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

Optimal. Leaf size=107 \[ -\frac {a^5 A}{4 x^4}-\frac {a^4 (a B+5 A b)}{3 x^3}-\frac {5 a^3 b (a B+2 A b)}{2 x^2}-\frac {10 a^2 b^2 (a B+A b)}{x}+b^4 x (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac {1}{2} b^5 B x^2 \]

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Rubi [A]  time = 0.07, antiderivative size = 107, 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 {10 a^2 b^2 (a B+A b)}{x}-\frac {a^4 (a B+5 A b)}{3 x^3}-\frac {5 a^3 b (a B+2 A b)}{2 x^2}-\frac {a^5 A}{4 x^4}+b^4 x (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac {1}{2} b^5 B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^5,x]

[Out]

-(a^5*A)/(4*x^4) - (a^4*(5*A*b + a*B))/(3*x^3) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) - (10*a^2*b^2*(A*b + a*B))/x
+ b^4*(A*b + 5*a*B)*x + (b^5*B*x^2)/2 + 5*a*b^3*(A*b + 2*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^5} \, dx &=\int \left (b^4 (A b+5 a B)+\frac {a^5 A}{x^5}+\frac {a^4 (5 A b+a B)}{x^4}+\frac {5 a^3 b (2 A b+a B)}{x^3}+\frac {10 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\right ) \, dx\\ &=-\frac {a^5 A}{4 x^4}-\frac {a^4 (5 A b+a B)}{3 x^3}-\frac {5 a^3 b (2 A b+a B)}{2 x^2}-\frac {10 a^2 b^2 (A b+a B)}{x}+b^4 (A b+5 a B) x+\frac {1}{2} b^5 B x^2+5 a b^3 (A b+2 a B) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 106, normalized size = 0.99 \begin {gather*} -\frac {a^5 (3 A+4 B x)}{12 x^4}-\frac {5 a^4 b (2 A+3 B x)}{6 x^3}-\frac {5 a^3 b^2 (A+2 B x)}{x^2}-\frac {10 a^2 A b^3}{x}+5 a b^3 \log (x) (2 a B+A b)+5 a b^4 B x+\frac {1}{2} b^5 x (2 A+B x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*a^2*A*b^3)/x + 5*a*b^4*B*x + (b^5*x*(2*A + B*x))/2 - (5*a^3*b^2*(A + 2*B*x))/x^2 - (5*a^4*b*(2*A + 3*B*x)
)/(6*x^3) - (a^5*(3*A + 4*B*x))/(12*x^4) + 5*a*b^3*(A*b + 2*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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.18, size = 121, normalized size = 1.13 \begin {gather*} \frac {6 \, B b^{5} x^{6} - 3 \, A a^{5} + 12 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 60 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} \log \relax (x) - 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*B*b^5*x^6 - 3*A*a^5 + 12*(5*B*a*b^4 + A*b^5)*x^5 + 60*(2*B*a^2*b^3 + A*a*b^4)*x^4*log(x) - 120*(B*a^3*
b^2 + A*a^2*b^3)*x^3 - 30*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 4*(B*a^5 + 5*A*a^4*b)*x)/x^4

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giac [A]  time = 1.25, size = 116, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, B b^{5} x^{2} + 5 \, B a b^{4} x + A b^{5} x + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{5} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*b^5*x^2 + 5*B*a*b^4*x + A*b^5*x + 5*(2*B*a^2*b^3 + A*a*b^4)*log(abs(x)) - 1/12*(3*A*a^5 + 120*(B*a^3*b^2
+ A*a^2*b^3)*x^3 + 30*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 4*(B*a^5 + 5*A*a^4*b)*x)/x^4

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maple [A]  time = 0.01, size = 119, normalized size = 1.11 \begin {gather*} \frac {B \,b^{5} x^{2}}{2}+5 A a \,b^{4} \ln \relax (x )+A \,b^{5} x +10 B \,a^{2} b^{3} \ln \relax (x )+5 B a \,b^{4} x -\frac {10 A \,a^{2} b^{3}}{x}-\frac {10 B \,a^{3} b^{2}}{x}-\frac {5 A \,a^{3} b^{2}}{x^{2}}-\frac {5 B \,a^{4} b}{2 x^{2}}-\frac {5 A \,a^{4} b}{3 x^{3}}-\frac {B \,a^{5}}{3 x^{3}}-\frac {A \,a^{5}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.12, size = 116, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, B b^{5} x^{2} + {\left (5 \, B a b^{4} + A b^{5}\right )} x + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \relax (x) - \frac {3 \, A a^{5} + 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*b^5*x^2 + (5*B*a*b^4 + A*b^5)*x + 5*(2*B*a^2*b^3 + A*a*b^4)*log(x) - 1/12*(3*A*a^5 + 120*(B*a^3*b^2 + A*
a^2*b^3)*x^3 + 30*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 4*(B*a^5 + 5*A*a^4*b)*x)/x^4

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mupad [B]  time = 0.07, size = 117, normalized size = 1.09 \begin {gather*} x\,\left (A\,b^5+5\,B\,a\,b^4\right )+\ln \relax (x)\,\left (10\,B\,a^2\,b^3+5\,A\,a\,b^4\right )-\frac {x\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )+\frac {A\,a^5}{4}+x^2\,\left (\frac {5\,B\,a^4\,b}{2}+5\,A\,a^3\,b^2\right )+x^3\,\left (10\,B\,a^3\,b^2+10\,A\,a^2\,b^3\right )}{x^4}+\frac {B\,b^5\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^5)/x^5,x)

[Out]

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

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sympy [A]  time = 1.49, size = 122, normalized size = 1.14 \begin {gather*} \frac {B b^{5} x^{2}}{2} + 5 a b^{3} \left (A b + 2 B a\right ) \log {\relax (x )} + x \left (A b^{5} + 5 B a b^{4}\right ) + \frac {- 3 A a^{5} + x^{3} \left (- 120 A a^{2} b^{3} - 120 B a^{3} b^{2}\right ) + x^{2} \left (- 60 A a^{3} b^{2} - 30 B a^{4} b\right ) + x \left (- 20 A a^{4} b - 4 B a^{5}\right )}{12 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**5*x**2/2 + 5*a*b**3*(A*b + 2*B*a)*log(x) + x*(A*b**5 + 5*B*a*b**4) + (-3*A*a**5 + x**3*(-120*A*a**2*b**3
- 120*B*a**3*b**2) + x**2*(-60*A*a**3*b**2 - 30*B*a**4*b) + x*(-20*A*a**4*b - 4*B*a**5))/(12*x**4)

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