+b x)^5}{x^7} \, dx}{7 a}\\ &=-\frac {A (a+b x)^6}{7 a x^7}+\frac {(A b-7 a B) (a+b x)^6}{42 a^2 x^6}\\ \end {align*}

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Mathematica [B]  time = 0.03, size = 104, normalized size = 2.36 \begin {gather*} -\frac {a^5 (6 A+7 B x)+7 a^4 b x (5 A+6 B x)+21 a^3 b^2 x^2 (4 A+5 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+35 a b^4 x^4 (2 A+3 B x)+21 b^5 x^5 (A+2 B x)}{42 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

-1/42*(21*b^5*x^5*(A + 2*B*x) + 35*a*b^4*x^4*(2*A + 3*B*x) + 35*a^2*b^3*x^3*(3*A + 4*B*x) + 21*a^3*b^2*x^2*(4*
A + 5*B*x) + 7*a^4*b*x*(5*A + 6*B*x) + a^5*(6*A + 7*B*x))/x^7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^5*(A + B*x))/x^8,x]

[Out]

IntegrateAlgebraic[((a + b*x)^5*(A + B*x))/x^8, x]

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fricas [B]  time = 1.33, size = 119, normalized size = 2.70 \begin {gather*} -\frac {42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 105*(B*a^3*b^2 +
A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 7*(B*a^5 + 5*A*a^4*b)*x)/x^7

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giac [B]  time = 1.21, size = 123, normalized size = 2.80 \begin {gather*} -\frac {42 \, B b^{5} x^{6} + 105 \, B a b^{4} x^{5} + 21 \, A b^{5} x^{5} + 140 \, B a^{2} b^{3} x^{4} + 70 \, A a b^{4} x^{4} + 105 \, B a^{3} b^{2} x^{3} + 105 \, A a^{2} b^{3} x^{3} + 42 \, B a^{4} b x^{2} + 84 \, A a^{3} b^{2} x^{2} + 7 \, B a^{5} x + 35 \, A a^{4} b x + 6 \, A a^{5}}{42 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/42*(42*B*b^5*x^6 + 105*B*a*b^4*x^5 + 21*A*b^5*x^5 + 140*B*a^2*b^3*x^4 + 70*A*a*b^4*x^4 + 105*B*a^3*b^2*x^3
+ 105*A*a^2*b^3*x^3 + 42*B*a^4*b*x^2 + 84*A*a^3*b^2*x^2 + 7*B*a^5*x + 35*A*a^4*b*x + 6*A*a^5)/x^7

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maple [B]  time = 0.01, size = 104, normalized size = 2.36 \begin {gather*} -\frac {B \,b^{5}}{x}-\frac {\left (A b +5 B a \right ) b^{4}}{2 x^{2}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{3 x^{3}}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{2 x^{4}}-\frac {A \,a^{5}}{7 x^{7}}-\frac {\left (2 A b +B a \right ) a^{3} b}{x^{5}}-\frac {\left (5 A b +B a \right ) a^{4}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^8,x)

[Out]

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

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maxima [B]  time = 1.01, size = 119, normalized size = 2.70 \begin {gather*} -\frac {42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 105*(B*a^3*b^2 +
A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 7*(B*a^5 + 5*A*a^4*b)*x)/x^7

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mupad [B]  time = 0.07, size = 118, normalized size = 2.68 \begin {gather*} -\frac {x\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+\frac {A\,a^5}{7}+x^2\,\left (B\,a^4\,b+2\,A\,a^3\,b^2\right )+x^4\,\left (\frac {10\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{3}\right )+x^5\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+x^3\,\left (\frac {5\,B\,a^3\,b^2}{2}+\frac {5\,A\,a^2\,b^3}{2}\right )+B\,b^5\,x^6}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^5)/x^8,x)

[Out]

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

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sympy [B]  time = 4.42, size = 133, normalized size = 3.02 \begin {gather*} \frac {- 6 A a^{5} - 42 B b^{5} x^{6} + x^{5} \left (- 21 A b^{5} - 105 B a b^{4}\right ) + x^{4} \left (- 70 A a b^{4} - 140 B a^{2} b^{3}\right ) + x^{3} \left (- 105 A a^{2} b^{3} - 105 B a^{3} b^{2}\right ) + x^{2} \left (- 84 A a^{3} b^{2} - 42 B a^{4} b\right ) + x \left (- 35 A a^{4} b - 7 B a^{5}\right )}{42 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**8,x)

[Out]

(-6*A*a**5 - 42*B*b**5*x**6 + x**5*(-21*A*b**5 - 105*B*a*b**4) + x**4*(-70*A*a*b**4 - 140*B*a**2*b**3) + x**3*
(-105*A*a**2*b**3 - 105*B*a**3*b**2) + x**2*(-84*A*a**3*b**2 - 42*B*a**4*b) + x*(-35*A*a**4*b - 7*B*a**5))/(42
*x**7)

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