3.9.5 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^{11}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {a^3 A}{10 x^{10}}-\frac {a^2 (a B+3 A b)}{9 x^9}-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{8 x^8}-\frac {3 c \left (a B c+A b c+b^2 B\right )}{5 x^5}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{6 x^6}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{7 x^7}-\frac {c^2 (A c+3 b B)}{4 x^4}-\frac {B c^3}{3 x^3} \]

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Rubi [A]  time = 0.10, 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)}{9 x^9}-\frac {a^3 A}{10 x^{10}}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{6 x^6}-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{8 x^8}-\frac {3 c \left (a B c+A b c+b^2 B\right )}{5 x^5}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{7 x^7}-\frac {c^2 (A c+3 b B)}{4 x^4}-\frac {B c^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^11,x]

[Out]

-(a^3*A)/(10*x^10) - (a^2*(3*A*b + a*B))/(9*x^9) - (3*a*(a*b*B + A*(b^2 + a*c)))/(8*x^8) - (3*a*B*(b^2 + a*c)
+ A*(b^3 + 6*a*b*c))/(7*x^7) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/(6*x^6) - (3*c*(b^2*B + A*b*c + a*B
*c))/(5*x^5) - (c^2*(3*b*B + A*c))/(4*x^4) - (B*c^3)/(3*x^3)

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^{11}} \, dx &=\int \left (\frac {a^3 A}{x^{11}}+\frac {a^2 (3 A b+a B)}{x^{10}}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^9}+\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^8}+\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^7}+\frac {3 c \left (b^2 B+A b c+a B c\right )}{x^6}+\frac {c^2 (3 b B+A c)}{x^5}+\frac {B c^3}{x^4}\right ) \, dx\\ &=-\frac {a^3 A}{10 x^{10}}-\frac {a^2 (3 A b+a B)}{9 x^9}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{8 x^8}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{7 x^7}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{6 x^6}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{5 x^5}-\frac {c^2 (3 b B+A c)}{4 x^4}-\frac {B c^3}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 176, normalized size = 1.06 \begin {gather*} -\frac {28 a^3 (9 A+10 B x)+15 a^2 x (7 A (8 b+9 c x)+9 B x (7 b+8 c x))+9 a x^2 \left (5 A \left (21 b^2+48 b c x+28 c^2 x^2\right )+8 B x \left (15 b^2+35 b c x+21 c^2 x^2\right )\right )+6 x^3 \left (3 A \left (20 b^3+70 b^2 c x+84 b c^2 x^2+35 c^3 x^3\right )+7 B x \left (10 b^3+36 b^2 c x+45 b c^2 x^2+20 c^3 x^3\right )\right )}{2520 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^11,x]

[Out]

-1/2520*(28*a^3*(9*A + 10*B*x) + 15*a^2*x*(9*B*x*(7*b + 8*c*x) + 7*A*(8*b + 9*c*x)) + 9*a*x^2*(8*B*x*(15*b^2 +
35*b*c*x + 21*c^2*x^2) + 5*A*(21*b^2 + 48*b*c*x + 28*c^2*x^2)) + 6*x^3*(7*B*x*(10*b^3 + 36*b^2*c*x + 45*b*c^2
*x^2 + 20*c^3*x^3) + 3*A*(20*b^3 + 70*b^2*c*x + 84*b*c^2*x^2 + 35*c^3*x^3)))/x^10

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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^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^11,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^11, x]

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fricas [A]  time = 0.40, size = 166, normalized size = 1.00 \begin {gather*} -\frac {840 \, B c^{3} x^{7} + 630 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1512 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 420 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 252 \, A a^{3} + 360 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 945 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 280 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \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^11,x, algorithm="fricas")

[Out]

-1/2520*(840*B*c^3*x^7 + 630*(3*B*b*c^2 + A*c^3)*x^6 + 1512*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 420*(B*b^3 + 3*A
*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 252*A*a^3 + 360*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 945*(B
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10

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giac [A]  time = 0.16, size = 191, normalized size = 1.15 \begin {gather*} -\frac {840 \, B c^{3} x^{7} + 1890 \, B b c^{2} x^{6} + 630 \, A c^{3} x^{6} + 1512 \, B b^{2} c x^{5} + 1512 \, B a c^{2} x^{5} + 1512 \, A b c^{2} x^{5} + 420 \, B b^{3} x^{4} + 2520 \, B a b c x^{4} + 1260 \, A b^{2} c x^{4} + 1260 \, A a c^{2} x^{4} + 1080 \, B a b^{2} x^{3} + 360 \, A b^{3} x^{3} + 1080 \, B a^{2} c x^{3} + 2160 \, A a b c x^{3} + 945 \, B a^{2} b x^{2} + 945 \, A a b^{2} x^{2} + 945 \, A a^{2} c x^{2} + 280 \, B a^{3} x + 840 \, A a^{2} b x + 252 \, A a^{3}}{2520 \, x^{10}} \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^11,x, algorithm="giac")

