3.103 \(\int \frac{(a+b x)^2 (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=55 \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{6 x^6}-\frac{b (2 a B+A b)}{5 x^5}-\frac{b^2 B}{4 x^4} \]

[Out]

-(a^2*A)/(7*x^7) - (a*(2*A*b + a*B))/(6*x^6) - (b*(A*b + 2*a*B))/(5*x^5) - (b^2*B)/(4*x^4)

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Rubi [A]  time = 0.0221769, antiderivative size = 55, 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} \[ -\frac{a^2 A}{7 x^7}-\frac{a (a B+2 A b)}{6 x^6}-\frac{b (2 a B+A b)}{5 x^5}-\frac{b^2 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/x^8,x]

[Out]

-(a^2*A)/(7*x^7) - (a*(2*A*b + a*B))/(6*x^6) - (b*(A*b + 2*a*B))/(5*x^5) - (b^2*B)/(4*x^4)

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)^2 (A+B x)}{x^8} \, dx &=\int \left (\frac{a^2 A}{x^8}+\frac{a (2 A b+a B)}{x^7}+\frac{b (A b+2 a B)}{x^6}+\frac{b^2 B}{x^5}\right ) \, dx\\ &=-\frac{a^2 A}{7 x^7}-\frac{a (2 A b+a B)}{6 x^6}-\frac{b (A b+2 a B)}{5 x^5}-\frac{b^2 B}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0156682, size = 50, normalized size = 0.91 \[ -\frac{10 a^2 (6 A+7 B x)+28 a b x (5 A+6 B x)+21 b^2 x^2 (4 A+5 B x)}{420 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*(A + B*x))/x^8,x]

[Out]

-(21*b^2*x^2*(4*A + 5*B*x) + 28*a*b*x*(5*A + 6*B*x) + 10*a^2*(6*A + 7*B*x))/(420*x^7)

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Maple [A]  time = 0.004, size = 48, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{7\,{x}^{7}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{6\,{x}^{6}}}-{\frac{b \left ( Ab+2\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{2}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/x^8,x)

[Out]

-1/7*a^2*A/x^7-1/6*a*(2*A*b+B*a)/x^6-1/5*b*(A*b+2*B*a)/x^5-1/4*b^2*B/x^4

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Maxima [A]  time = 1.01831, size = 69, normalized size = 1.25 \begin{align*} -\frac{105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \,{\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/420*(105*B*b^2*x^3 + 60*A*a^2 + 84*(2*B*a*b + A*b^2)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7

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Fricas [A]  time = 1.85972, size = 123, normalized size = 2.24 \begin{align*} -\frac{105 \, B b^{2} x^{3} + 60 \, A a^{2} + 84 \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + 70 \,{\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

-1/420*(105*B*b^2*x^3 + 60*A*a^2 + 84*(2*B*a*b + A*b^2)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7

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Sympy [A]  time = 1.51608, size = 54, normalized size = 0.98 \begin{align*} - \frac{60 A a^{2} + 105 B b^{2} x^{3} + x^{2} \left (84 A b^{2} + 168 B a b\right ) + x \left (140 A a b + 70 B a^{2}\right )}{420 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)/x**8,x)

[Out]

-(60*A*a**2 + 105*B*b**2*x**3 + x**2*(84*A*b**2 + 168*B*a*b) + x*(140*A*a*b + 70*B*a**2))/(420*x**7)

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Giac [A]  time = 1.24168, size = 69, normalized size = 1.25 \begin{align*} -\frac{105 \, B b^{2} x^{3} + 168 \, B a b x^{2} + 84 \, A b^{2} x^{2} + 70 \, B a^{2} x + 140 \, A a b x + 60 \, A a^{2}}{420 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/420*(105*B*b^2*x^3 + 168*B*a*b*x^2 + 84*A*b^2*x^2 + 70*B*a^2*x + 140*A*a*b*x + 60*A*a^2)/x^7