3.9.77 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^8} \, dx\) [877]

Optimal. Leaf size=160 \[ -\frac {a^3 A}{7 x^7}-\frac {a^2 (3 A b+a B)}{6 x^6}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{5 x^5}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{4 x^4}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{3 x^3}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{2 x^2}-\frac {c^2 (3 b B+A c)}{x}+B c^3 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 160, 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^3 A}{7 x^7}-\frac {a^2 (a B+3 A b)}{6 x^6}-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^5}-\frac {3 c \left (a B c+A b c+b^2 B\right )}{2 x^2}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{3 x^3}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{4 x^4}-\frac {c^2 (A c+3 b B)}{x}+B c^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x]

[Out]

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

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

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Mathematica [A]
time = 0.06, size = 175, normalized size = 1.09 \begin {gather*} -\frac {10 a^3 (6 A+7 B x)+21 a^2 x (3 B x (4 b+5 c x)+2 A (5 b+6 c x))+21 a x^2 \left (5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )+2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+35 x^3 \left (2 b B x \left (2 b^2+9 b c x+18 c^2 x^2\right )+3 A \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )\right )-420 B c^3 x^7 \log (x)}{420 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^8,x]

[Out]

-1/420*(10*a^3*(6*A + 7*B*x) + 21*a^2*x*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x)) + 21*a*x^2*(5*B*x*(3*b^2 + 8
*b*c*x + 6*c^2*x^2) + 2*A*(6*b^2 + 15*b*c*x + 10*c^2*x^2)) + 35*x^3*(2*b*B*x*(2*b^2 + 9*b*c*x + 18*c^2*x^2) +
3*A*(b^3 + 4*b^2*c*x + 6*b*c^2*x^2 + 4*c^3*x^3)) - 420*B*c^3*x^7*Log[x])/x^7

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Maple [A]
time = 0.68, size = 152, normalized size = 0.95

method	result	size
default	\(-\frac {3 a \left (A a c +b^{2} A +a b B \right )}{5 x^{5}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{4 x^{4}}-\frac {a^{2} \left (3 A b +B a \right )}{6 x^{6}}-\frac {3 c \left (A b c +a B c +b^{2} B \right )}{2 x^{2}}-\frac {a^{3} A}{7 x^{7}}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 a b B c +B \,b^{3}}{3 x^{3}}+B \,c^{3} \ln \left (x \right )-\frac {c^{2} \left (A c +3 B b \right )}{x}\)	\(152\)
norman	\(\frac {\left (-\frac {1}{2} A \,a^{2} b -\frac {1}{6} B \,a^{3}\right ) x +\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B a \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-\frac {3}{5} a^{2} A c -\frac {3}{5} A a \,b^{2}-\frac {3}{5} B \,a^{2} b \right ) x^{2}+\left (-A a \,c^{2}-A \,b^{2} c -2 a b B c -\frac {1}{3} B \,b^{3}\right ) x^{4}+\left (-\frac {3}{2} A a b c -\frac {1}{4} A \,b^{3}-\frac {3}{4} B \,a^{2} c -\frac {3}{4} B a \,b^{2}\right ) x^{3}+\left (-A \,c^{3}-3 B b \,c^{2}\right ) x^{6}-\frac {A \,a^{3}}{7}}{x^{7}}+B \,c^{3} \ln \left (x \right )\)	\(168\)
risch	\(\frac {\left (-\frac {1}{2} A \,a^{2} b -\frac {1}{6} B \,a^{3}\right ) x +\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B a \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-\frac {3}{5} a^{2} A c -\frac {3}{5} A a \,b^{2}-\frac {3}{5} B \,a^{2} b \right ) x^{2}+\left (-A a \,c^{2}-A \,b^{2} c -2 a b B c -\frac {1}{3} B \,b^{3}\right ) x^{4}+\left (-\frac {3}{2} A a b c -\frac {1}{4} A \,b^{3}-\frac {3}{4} B \,a^{2} c -\frac {3}{4} B a \,b^{2}\right ) x^{3}+\left (-A \,c^{3}-3 B b \,c^{2}\right ) x^{6}-\frac {A \,a^{3}}{7}}{x^{7}}+B \,c^{3} \ln \left (x \right )\)	\(168\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^3/x^8,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 165, normalized size = 1.03 \begin {gather*} B c^{3} \log \left (x\right ) - \frac {420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 140 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

B*c^3*log(x) - 1/420*(420*(3*B*b*c^2 + A*c^3)*x^6 + 630*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 140*(B*b^3 + 3*A*a*c
^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 252*(B*a^2*
b + A*a*b^2 + A*a^2*c)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7

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Fricas [A]
time = 1.04, size = 168, normalized size = 1.05 \begin {gather*} \frac {420 \, B c^{3} x^{7} \log \left (x\right ) - 420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 630 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} - 140 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 60 \, A a^{3} - 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

1/420*(420*B*c^3*x^7*log(x) - 420*(3*B*b*c^2 + A*c^3)*x^6 - 630*(B*b^2*c + (B*a + A*b)*c^2)*x^5 - 140*(B*b^3 +
3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 60*A*a^3 - 105*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 252
*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 70*(B*a^3 + 3*A*a^2*b)*x)/x^7

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Sympy [F(-1)] Timed out
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**8,x)

[Out]

Timed out

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Giac [A]
time = 1.26, size = 165, normalized size = 1.03 \begin {gather*} B c^{3} \log \left ({\left | x \right |}\right ) - \frac {420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \, {\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} x^{5} + 140 \, {\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

B*c^3*log(abs(x)) - 1/420*(420*(3*B*b*c^2 + A*c^3)*x^6 + 630*(B*b^2*c + B*a*c^2 + A*b*c^2)*x^5 + 140*(B*b^3 +
6*B*a*b*c + 3*A*b^2*c + 3*A*a*c^2)*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*x^3 + 252*
(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7

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Mupad [B]
time = 1.22, size = 165, normalized size = 1.03 \begin {gather*} B\,c^3\,\ln \left (x\right )-\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{4}+\frac {3\,B\,a\,b^2}{4}+\frac {3\,A\,c\,a\,b}{2}+\frac {A\,b^3}{4}\right )+x^4\,\left (\frac {B\,b^3}{3}+A\,b^2\,c+2\,B\,a\,b\,c+A\,a\,c^2\right )+x\,\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )+\frac {A\,a^3}{7}+x^6\,\left (A\,c^3+3\,B\,b\,c^2\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,c\,a^2}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{2}+\frac {3\,A\,b\,c^2}{2}+\frac {3\,B\,a\,c^2}{2}\right )}{x^7} \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^8,x)

[Out]

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

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