[Out]

-1/2520*(840*B*c^3*x^7 + 1890*B*b*c^2*x^6 + 630*A*c^3*x^6 + 1512*B*b^2*c*x^5 + 1512*B*a*c^2*x^5 + 1512*A*b*c^2
*x^5 + 420*B*b^3*x^4 + 2520*B*a*b*c*x^4 + 1260*A*b^2*c*x^4 + 1260*A*a*c^2*x^4 + 1080*B*a*b^2*x^3 + 360*A*b^3*x
^3 + 1080*B*a^2*c*x^3 + 2160*A*a*b*c*x^3 + 945*B*a^2*b*x^2 + 945*A*a*b^2*x^2 + 945*A*a^2*c*x^2 + 280*B*a^3*x +
840*A*a^2*b*x + 252*A*a^3)/x^10

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maple [A]  time = 0.06, size = 154, normalized size = 0.93 \begin {gather*} -\frac {B \,c^{3}}{3 x^{3}}-\frac {\left (A c +3 b B \right ) c^{2}}{4 x^{4}}-\frac {3 \left (A b c +a B c +b^{2} B \right ) c}{5 x^{5}}-\frac {A \,a^{3}}{10 x^{10}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 a b B c +b^{3} B}{6 x^{6}}-\frac {\left (3 A b +B a \right ) a^{2}}{9 x^{9}}-\frac {3 \left (A a c +A \,b^{2}+B a b \right ) a}{8 x^{8}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{7 x^{7}} \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^11,x)

[Out]

-3/5*c*(A*b*c+B*a*c+B*b^2)/x^5-1/4*(A*c+3*B*b)*c^2/x^4-1/3*B*c^3/x^3-3/8*a*(A*a*c+A*b^2+B*a*b)/x^8-1/9*a^2*(3*
A*b+B*a)/x^9-1/10*a^3*A/x^10-1/7*(6*A*a*b*c+A*b^3+3*B*a^2*c+3*B*a*b^2)/x^7-1/6*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+
B*b^3)/x^6

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maxima [A]  time = 0.46, size = 166, normalized size = 1.00 \begin {gather*} -\frac {840 \, B c^{3} x^{7} + 630 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1512 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 420 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 252 \, A a^{3} + 360 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 945 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 280 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \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^11,x, algorithm="maxima")

[Out]

-1/2520*(840*B*c^3*x^7 + 630*(3*B*b*c^2 + A*c^3)*x^6 + 1512*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 420*(B*b^3 + 3*A
*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 252*A*a^3 + 360*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 945*(B
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10

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mupad [B]  time = 1.19, size = 168, normalized size = 1.01 \begin {gather*} -\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{7}+\frac {3\,B\,a\,b^2}{7}+\frac {6\,A\,c\,a\,b}{7}+\frac {A\,b^3}{7}\right )+x^4\,\left (\frac {B\,b^3}{6}+\frac {A\,b^2\,c}{2}+B\,a\,b\,c+\frac {A\,a\,c^2}{2}\right )+x\,\left (\frac {B\,a^3}{9}+\frac {A\,b\,a^2}{3}\right )+\frac {A\,a^3}{10}+x^6\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{8}+\frac {3\,A\,c\,a^2}{8}+\frac {3\,A\,a\,b^2}{8}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{5}+\frac {3\,A\,b\,c^2}{5}+\frac {3\,B\,a\,c^2}{5}\right )+\frac {B\,c^3\,x^7}{3}}{x^{10}} \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^11,x)

[Out]

-(x^3*((A*b^3)/7 + (3*B*a*b^2)/7 + (3*B*a^2*c)/7 + (6*A*a*b*c)/7) + x^4*((B*b^3)/6 + (A*a*c^2)/2 + (A*b^2*c)/2
+ B*a*b*c) + x*((B*a^3)/9 + (A*a^2*b)/3) + (A*a^3)/10 + x^6*((A*c^3)/4 + (3*B*b*c^2)/4) + x^2*((3*A*a*b^2)/8
+ (3*A*a^2*c)/8 + (3*B*a^2*b)/8) + x^5*((3*A*b*c^2)/5 + (3*B*a*c^2)/5 + (3*B*b^2*c)/5) + (B*c^3*x^7)/3)/x^10

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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**11,x)

[Out]

Timed out

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