3.9.66 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{x^9} \, dx\) [866]

Optimal. Leaf size=101 \[ -\frac {a^2 A}{8 x^8}-\frac {a (2 A b+a B)}{7 x^7}-\frac {2 a b B+A \left (b^2+2 a c\right )}{6 x^6}-\frac {b^2 B+2 A b c+2 a B c}{5 x^5}-\frac {c (2 b B+A c)}{4 x^4}-\frac {B c^2}{3 x^3} \]

[Out]

-1/8*a^2*A/x^8-1/7*a*(2*A*b+B*a)/x^7+1/6*(-2*a*b*B-A*(2*a*c+b^2))/x^6+1/5*(-2*A*b*c-2*B*a*c-B*b^2)/x^5-1/4*c*(
A*c+2*B*b)/x^4-1/3*B*c^2/x^3

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Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {779} \begin {gather*} -\frac {a^2 A}{8 x^8}-\frac {A \left (2 a c+b^2\right )+2 a b B}{6 x^6}-\frac {2 a B c+2 A b c+b^2 B}{5 x^5}-\frac {a (a B+2 A b)}{7 x^7}-\frac {c (A c+2 b B)}{4 x^4}-\frac {B c^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^9,x]

[Out]

-1/8*(a^2*A)/x^8 - (a*(2*A*b + a*B))/(7*x^7) - (2*a*b*B + A*(b^2 + 2*a*c))/(6*x^6) - (b^2*B + 2*A*b*c + 2*a*B*
c)/(5*x^5) - (c*(2*b*B + A*c))/(4*x^4) - (B*c^2)/(3*x^3)

Rule 779

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 (a+b x+c x^2\right )^2}{x^9} \, dx &=\int \left (\frac {a^2 A}{x^9}+\frac {a (2 A b+a B)}{x^8}+\frac {2 a b B+A \left (b^2+2 a c\right )}{x^7}+\frac {b^2 B+2 A b c+2 a B c}{x^6}+\frac {c (2 b B+A c)}{x^5}+\frac {B c^2}{x^4}\right ) \, dx\\ &=-\frac {a^2 A}{8 x^8}-\frac {a (2 A b+a B)}{7 x^7}-\frac {2 a b B+A \left (b^2+2 a c\right )}{6 x^6}-\frac {b^2 B+2 A b c+2 a B c}{5 x^5}-\frac {c (2 b B+A c)}{4 x^4}-\frac {B c^2}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 99, normalized size = 0.98 \begin {gather*} -\frac {15 a^2 (7 A+8 B x)+8 a x (7 B x (5 b+6 c x)+5 A (6 b+7 c x))+14 x^2 \left (2 B x \left (6 b^2+15 b c x+10 c^2 x^2\right )+A \left (10 b^2+24 b c x+15 c^2 x^2\right )\right )}{840 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^9,x]

[Out]

-1/840*(15*a^2*(7*A + 8*B*x) + 8*a*x*(7*B*x*(5*b + 6*c*x) + 5*A*(6*b + 7*c*x)) + 14*x^2*(2*B*x*(6*b^2 + 15*b*c
*x + 10*c^2*x^2) + A*(10*b^2 + 24*b*c*x + 15*c^2*x^2)))/x^8

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Maple [A]
time = 0.85, size = 90, normalized size = 0.89

method	result	size
default	\(-\frac {2 A b c +2 a B c +b^{2} B}{5 x^{5}}-\frac {c \left (A c +2 B b \right )}{4 x^{4}}-\frac {2 A a c +b^{2} A +2 a b B}{6 x^{6}}-\frac {a^{2} A}{8 x^{8}}-\frac {a \left (2 A b +B a \right )}{7 x^{7}}-\frac {B \,c^{2}}{3 x^{3}}\)	\(90\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{3}+\left (-\frac {1}{4} A \,c^{2}-\frac {1}{2} b B c \right ) x^{4}+\left (-\frac {2}{5} A b c -\frac {2}{5} a B c -\frac {1}{5} b^{2} B \right ) x^{3}+\left (-\frac {1}{3} A a c -\frac {1}{6} b^{2} A -\frac {1}{3} a b B \right ) x^{2}+\left (-\frac {2}{7} a b A -\frac {1}{7} a^{2} B \right ) x -\frac {a^{2} A}{8}}{x^{8}}\)	\(93\)
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{3}+\left (-\frac {1}{4} A \,c^{2}-\frac {1}{2} b B c \right ) x^{4}+\left (-\frac {2}{5} A b c -\frac {2}{5} a B c -\frac {1}{5} b^{2} B \right ) x^{3}+\left (-\frac {1}{3} A a c -\frac {1}{6} b^{2} A -\frac {1}{3} a b B \right ) x^{2}+\left (-\frac {2}{7} a b A -\frac {1}{7} a^{2} B \right ) x -\frac {a^{2} A}{8}}{x^{8}}\)	\(93\)
gosper	\(-\frac {280 B \,c^{2} x^{5}+210 A \,c^{2} x^{4}+420 B b c \,x^{4}+336 A b c \,x^{3}+336 a B c \,x^{3}+168 b^{2} B \,x^{3}+280 a A c \,x^{2}+140 b^{2} A \,x^{2}+280 B a b \,x^{2}+240 a b A x +120 a^{2} B x +105 a^{2} A}{840 x^{8}}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^2/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/5*(2*A*b*c+2*B*a*c+B*b^2)/x^5-1/4*c*(A*c+2*B*b)/x^4-1/6*(2*A*a*c+A*b^2+2*B*a*b)/x^6-1/8*a^2*A/x^8-1/7*a*(2*
A*b+B*a)/x^7-1/3*B*c^2/x^3

