3.2.17 \(\int \frac {(a+b x)^5 (A+B x)}{x^7} \, dx\)

Optimal. Leaf size=85 \[ -\frac {a^5 B}{5 x^5}-\frac {5 a^4 b B}{4 x^4}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^2 b^3 B}{x^2}-\frac {A (a+b x)^6}{6 a x^6}-\frac {5 a b^4 B}{x}+b^5 B \log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 43} \begin {gather*} -\frac {5 a^2 b^3 B}{x^2}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^4 b B}{4 x^4}-\frac {a^5 B}{5 x^5}-\frac {A (a+b x)^6}{6 a x^6}-\frac {5 a b^4 B}{x}+b^5 B \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^7,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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^7} \, dx &=-\frac {A (a+b x)^6}{6 a x^6}+B \int \frac {(a+b x)^5}{x^6} \, dx\\ &=-\frac {A (a+b x)^6}{6 a x^6}+B \int \left (\frac {a^5}{x^6}+\frac {5 a^4 b}{x^5}+\frac {10 a^3 b^2}{x^4}+\frac {10 a^2 b^3}{x^3}+\frac {5 a b^4}{x^2}+\frac {b^5}{x}\right ) \, dx\\ &=-\frac {a^5 B}{5 x^5}-\frac {5 a^4 b B}{4 x^4}-\frac {10 a^3 b^2 B}{3 x^3}-\frac {5 a^2 b^3 B}{x^2}-\frac {5 a b^4 B}{x}-\frac {A (a+b x)^6}{6 a x^6}+b^5 B \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 109, normalized size = 1.28 \begin {gather*} -\frac {2 a^5 (5 A+6 B x)+15 a^4 b x (4 A+5 B x)+50 a^3 b^2 x^2 (3 A+4 B x)+100 a^2 b^3 x^3 (2 A+3 B x)+150 a b^4 x^4 (A+2 B x)+60 A b^5 x^5-60 b^5 B x^6 \log (x)}{60 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^7,x]

[Out]

-1/60*(60*A*b^5*x^5 + 150*a*b^4*x^4*(A + 2*B*x) + 100*a^2*b^3*x^3*(2*A + 3*B*x) + 50*a^3*b^2*x^2*(3*A + 4*B*x)
+ 15*a^4*b*x*(4*A + 5*B*x) + 2*a^5*(5*A + 6*B*x) - 60*b^5*B*x^6*Log[x])/x^6

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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^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^5*(A + B*x))/x^7,x]

[Out]

IntegrateAlgebraic[((a + b*x)^5*(A + B*x))/x^7, x]

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fricas [A]  time = 1.04, size = 121, normalized size = 1.42 \begin {gather*} \frac {60 \, B b^{5} x^{6} \log \relax (x) - 10 \, A a^{5} - 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} - 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^7,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*x^6*log(x) - 10*A*a^5 - 60*(5*B*a*b^4 + A*b^5)*x^5 - 150*(2*B*a^2*b^3 + A*a*b^4)*x^4 - 200*(B*a
^3*b^2 + A*a^2*b^3)*x^3 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 12*(B*a^5 + 5*A*a^4*b)*x)/x^6

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giac [A]  time = 1.30, size = 119, normalized size = 1.40 \begin {gather*} B b^{5} \log \left ({\left | x \right |}\right ) - \frac {10 \, A a^{5} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^7,x, algorithm="giac")

[Out]

B*b^5*log(abs(x)) - 1/60*(10*A*a^5 + 60*(5*B*a*b^4 + A*b^5)*x^5 + 150*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 200*(B*a^3
*b^2 + A*a^2*b^3)*x^3 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 12*(B*a^5 + 5*A*a^4*b)*x)/x^6

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maple [A]  time = 0.01, size = 124, normalized size = 1.46 \begin {gather*} B \,b^{5} \ln \relax (x )-\frac {A \,b^{5}}{x}-\frac {5 B a \,b^{4}}{x}-\frac {5 A a \,b^{4}}{2 x^{2}}-\frac {5 B \,a^{2} b^{3}}{x^{2}}-\frac {10 A \,a^{2} b^{3}}{3 x^{3}}-\frac {10 B \,a^{3} b^{2}}{3 x^{3}}-\frac {5 A \,a^{3} b^{2}}{2 x^{4}}-\frac {5 B \,a^{4} b}{4 x^{4}}-\frac {A \,a^{4} b}{x^{5}}-\frac {B \,a^{5}}{5 x^{5}}-\frac {A \,a^{5}}{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^7,x)

[Out]

-1/6*A*a^5/x^6-a^4/x^5*A*b-1/5*a^5*B/x^5-b^5/x*A-5*a*b^4*B/x+b^5*B*ln(x)-5/2*a*b^4/x^2*A-5*a^2*b^3*B/x^2-10/3*
a^2*b^3/x^3*A-10/3*a^3*b^2*B/x^3-5/2*a^3*b^2/x^4*A-5/4*a^4*b*B/x^4

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maxima [A]  time = 1.01, size = 118, normalized size = 1.39 \begin {gather*} B b^{5} \log \relax (x) - \frac {10 \, A a^{5} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 150 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 200 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 75 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^7,x, algorithm="maxima")

[Out]

B*b^5*log(x) - 1/60*(10*A*a^5 + 60*(5*B*a*b^4 + A*b^5)*x^5 + 150*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 200*(B*a^3*b^2
+ A*a^2*b^3)*x^3 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 12*(B*a^5 + 5*A*a^4*b)*x)/x^6

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mupad [B]  time = 0.36, size = 117, normalized size = 1.38 \begin {gather*} B\,b^5\,\ln \relax (x)-\frac {x\,\left (\frac {B\,a^5}{5}+A\,b\,a^4\right )+\frac {A\,a^5}{6}+x^4\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^5\,\left (A\,b^5+5\,B\,a\,b^4\right )+x^3\,\left (\frac {10\,B\,a^3\,b^2}{3}+\frac {10\,A\,a^2\,b^3}{3}\right )}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^5)/x^7,x)

[Out]

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

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sympy [A]  time = 3.56, size = 131, normalized size = 1.54 \begin {gather*} B b^{5} \log {\relax (x )} + \frac {- 10 A a^{5} + x^{5} \left (- 60 A b^{5} - 300 B a b^{4}\right ) + x^{4} \left (- 150 A a b^{4} - 300 B a^{2} b^{3}\right ) + x^{3} \left (- 200 A a^{2} b^{3} - 200 B a^{3} b^{2}\right ) + x^{2} \left (- 150 A a^{3} b^{2} - 75 B a^{4} b\right ) + x \left (- 60 A a^{4} b - 12 B a^{5}\right )}{60 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**7,x)

[Out]

B*b**5*log(x) + (-10*A*a**5 + x**5*(-60*A*b**5 - 300*B*a*b**4) + x**4*(-150*A*a*b**4 - 300*B*a**2*b**3) + x**3
*(-200*A*a**2*b**3 - 200*B*a**3*b**2) + x**2*(-150*A*a**3*b**2 - 75*B*a**4*b) + x*(-60*A*a**4*b - 12*B*a**5))/
(60*x**6)

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