3.127 \(\int \frac{(a+b x)^5 (A+B x)}{x^3} \, dx\)

Optimal. Leaf size=108 \[ 10 a^2 b^2 x (a B+A b)-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{2 x^2}+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{3} b^4 x^3 (5 a B+A b)+\frac{1}{4} b^5 B x^4 \]

[Out]

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

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Rubi [A]  time = 0.0614029, antiderivative size = 108, 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} \[ 10 a^2 b^2 x (a B+A b)-\frac{a^4 (a B+5 A b)}{x}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{2 x^2}+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{3} b^4 x^3 (5 a B+A b)+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^5*(A + B*x))/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0379213, size = 106, normalized size = 0.98 \[ 5 a^2 b^3 x (2 A+B x)+5 a^3 b \log (x) (a B+2 A b)-\frac{5 a^4 A b}{x}-\frac{a^5 (A+2 B x)}{2 x^2}+10 a^3 b^2 B x+\frac{5}{6} a b^4 x^2 (3 A+2 B x)+\frac{1}{12} b^5 x^3 (4 A+3 B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^3,x]

[Out]

(-5*a^4*A*b)/x + 10*a^3*b^2*B*x + 5*a^2*b^3*x*(2*A + B*x) - (a^5*(A + 2*B*x))/(2*x^2) + (5*a*b^4*x^2*(3*A + 2*
B*x))/6 + (b^5*x^3*(4*A + 3*B*x))/12 + 5*a^3*b*(2*A*b + a*B)*Log[x]

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Maple [A]  time = 0.006, size = 120, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{4}}{4}}+{\frac{A{x}^{3}{b}^{5}}{3}}+{\frac{5\,B{x}^{3}a{b}^{4}}{3}}+{\frac{5\,A{x}^{2}a{b}^{4}}{2}}+5\,B{x}^{2}{a}^{2}{b}^{3}+10\,{a}^{2}{b}^{3}Ax+10\,{a}^{3}{b}^{2}Bx+10\,A\ln \left ( x \right ){a}^{3}{b}^{2}+5\,B\ln \left ( x \right ){a}^{4}b-{\frac{A{a}^{5}}{2\,{x}^{2}}}-5\,{\frac{{a}^{4}bA}{x}}-{\frac{{a}^{5}B}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(B*x+A)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.02934, size = 157, normalized size = 1.45 \begin{align*} \frac{1}{4} \, B b^{5} x^{4} + \frac{1}{3} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{3} + \frac{5}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.68313, size = 265, normalized size = 2.45 \begin{align*} \frac{3 \, B b^{5} x^{6} - 6 \, A a^{5} + 4 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 60 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} \log \left (x\right ) - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.634344, size = 121, normalized size = 1.12 \begin{align*} \frac{B b^{5} x^{4}}{4} + 5 a^{3} b \left (2 A b + B a\right ) \log{\left (x \right )} + x^{3} \left (\frac{A b^{5}}{3} + \frac{5 B a b^{4}}{3}\right ) + x^{2} \left (\frac{5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) - \frac{A a^{5} + x \left (10 A a^{4} b + 2 B a^{5}\right )}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(B*x+A)/x**3,x)

[Out]

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

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Giac [A]  time = 1.18103, size = 161, normalized size = 1.49 \begin{align*} \frac{1}{4} \, B b^{5} x^{4} + \frac{5}{3} \, B a b^{4} x^{3} + \frac{1}{3} \, A b^{5} x^{3} + 5 \, B a^{2} b^{3} x^{2} + \frac{5}{2} \, A a b^{4} x^{2} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(B*x+A)/x^3,x, algorithm="giac")

[Out]

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