3.9.4 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^{10}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {a^3 A}{9 x^9}-\frac {a^2 (a B+3 A b)}{8 x^8}-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{7 x^7}-\frac {3 c \left (a B c+A b c+b^2 B\right )}{4 x^4}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{5 x^5}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{6 x^6}-\frac {c^2 (A c+3 b B)}{3 x^3}-\frac {B c^3}{2 x^2} \]

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Rubi [A]  time = 0.11, antiderivative size = 166, 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)}{8 x^8}-\frac {a^3 A}{9 x^9}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{5 x^5}-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{7 x^7}-\frac {3 c \left (a B c+A b c+b^2 B\right )}{4 x^4}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{6 x^6}-\frac {c^2 (A c+3 b B)}{3 x^3}-\frac {B c^3}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]

[Out]

-(a^3*A)/(9*x^9) - (a^2*(3*A*b + a*B))/(8*x^8) - (3*a*(a*b*B + A*(b^2 + a*c)))/(7*x^7) - (3*a*B*(b^2 + a*c) +
A*(b^3 + 6*a*b*c))/(6*x^6) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/(5*x^5) - (3*c*(b^2*B + A*b*c + a*B*c
))/(4*x^4) - (c^2*(3*b*B + A*c))/(3*x^3) - (B*c^3)/(2*x^2)

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^{10}} \, dx &=\int \left (\frac {a^3 A}{x^{10}}+\frac {a^2 (3 A b+a B)}{x^9}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^8}+\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^7}+\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^6}+\frac {3 c \left (b^2 B+A b c+a B c\right )}{x^5}+\frac {c^2 (3 b B+A c)}{x^4}+\frac {B c^3}{x^3}\right ) \, dx\\ &=-\frac {a^3 A}{9 x^9}-\frac {a^2 (3 A b+a B)}{8 x^8}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{7 x^7}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{6 x^6}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{5 x^5}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{4 x^4}-\frac {c^2 (3 b B+A c)}{3 x^3}-\frac {B c^3}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 175, normalized size = 1.05 \begin {gather*} -\frac {35 a^3 (8 A+9 B x)+45 a^2 x (3 A (7 b+8 c x)+4 B x (6 b+7 c x))+18 a x^2 \left (4 A \left (15 b^2+35 b c x+21 c^2 x^2\right )+7 B x \left (10 b^2+24 b c x+15 c^2 x^2\right )\right )+42 x^3 \left (A \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )+3 B x \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )\right )}{2520 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]

[Out]

-1/2520*(35*a^3*(8*A + 9*B*x) + 45*a^2*x*(4*B*x*(6*b + 7*c*x) + 3*A*(7*b + 8*c*x)) + 18*a*x^2*(7*B*x*(10*b^2 +
24*b*c*x + 15*c^2*x^2) + 4*A*(15*b^2 + 35*b*c*x + 21*c^2*x^2)) + 42*x^3*(3*B*x*(4*b^3 + 15*b^2*c*x + 20*b*c^2
*x^2 + 10*c^3*x^3) + A*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 + 20*c^3*x^3)))/x^9

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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^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^10, x]

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fricas [A]  time = 0.41, size = 166, normalized size = 1.00 \begin {gather*} -\frac {1260 \, B c^{3} x^{7} + 840 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1890 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 504 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 315 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/2520*(1260*B*c^3*x^7 + 840*(3*B*b*c^2 + A*c^3)*x^6 + 1890*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 504*(B*b^3 + 3*
A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 1080*
(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

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giac [A]  time = 0.16, size = 191, normalized size = 1.15 \begin {gather*} -\frac {1260 \, B c^{3} x^{7} + 2520 \, B b c^{2} x^{6} + 840 \, A c^{3} x^{6} + 1890 \, B b^{2} c x^{5} + 1890 \, B a c^{2} x^{5} + 1890 \, A b c^{2} x^{5} + 504 \, B b^{3} x^{4} + 3024 \, B a b c x^{4} + 1512 \, A b^{2} c x^{4} + 1512 \, A a c^{2} x^{4} + 1260 \, B a b^{2} x^{3} + 420 \, A b^{3} x^{3} + 1260 \, B a^{2} c x^{3} + 2520 \, A a b c x^{3} + 1080 \, B a^{2} b x^{2} + 1080 \, A a b^{2} x^{2} + 1080 \, A a^{2} c x^{2} + 315 \, B a^{3} x + 945 \, A a^{2} b x + 280 \, A a^{3}}{2520 \, x^{9}} \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^10,x, algorithm="giac")

