3.2.16 \(\int \frac {(a+b x)^5 (A+B x)}{x^6} \, dx\)

Optimal. Leaf size=104 \[ -\frac {a^5 A}{5 x^5}-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{3 x^3}-\frac {5 a^2 b^2 (a B+A b)}{x^2}+b^4 \log (x) (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{x}+b^5 B x \]

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Rubi [A]  time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} -\frac {5 a^2 b^2 (a B+A b)}{x^2}-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{3 x^3}-\frac {a^5 A}{5 x^5}-\frac {5 a b^3 (2 a B+A b)}{x}+b^4 \log (x) (5 a B+A b)+b^5 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^6,x]

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 106, normalized size = 1.02 \begin {gather*} -\frac {a^5 (4 A+5 B x)}{20 x^5}-\frac {5 a^4 b (3 A+4 B x)}{12 x^4}-\frac {5 a^3 b^2 (2 A+3 B x)}{3 x^3}-\frac {5 a^2 b^3 (A+2 B x)}{x^2}+b^4 \log (x) (5 a B+A b)-\frac {5 a A b^4}{x}+b^5 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^6,x]

[Out]

(-5*a*A*b^4)/x + b^5*B*x - (5*a^2*b^3*(A + 2*B*x))/x^2 - (5*a^3*b^2*(2*A + 3*B*x))/(3*x^3) - (5*a^4*b*(3*A + 4
*B*x))/(12*x^4) - (a^5*(4*A + 5*B*x))/(20*x^5) + b^4*(A*b + 5*a*B)*Log[x]

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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^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^5*(A + B*x))/x^6,x]

[Out]

IntegrateAlgebraic[((a + b*x)^5*(A + B*x))/x^6, x]

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fricas [A]  time = 1.19, size = 121, normalized size = 1.16 \begin {gather*} \frac {60 \, B b^{5} x^{6} + 60 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} \log \relax (x) - 12 \, A a^{5} - 300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} - 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^6,x, algorithm="fricas")

[Out]

1/60*(60*B*b^5*x^6 + 60*(5*B*a*b^4 + A*b^5)*x^5*log(x) - 12*A*a^5 - 300*(2*B*a^2*b^3 + A*a*b^4)*x^4 - 300*(B*a
^3*b^2 + A*a^2*b^3)*x^3 - 100*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 15*(B*a^5 + 5*A*a^4*b)*x)/x^5

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giac [A]  time = 1.23, size = 116, normalized size = 1.12 \begin {gather*} B b^{5} x + {\left (5 \, B a b^{4} + A b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac {12 \, A a^{5} + 300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^6,x, algorithm="giac")

[Out]

B*b^5*x + (5*B*a*b^4 + A*b^5)*log(abs(x)) - 1/60*(12*A*a^5 + 300*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 300*(B*a^3*b^2
+ A*a^2*b^3)*x^3 + 100*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 15*(B*a^5 + 5*A*a^4*b)*x)/x^5

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maple [A]  time = 0.01, size = 120, normalized size = 1.15 \begin {gather*} A \,b^{5} \ln \relax (x )+5 B a \,b^{4} \ln \relax (x )+B \,b^{5} x -\frac {5 A a \,b^{4}}{x}-\frac {10 B \,a^{2} b^{3}}{x}-\frac {5 A \,a^{2} b^{3}}{x^{2}}-\frac {5 B \,a^{3} b^{2}}{x^{2}}-\frac {10 A \,a^{3} b^{2}}{3 x^{3}}-\frac {5 B \,a^{4} b}{3 x^{3}}-\frac {5 A \,a^{4} b}{4 x^{4}}-\frac {B \,a^{5}}{4 x^{4}}-\frac {A \,a^{5}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^6,x)

[Out]

b^5*B*x-1/5*a^5*A/x^5-5*a*b^4/x*A-10*a^2*b^3/x*B+A*ln(x)*b^5+5*B*ln(x)*a*b^4-5*a^2*b^3/x^2*A-5*a^3*b^2/x^2*B-1
0/3*a^3*b^2/x^3*A-5/3*a^4*b/x^3*B-5/4*a^4/x^4*A*b-1/4*a^5/x^4*B

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maxima [A]  time = 1.11, size = 115, normalized size = 1.11 \begin {gather*} B b^{5} x + {\left (5 \, B a b^{4} + A b^{5}\right )} \log \relax (x) - \frac {12 \, A a^{5} + 300 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 100 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 15 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^6,x, algorithm="maxima")

[Out]

B*b^5*x + (5*B*a*b^4 + A*b^5)*log(x) - 1/60*(12*A*a^5 + 300*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 300*(B*a^3*b^2 + A*a
^2*b^3)*x^3 + 100*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 15*(B*a^5 + 5*A*a^4*b)*x)/x^5

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mupad [B]  time = 0.35, size = 116, normalized size = 1.12 \begin {gather*} \ln \relax (x)\,\left (A\,b^5+5\,B\,a\,b^4\right )-\frac {x\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )+\frac {A\,a^5}{5}+x^4\,\left (10\,B\,a^2\,b^3+5\,A\,a\,b^4\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{3}+\frac {10\,A\,a^3\,b^2}{3}\right )+x^3\,\left (5\,B\,a^3\,b^2+5\,A\,a^2\,b^3\right )}{x^5}+B\,b^5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^5)/x^6,x)

[Out]

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

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sympy [A]  time = 2.46, size = 124, normalized size = 1.19 \begin {gather*} B b^{5} x + b^{4} \left (A b + 5 B a\right ) \log {\relax (x )} + \frac {- 12 A a^{5} + x^{4} \left (- 300 A a b^{4} - 600 B a^{2} b^{3}\right ) + x^{3} \left (- 300 A a^{2} b^{3} - 300 B a^{3} b^{2}\right ) + x^{2} \left (- 200 A a^{3} b^{2} - 100 B a^{4} b\right ) + x \left (- 75 A a^{4} b - 15 B a^{5}\right )}{60 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**6,x)

[Out]

B*b**5*x + b**4*(A*b + 5*B*a)*log(x) + (-12*A*a**5 + x**4*(-300*A*a*b**4 - 600*B*a**2*b**3) + x**3*(-300*A*a**
2*b**3 - 300*B*a**3*b**2) + x**2*(-200*A*a**3*b**2 - 100*B*a**4*b) + x*(-75*A*a**4*b - 15*B*a**5))/(60*x**5)

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