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

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

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Rubi [A]  time = 0.06, antiderivative size = 108, 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*} 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 \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 106, normalized size = 0.98 \begin {gather*} -\frac {a^5 (A+2 B x)}{2 x^2}-\frac {5 a^4 A b}{x}+5 a^3 b \log (x) (a B+2 A b)+10 a^3 b^2 B x+5 a^2 b^3 x (2 A+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) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.32, size = 121, normalized size = 1.12 \begin {gather*} \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 \relax (x) - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{2}} \end {gather*}

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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giac [A]  time = 1.22, size = 119, normalized size = 1.10 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 120, normalized size = 1.11 \begin {gather*} \frac {B \,b^{5} x^{4}}{4}+\frac {A \,b^{5} x^{3}}{3}+\frac {5 B a \,b^{4} x^{3}}{3}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}+10 A \,a^{3} b^{2} \ln \relax (x )+10 A \,a^{2} b^{3} x +5 B \,a^{4} b \ln \relax (x )+10 B \,a^{3} b^{2} x -\frac {5 A \,a^{4} b}{x}-\frac {B \,a^{5}}{x}-\frac {A \,a^{5}}{2 x^{2}} \end {gather*}

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

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maxima [A]  time = 1.09, size = 116, normalized size = 1.07 \begin {gather*} \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 \relax (x) - \frac {A a^{5} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2 \, x^{2}} \end {gather*}

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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mupad [B]  time = 0.31, size = 108, normalized size = 1.00 \begin {gather*} \ln \relax (x)\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )-\frac {x\,\left (B\,a^5+5\,A\,b\,a^4\right )+\frac {A\,a^5}{2}}{x^2}+x^3\,\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )+\frac {B\,b^5\,x^4}{4}+10\,a^2\,b^2\,x\,\left (A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^2\,\left (A\,b+2\,B\,a\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^5)/x^3,x)

[Out]

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

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sympy [A]  time = 0.65, size = 122, normalized size = 1.13 \begin {gather*} \frac {B b^{5} x^{4}}{4} + 5 a^{3} b \left (2 A b + B a\right ) \log {\relax (x )} + 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 {gather*}

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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