3.2.2 \(\int \frac {(a+b x)^2 (A+B x)}{x^7} \, dx\) [102]

Optimal. Leaf size=55 \[ -\frac {a^2 A}{6 x^6}-\frac {a (2 A b+a B)}{5 x^5}-\frac {b (A b+2 a B)}{4 x^4}-\frac {b^2 B}{3 x^3} \]

[Out]

-1/6*a^2*A/x^6-1/5*a*(2*A*b+B*a)/x^5-1/4*b*(A*b+2*B*a)/x^4-1/3*b^2*B/x^3

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Rubi [A]
time = 0.02, antiderivative size = 55, 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 = {77} \begin {gather*} -\frac {a^2 A}{6 x^6}-\frac {a (a B+2 A b)}{5 x^5}-\frac {b (2 a B+A b)}{4 x^4}-\frac {b^2 B}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/x^7,x]

[Out]

-1/6*(a^2*A)/x^6 - (a*(2*A*b + a*B))/(5*x^5) - (b*(A*b + 2*a*B))/(4*x^4) - (b^2*B)/(3*x^3)

Rule 77

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

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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.91 \begin {gather*} -\frac {5 b^2 x^2 (3 A+4 B x)+6 a b x (4 A+5 B x)+2 a^2 (5 A+6 B x)}{60 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.06, size = 48, normalized size = 0.87

method	result	size
default	\(-\frac {a^{2} A}{6 x^{6}}-\frac {a \left (2 A b +B a \right )}{5 x^{5}}-\frac {b \left (A b +2 B a \right )}{4 x^{4}}-\frac {b^{2} B}{3 x^{3}}\)	\(48\)
norman	\(\frac {-\frac {b^{2} B \,x^{3}}{3}+\left (-\frac {1}{4} b^{2} A -\frac {1}{2} a b B \right ) x^{2}+\left (-\frac {2}{5} a b A -\frac {1}{5} a^{2} B \right ) x -\frac {a^{2} A}{6}}{x^{6}}\)	\(51\)
risch	\(\frac {-\frac {b^{2} B \,x^{3}}{3}+\left (-\frac {1}{4} b^{2} A -\frac {1}{2} a b B \right ) x^{2}+\left (-\frac {2}{5} a b A -\frac {1}{5} a^{2} B \right ) x -\frac {a^{2} A}{6}}{x^{6}}\)	\(51\)
gosper	\(-\frac {20 b^{2} B \,x^{3}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+24 a A b x +12 a^{2} B x +10 a^{2} A}{60 x^{6}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^2*A/x^6-1/5*a*(2*A*b+B*a)/x^5-1/4*b*(A*b+2*B*a)/x^4-1/3*b^2*B/x^3

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Maxima [A]
time = 0.27, size = 51, normalized size = 0.93 \begin {gather*} -\frac {20 \, B b^{2} x^{3} + 10 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*B*b^2*x^3 + 10*A*a^2 + 15*(2*B*a*b + A*b^2)*x^2 + 12*(B*a^2 + 2*A*a*b)*x)/x^6

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Fricas [A]
time = 0.81, size = 51, normalized size = 0.93 \begin {gather*} -\frac {20 \, B b^{2} x^{3} + 10 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*B*b^2*x^3 + 10*A*a^2 + 15*(2*B*a*b + A*b^2)*x^2 + 12*(B*a^2 + 2*A*a*b)*x)/x^6

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Sympy [A]
time = 0.57, size = 56, normalized size = 1.02 \begin {gather*} \frac {- 10 A a^{2} - 20 B b^{2} x^{3} + x^{2} \left (- 15 A b^{2} - 30 B a b\right ) + x \left (- 24 A a b - 12 B a^{2}\right )}{60 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-10*A*a**2 - 20*B*b**2*x**3 + x**2*(-15*A*b**2 - 30*B*a*b) + x*(-24*A*a*b - 12*B*a**2))/(60*x**6)

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Giac [A]
time = 1.13, size = 51, normalized size = 0.93 \begin {gather*} -\frac {20 \, B b^{2} x^{3} + 30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 12 \, B a^{2} x + 24 \, A a b x + 10 \, A a^{2}}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*B*b^2*x^3 + 30*B*a*b*x^2 + 15*A*b^2*x^2 + 12*B*a^2*x + 24*A*a*b*x + 10*A*a^2)/x^6

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Mupad [B]
time = 0.04, size = 51, normalized size = 0.93 \begin {gather*} -\frac {x^2\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}\right )+\frac {A\,a^2}{6}+x\,\left (\frac {B\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+\frac {B\,b^2\,x^3}{3}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^2)/x^7,x)

[Out]

-(x^2*((A*b^2)/4 + (B*a*b)/2) + (A*a^2)/6 + x*((B*a^2)/5 + (2*A*a*b)/5) + (B*b^2*x^3)/3)/x^6

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