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

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

[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]

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Rubi [A]  time = 0.0656726, antiderivative size = 107, normalized size of antiderivative = 1., 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} \[ -\frac{10 a^2 b^2 (a B+A b)}{x}-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{a^5 A}{4 x^4}+b^4 x (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac{1}{2} b^5 B x^2 \]

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*}

Mathematica [A]  time = 0.0428567, size = 106, normalized size = 0.99 \[ -\frac{5 a^3 b^2 (A+2 B x)}{x^2}-\frac{10 a^2 A b^3}{x}-\frac{5 a^4 b (2 A+3 B x)}{6 x^3}-\frac{a^5 (3 A+4 B x)}{12 x^4}+5 a b^3 \log (x) (2 a B+A b)+5 a b^4 B x+\frac{1}{2} b^5 x (2 A+B x) \]

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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Maple [A]  time = 0.006, size = 119, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{2}}{2}}+{b}^{5}Ax+5\,a{b}^{4}Bx+5\,A\ln \left ( x \right ) a{b}^{4}+10\,B\ln \left ( x \right ){a}^{2}{b}^{3}-{\frac{5\,{a}^{4}bA}{3\,{x}^{3}}}-{\frac{{a}^{5}B}{3\,{x}^{3}}}-{\frac{A{a}^{5}}{4\,{x}^{4}}}-5\,{\frac{{a}^{3}{b}^{2}A}{{x}^{2}}}-{\frac{5\,{a}^{4}bB}{2\,{x}^{2}}}-10\,{\frac{{a}^{2}{b}^{3}A}{x}}-10\,{\frac{{a}^{3}{b}^{2}B}{x}} \end{align*}

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

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Maxima [A]  time = 1.01192, size = 157, normalized size = 1.47 \begin{align*} \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 \left (x\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{align*}

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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Fricas [A]  time = 1.63278, size = 265, normalized size = 2.48 \begin{align*} \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 \left (x\right ) - 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{align*}

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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Sympy [A]  time = 1.56394, size = 117, normalized size = 1.09 \begin{align*} \frac{B b^{5} x^{2}}{2} + 5 a b^{3} \left (A b + 2 B a\right ) \log{\left (x \right )} + 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{align*}

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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Giac [A]  time = 1.15011, size = 157, normalized size = 1.47 \begin{align*} \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{align*}

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