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

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

[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]

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Rubi [A]  time = 0.0601232, antiderivative size = 104, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{x^2}-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{a^5 A}{5 x^5}-\frac{5 a b^3 (2 a B+A b)}{x}+b^4 \log (x) (5 a B+A b)+b^5 B x \]

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*}

Mathematica [A]  time = 0.0464878, size = 106, normalized size = 1.02 \[ -\frac{5 a^3 b^2 (2 A+3 B x)}{3 x^3}-\frac{5 a^2 b^3 (A+2 B x)}{x^2}-\frac{5 a^4 b (3 A+4 B x)}{12 x^4}-\frac{a^5 (4 A+5 B x)}{20 x^5}+b^4 \log (x) (5 a B+A b)-\frac{5 a A b^4}{x}+b^5 B x \]

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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Maple [A]  time = 0.006, size = 120, normalized size = 1.2 \begin{align*}{b}^{5}Bx+A\ln \left ( x \right ){b}^{5}+5\,B\ln \left ( x \right ) a{b}^{4}-{\frac{10\,{a}^{3}{b}^{2}A}{3\,{x}^{3}}}-{\frac{5\,{a}^{4}bB}{3\,{x}^{3}}}-{\frac{A{a}^{5}}{5\,{x}^{5}}}-{\frac{5\,{a}^{4}bA}{4\,{x}^{4}}}-{\frac{{a}^{5}B}{4\,{x}^{4}}}-5\,{\frac{{a}^{2}{b}^{3}A}{{x}^{2}}}-5\,{\frac{{a}^{3}{b}^{2}B}{{x}^{2}}}-5\,{\frac{a{b}^{4}A}{x}}-10\,{\frac{{a}^{2}{b}^{3}B}{x}} \end{align*}

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

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Maxima [A]  time = 1.0087, size = 155, normalized size = 1.49 \begin{align*} B b^{5} x +{\left (5 \, B a b^{4} + A b^{5}\right )} \log \left (x\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{align*}

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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Fricas [A]  time = 1.72268, size = 271, normalized size = 2.61 \begin{align*} \frac{60 \, B b^{5} x^{6} + 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} \log \left (x\right ) - 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{align*}

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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Sympy [A]  time = 2.47467, size = 117, normalized size = 1.12 \begin{align*} B b^{5} x + b^{4} \left (A b + 5 B a\right ) \log{\left (x \right )} - \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{align*}

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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Giac [A]  time = 1.20004, size = 157, normalized size = 1.51 \begin{align*} 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{align*}

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