3.126 \(\int \frac{(a+b x)^5 (A+B x)}{x^2} \, dx\)

Optimal. Leaf size=105 \[ 5 a^2 b^2 x^2 (a B+A b)+5 a^3 b x (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac{a^5 A}{x}+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{4} b^4 x^4 (5 a B+A b)+\frac{1}{5} b^5 B x^5 \]

[Out]

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

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Rubi [A]  time = 0.0617309, antiderivative size = 105, 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} \[ 5 a^2 b^2 x^2 (a B+A b)+5 a^3 b x (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac{a^5 A}{x}+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{4} b^4 x^4 (5 a B+A b)+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^2,x]

[Out]

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

Mathematica [A]  time = 0.038931, size = 107, normalized size = 1.02 \[ 5 a^2 b^2 x^2 (a B+A b)+5 a^3 b x (a B+2 A b)+\log (x) \left (5 a^4 A b+a^5 B\right )-\frac{a^5 A}{x}+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{4} b^4 x^4 (5 a B+A b)+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 119, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{5}}{5}}+{\frac{A{x}^{4}{b}^{5}}{4}}+{\frac{5\,B{x}^{4}a{b}^{4}}{4}}+{\frac{5\,A{x}^{3}a{b}^{4}}{3}}+{\frac{10\,B{x}^{3}{a}^{2}{b}^{3}}{3}}+5\,A{x}^{2}{a}^{2}{b}^{3}+5\,B{x}^{2}{a}^{3}{b}^{2}+10\,{a}^{3}{b}^{2}Ax+5\,{a}^{4}bBx+5\,A\ln \left ( x \right ){a}^{4}b+B\ln \left ( x \right ){a}^{5}-{\frac{A{a}^{5}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.02573, size = 155, normalized size = 1.48 \begin{align*} \frac{1}{5} \, B b^{5} x^{5} - \frac{A a^{5}}{x} + \frac{1}{4} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + \frac{5}{3} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 5 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x +{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.65606, size = 269, normalized size = 2.56 \begin{align*} \frac{12 \, B b^{5} x^{6} - 60 \, A a^{5} + 15 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 100 \,{\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} + 300 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 60 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x \log \left (x\right )}{60 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.469449, size = 121, normalized size = 1.15 \begin{align*} - \frac{A a^{5}}{x} + \frac{B b^{5} x^{5}}{5} + a^{4} \left (5 A b + B a\right ) \log{\left (x \right )} + x^{4} \left (\frac{A b^{5}}{4} + \frac{5 B a b^{4}}{4}\right ) + x^{3} \left (\frac{5 A a b^{4}}{3} + \frac{10 B a^{2} b^{3}}{3}\right ) + x^{2} \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**2,x)

[Out]

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

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Giac [A]  time = 1.14565, size = 161, normalized size = 1.53 \begin{align*} \frac{1}{5} \, B b^{5} x^{5} + \frac{5}{4} \, B a b^{4} x^{4} + \frac{1}{4} \, A b^{5} x^{4} + \frac{10}{3} \, B a^{2} b^{3} x^{3} + \frac{5}{3} \, A a b^{4} x^{3} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac{A a^{5}}{x} +{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^2,x, algorithm="giac")

[Out]

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