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Maxima [A]
time = 0.27, size = 93, normalized size = 0.92 \begin {gather*} -\frac {280 \, B c^{2} x^{5} + 210 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 168 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 105 \, A a^{2} + 140 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 120 \, {\left (B a^{2} + 2 \, A a b\right )} x}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*c^2*x^5 + 210*(2*B*b*c + A*c^2)*x^4 + 168*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 105*A*a^2 + 140*(2*B*a
*b + A*b^2 + 2*A*a*c)*x^2 + 120*(B*a^2 + 2*A*a*b)*x)/x^8

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Fricas [A]
time = 3.19, size = 93, normalized size = 0.92 \begin {gather*} -\frac {280 \, B c^{2} x^{5} + 210 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 168 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 105 \, A a^{2} + 140 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 120 \, {\left (B a^{2} + 2 \, A a b\right )} x}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*c^2*x^5 + 210*(2*B*b*c + A*c^2)*x^4 + 168*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 105*A*a^2 + 140*(2*B*a
*b + A*b^2 + 2*A*a*c)*x^2 + 120*(B*a^2 + 2*A*a*b)*x)/x^8

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Sympy [A]
time = 210.21, size = 107, normalized size = 1.06 \begin {gather*} \frac {- 105 A a^{2} - 280 B c^{2} x^{5} + x^{4} \left (- 210 A c^{2} - 420 B b c\right ) + x^{3} \left (- 336 A b c - 336 B a c - 168 B b^{2}\right ) + x^{2} \left (- 280 A a c - 140 A b^{2} - 280 B a b\right ) + x \left (- 240 A a b - 120 B a^{2}\right )}{840 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**2/x**9,x)

[Out]

(-105*A*a**2 - 280*B*c**2*x**5 + x**4*(-210*A*c**2 - 420*B*b*c) + x**3*(-336*A*b*c - 336*B*a*c - 168*B*b**2) +
x**2*(-280*A*a*c - 140*A*b**2 - 280*B*a*b) + x*(-240*A*a*b - 120*B*a**2))/(840*x**8)

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Giac [A]
time = 1.31, size = 101, normalized size = 1.00 \begin {gather*} -\frac {280 \, B c^{2} x^{5} + 420 \, B b c x^{4} + 210 \, A c^{2} x^{4} + 168 \, B b^{2} x^{3} + 336 \, B a c x^{3} + 336 \, A b c x^{3} + 280 \, B a b x^{2} + 140 \, A b^{2} x^{2} + 280 \, A a c x^{2} + 120 \, B a^{2} x + 240 \, A a b x + 105 \, A a^{2}}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^9,x, algorithm="giac")

[Out]

-1/840*(280*B*c^2*x^5 + 420*B*b*c*x^4 + 210*A*c^2*x^4 + 168*B*b^2*x^3 + 336*B*a*c*x^3 + 336*A*b*c*x^3 + 280*B*
a*b*x^2 + 140*A*b^2*x^2 + 280*A*a*c*x^2 + 120*B*a^2*x + 240*A*a*b*x + 105*A*a^2)/x^8

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Mupad [B]
time = 0.05, size = 93, normalized size = 0.92 \begin {gather*} -\frac {x^4\,\left (\frac {A\,c^2}{4}+\frac {B\,b\,c}{2}\right )+\frac {A\,a^2}{8}+x^2\,\left (\frac {A\,b^2}{6}+\frac {B\,a\,b}{3}+\frac {A\,a\,c}{3}\right )+x^3\,\left (\frac {B\,b^2}{5}+\frac {2\,A\,c\,b}{5}+\frac {2\,B\,a\,c}{5}\right )+x\,\left (\frac {B\,a^2}{7}+\frac {2\,A\,b\,a}{7}\right )+\frac {B\,c^2\,x^5}{3}}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^2)/x^9,x)

[Out]

-(x^4*((A*c^2)/4 + (B*b*c)/2) + (A*a^2)/8 + x^2*((A*b^2)/6 + (A*a*c)/3 + (B*a*b)/3) + x^3*((B*b^2)/5 + (2*A*b*
c)/5 + (2*B*a*c)/5) + x*((B*a^2)/7 + (2*A*a*b)/7) + (B*c^2*x^5)/3)/x^8

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