3.1.12 \(\int \frac {(A+B x) (b x+c x^2)}{x^7} \, dx\)

Optimal. Leaf size=33 \[ -\frac {A c+b B}{4 x^4}-\frac {A b}{5 x^5}-\frac {B c}{3 x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} -\frac {A c+b B}{4 x^4}-\frac {A b}{5 x^5}-\frac {B c}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^7,x]

[Out]

-(A*b)/(5*x^5) - (b*B + A*c)/(4*x^4) - (B*c)/(3*x^3)

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x^7} \, dx &=\int \left (\frac {A b}{x^6}+\frac {b B+A c}{x^5}+\frac {B c}{x^4}\right ) \, dx\\ &=-\frac {A b}{5 x^5}-\frac {b B+A c}{4 x^4}-\frac {B c}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 31, normalized size = 0.94 \begin {gather*} -\frac {3 A (4 b+5 c x)+5 B x (3 b+4 c x)}{60 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^7,x]

[Out]

-1/60*(5*B*x*(3*b + 4*c*x) + 3*A*(4*b + 5*c*x))/x^5

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^7,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^7, x]

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 27, normalized size = 0.82 \begin {gather*} -\frac {20 \, B c x^{2} + 12 \, A b + 15 \, {\left (B b + A c\right )} x}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^7,x, algorithm="fricas")

[Out]

-1/60*(20*B*c*x^2 + 12*A*b + 15*(B*b + A*c)*x)/x^5

________________________________________________________________________________________

giac [A]  time = 0.15, size = 27, normalized size = 0.82 \begin {gather*} -\frac {20 \, B c x^{2} + 15 \, B b x + 15 \, A c x + 12 \, A b}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^7,x, algorithm="giac")

[Out]

-1/60*(20*B*c*x^2 + 15*B*b*x + 15*A*c*x + 12*A*b)/x^5

________________________________________________________________________________________

maple [A]  time = 0.05, size = 28, normalized size = 0.85 \begin {gather*} -\frac {B c}{3 x^{3}}-\frac {A b}{5 x^{5}}-\frac {A c +b B}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^7,x)

[Out]

-1/5*A*b/x^5-1/4*(A*c+B*b)/x^4-1/3*B*c/x^3

________________________________________________________________________________________

maxima [A]  time = 0.91, size = 27, normalized size = 0.82 \begin {gather*} -\frac {20 \, B c x^{2} + 12 \, A b + 15 \, {\left (B b + A c\right )} x}{60 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^7,x, algorithm="maxima")

[Out]

-1/60*(20*B*c*x^2 + 12*A*b + 15*(B*b + A*c)*x)/x^5

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} -\frac {\frac {B\,c\,x^2}{3}+\left (\frac {A\,c}{4}+\frac {B\,b}{4}\right )\,x+\frac {A\,b}{5}}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^7,x)

[Out]

-((A*b)/5 + x*((A*c)/4 + (B*b)/4) + (B*c*x^2)/3)/x^5

________________________________________________________________________________________

sympy [A]  time = 0.41, size = 31, normalized size = 0.94 \begin {gather*} \frac {- 12 A b - 20 B c x^{2} + x \left (- 15 A c - 15 B b\right )}{60 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**7,x)

[Out]

(-12*A*b - 20*B*c*x**2 + x*(-15*A*c - 15*B*b))/(60*x**5)

________________________________________________________________________________________