3.9.3 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^9} \, dx\)

Optimal. Leaf size=162 \[ -\frac {a^3 A}{8 x^8}-\frac {a^2 (a B+3 A b)}{7 x^7}-\frac {a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac {c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac {c^2 (A c+3 b B)}{2 x^2}-\frac {B c^3}{x} \]

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Rubi [A]  time = 0.10, antiderivative size = 162, 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 = {765} \begin {gather*} -\frac {a^2 (a B+3 A b)}{7 x^7}-\frac {a^3 A}{8 x^8}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac {a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac {c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac {c^2 (A c+3 b B)}{2 x^2}-\frac {B c^3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^9,x]

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 172, normalized size = 1.06 \begin {gather*} -\frac {5 a^3 (7 A+8 B x)+4 a^2 x (5 A (6 b+7 c x)+7 B x (5 b+6 c x))+14 a x^2 \left (A \left (10 b^2+24 b c x+15 c^2 x^2\right )+2 B x \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+14 x^3 \left (A \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )\right )}{280 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^9,x]

[Out]

-1/280*(5*a^3*(7*A + 8*B*x) + 4*a^2*x*(7*B*x*(5*b + 6*c*x) + 5*A*(6*b + 7*c*x)) + 14*a*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)) + 14*x^3*(5*B*x*(b^3 + 4*b^2*c*x + 6*b*c^2*x^2 + 4*c
^3*x^3) + A*(4*b^3 + 15*b^2*c*x + 20*b*c^2*x^2 + 10*c^3*x^3)))/x^8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^9,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^9, x]

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fricas [A]  time = 0.41, size = 166, normalized size = 1.02 \begin {gather*} -\frac {280 \, B c^{3} x^{7} + 140 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, 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)^3/x^9,x, algorithm="fricas")

[Out]

-1/280*(280*B*c^3*x^7 + 140*(3*B*b*c^2 + A*c^3)*x^6 + 280*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 70*(B*b^3 + 3*A*a*
c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 140*(B*a^2*
b + A*a*b^2 + A*a^2*c)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^8

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giac [A]  time = 0.17, size = 191, normalized size = 1.18 \begin {gather*} -\frac {280 \, B c^{3} x^{7} + 420 \, B b c^{2} x^{6} + 140 \, A c^{3} x^{6} + 280 \, B b^{2} c x^{5} + 280 \, B a c^{2} x^{5} + 280 \, A b c^{2} x^{5} + 70 \, B b^{3} x^{4} + 420 \, B a b c x^{4} + 210 \, A b^{2} c x^{4} + 210 \, A a c^{2} x^{4} + 168 \, B a b^{2} x^{3} + 56 \, A b^{3} x^{3} + 168 \, B a^{2} c x^{3} + 336 \, A a b c x^{3} + 140 \, B a^{2} b x^{2} + 140 \, A a b^{2} x^{2} + 140 \, A a^{2} c x^{2} + 40 \, B a^{3} x + 120 \, A a^{2} b x + 35 \, A a^{3}}{280 \, 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)^3/x^9,x, algorithm="giac")

[Out]

-1/280*(280*B*c^3*x^7 + 420*B*b*c^2*x^6 + 140*A*c^3*x^6 + 280*B*b^2*c*x^5 + 280*B*a*c^2*x^5 + 280*A*b*c^2*x^5
+ 70*B*b^3*x^4 + 420*B*a*b*c*x^4 + 210*A*b^2*c*x^4 + 210*A*a*c^2*x^4 + 168*B*a*b^2*x^3 + 56*A*b^3*x^3 + 168*B*
a^2*c*x^3 + 336*A*a*b*c*x^3 + 140*B*a^2*b*x^2 + 140*A*a*b^2*x^2 + 140*A*a^2*c*x^2 + 40*B*a^3*x + 120*A*a^2*b*x
+ 35*A*a^3)/x^8

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maple [A]  time = 0.05, size = 154, normalized size = 0.95 \begin {gather*} -\frac {B \,c^{3}}{x}-\frac {\left (A c +3 b B \right ) c^{2}}{2 x^{2}}-\frac {\left (A b c +a B c +b^{2} B \right ) c}{x^{3}}-\frac {A \,a^{3}}{8 x^{8}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 a b B c +b^{3} B}{4 x^{4}}-\frac {\left (3 A b +B a \right ) a^{2}}{7 x^{7}}-\frac {\left (A a c +A \,b^{2}+B a b \right ) a}{2 x^{6}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^3/x^9,x)

[Out]

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

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maxima [A]  time = 0.56, size = 166, normalized size = 1.02 \begin {gather*} -\frac {280 \, B c^{3} x^{7} + 140 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, 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)^3/x^9,x, algorithm="maxima")

[Out]

-1/280*(280*B*c^3*x^7 + 140*(3*B*b*c^2 + A*c^3)*x^6 + 280*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 70*(B*b^3 + 3*A*a*
c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 140*(B*a^2*
b + A*a*b^2 + A*a^2*c)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^8

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mupad [B]  time = 1.18, size = 165, normalized size = 1.02 \begin {gather*} -\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{5}+\frac {3\,B\,a\,b^2}{5}+\frac {6\,A\,c\,a\,b}{5}+\frac {A\,b^3}{5}\right )+x^4\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,b^2\,c}{4}+\frac {3\,B\,a\,b\,c}{2}+\frac {3\,A\,a\,c^2}{4}\right )+x\,\left (\frac {B\,a^3}{7}+\frac {3\,A\,b\,a^2}{7}\right )+\frac {A\,a^3}{8}+x^6\,\left (\frac {A\,c^3}{2}+\frac {3\,B\,b\,c^2}{2}\right )+x^2\,\left (\frac {B\,a^2\,b}{2}+\frac {A\,c\,a^2}{2}+\frac {A\,a\,b^2}{2}\right )+x^5\,\left (B\,b^2\,c+A\,b\,c^2+B\,a\,c^2\right )+B\,c^3\,x^7}{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)^3)/x^9,x)

[Out]

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

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sympy [A]  time = 119.40, size = 196, normalized size = 1.21 \begin {gather*} \frac {- 35 A a^{3} - 280 B c^{3} x^{7} + x^{6} \left (- 140 A c^{3} - 420 B b c^{2}\right ) + x^{5} \left (- 280 A b c^{2} - 280 B a c^{2} - 280 B b^{2} c\right ) + x^{4} \left (- 210 A a c^{2} - 210 A b^{2} c - 420 B a b c - 70 B b^{3}\right ) + x^{3} \left (- 336 A a b c - 56 A b^{3} - 168 B a^{2} c - 168 B a b^{2}\right ) + x^{2} \left (- 140 A a^{2} c - 140 A a b^{2} - 140 B a^{2} b\right ) + x \left (- 120 A a^{2} b - 40 B a^{3}\right )}{280 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)**3/x**9,x)

[Out]

(-35*A*a**3 - 280*B*c**3*x**7 + x**6*(-140*A*c**3 - 420*B*b*c**2) + x**5*(-280*A*b*c**2 - 280*B*a*c**2 - 280*B
*b**2*c) + x**4*(-210*A*a*c**2 - 210*A*b**2*c - 420*B*a*b*c - 70*B*b**3) + x**3*(-336*A*a*b*c - 56*A*b**3 - 16
8*B*a**2*c - 168*B*a*b**2) + x**2*(-140*A*a**2*c - 140*A*a*b**2 - 140*B*a**2*b) + x*(-120*A*a**2*b - 40*B*a**3
))/(280*x**8)

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