[Out]

-1/2520*(1260*B*c^3*x^7 + 2520*B*b*c^2*x^6 + 840*A*c^3*x^6 + 1890*B*b^2*c*x^5 + 1890*B*a*c^2*x^5 + 1890*A*b*c^
2*x^5 + 504*B*b^3*x^4 + 3024*B*a*b*c*x^4 + 1512*A*b^2*c*x^4 + 1512*A*a*c^2*x^4 + 1260*B*a*b^2*x^3 + 420*A*b^3*
x^3 + 1260*B*a^2*c*x^3 + 2520*A*a*b*c*x^3 + 1080*B*a^2*b*x^2 + 1080*A*a*b^2*x^2 + 1080*A*a^2*c*x^2 + 315*B*a^3
*x + 945*A*a^2*b*x + 280*A*a^3)/x^9

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maple [A]  time = 0.06, size = 154, normalized size = 0.93 \begin {gather*} -\frac {B \,c^{3}}{2 x^{2}}-\frac {\left (A c +3 b B \right ) c^{2}}{3 x^{3}}-\frac {3 \left (A b c +a B c +b^{2} B \right ) c}{4 x^{4}}-\frac {A \,a^{3}}{9 x^{9}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 a b B c +b^{3} B}{5 x^{5}}-\frac {\left (3 A b +B a \right ) a^{2}}{8 x^{8}}-\frac {3 \left (A a c +A \,b^{2}+B a b \right ) a}{7 x^{7}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{6 x^{6}} \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^10,x)

[Out]

-1/5*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)/x^5-3/4*c*(A*b*c+B*a*c+B*b^2)/x^4-1/3*(A*c+3*B*b)*c^2/x^3-1/8*a^2*(
3*A*b+B*a)/x^8-1/2*B*c^3/x^2-1/9*a^3*A/x^9-3/7*a*(A*a*c+A*b^2+B*a*b)/x^7-1/6*(6*A*a*b*c+A*b^3+3*B*a^2*c+3*B*a*
b^2)/x^6

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maxima [A]  time = 0.52, size = 166, normalized size = 1.00 \begin {gather*} -\frac {1260 \, B c^{3} x^{7} + 840 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1890 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 504 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 315 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \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^10,x, algorithm="maxima")

[Out]

-1/2520*(1260*B*c^3*x^7 + 840*(3*B*b*c^2 + A*c^3)*x^6 + 1890*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 504*(B*b^3 + 3*
A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 1080*
(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

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mupad [B]  time = 1.19, size = 167, normalized size = 1.01 \begin {gather*} -\frac {x^3\,\left (\frac {B\,c\,a^2}{2}+\frac {B\,a\,b^2}{2}+A\,c\,a\,b+\frac {A\,b^3}{6}\right )+x^4\,\left (\frac {B\,b^3}{5}+\frac {3\,A\,b^2\,c}{5}+\frac {6\,B\,a\,b\,c}{5}+\frac {3\,A\,a\,c^2}{5}\right )+x\,\left (\frac {B\,a^3}{8}+\frac {3\,A\,b\,a^2}{8}\right )+\frac {A\,a^3}{9}+x^6\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{7}+\frac {3\,A\,c\,a^2}{7}+\frac {3\,A\,a\,b^2}{7}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{4}+\frac {3\,A\,b\,c^2}{4}+\frac {3\,B\,a\,c^2}{4}\right )+\frac {B\,c^3\,x^7}{2}}{x^9} \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^10,x)

[Out]

-(x^3*((A*b^3)/6 + (B*a*b^2)/2 + (B*a^2*c)/2 + A*a*b*c) + x^4*((B*b^3)/5 + (3*A*a*c^2)/5 + (3*A*b^2*c)/5 + (6*
B*a*b*c)/5) + x*((B*a^3)/8 + (3*A*a^2*b)/8) + (A*a^3)/9 + x^6*((A*c^3)/3 + B*b*c^2) + x^2*((3*A*a*b^2)/7 + (3*
A*a^2*c)/7 + (3*B*a^2*b)/7) + x^5*((3*A*b*c^2)/4 + (3*B*a*c^2)/4 + (3*B*b^2*c)/4) + (B*c^3*x^7)/2)/x^9

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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**10,x)

[Out]

Timed out

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