3.8.100 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{x^6} \, dx\)

Optimal. Leaf size=154 \[ -\frac {a^3 A}{5 x^5}-\frac {a^2 (a B+3 A b)}{4 x^4}-\frac {a \left (A \left (a c+b^2\right )+a b B\right )}{x^3}+3 c \log (x) \left (a B c+A b c+b^2 B\right )-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{2 x^2}+c^2 x (A c+3 b B)+\frac {1}{2} B c^3 x^2 \]

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Rubi [A]  time = 0.12, antiderivative size = 154, 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)}{4 x^4}-\frac {a^3 A}{5 x^5}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{x}-\frac {a \left (A \left (a c+b^2\right )+a b B\right )}{x^3}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{2 x^2}+3 c \log (x) \left (a B c+A b c+b^2 B\right )+c^2 x (A c+3 b B)+\frac {1}{2} B c^3 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]

[Out]

-(a^3*A)/(5*x^5) - (a^2*(3*A*b + a*B))/(4*x^4) - (a*(a*b*B + A*(b^2 + a*c)))/x^3 - (3*a*B*(b^2 + a*c) + A*(b^3
+ 6*a*b*c))/(2*x^2) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/x + c^2*(3*b*B + A*c)*x + (B*c^3*x^2)/2 + 3
*c*(b^2*B + A*b*c + a*B*c)*Log[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^6} \, dx &=\int \left (c^2 (3 b B+A c)+\frac {a^3 A}{x^6}+\frac {a^2 (3 A b+a B)}{x^5}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^4}+\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^3}+\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^2}+\frac {3 c \left (b^2 B+A b c+a B c\right )}{x}+B c^3 x\right ) \, dx\\ &=-\frac {a^3 A}{5 x^5}-\frac {a^2 (3 A b+a B)}{4 x^4}-\frac {a \left (a b B+A \left (b^2+a c\right )\right )}{x^3}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{2 x^2}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x}+c^2 (3 b B+A c) x+\frac {1}{2} B c^3 x^2+3 c \left (b^2 B+A b c+a B c\right ) \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 161, normalized size = 1.05 \begin {gather*} -\frac {a^3 (4 A+5 B x)+5 a^2 x \left (3 A b+4 A c x+4 b B x+6 B c x^2\right )+10 a x^2 \left (2 A \left (b^2+3 b c x+3 c^2 x^2\right )+3 b B x (b+4 c x)\right )-60 c x^5 \log (x) \left (a B c+A b c+b^2 B\right )+10 x^3 \left (A \left (b^3+6 b^2 c x-2 c^3 x^3\right )-B x \left (-2 b^3+6 b c^2 x^2+c^3 x^3\right )\right )}{20 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]

[Out]

-1/20*(a^3*(4*A + 5*B*x) + 5*a^2*x*(3*A*b + 4*b*B*x + 4*A*c*x + 6*B*c*x^2) + 10*a*x^2*(3*b*B*x*(b + 4*c*x) + 2
*A*(b^2 + 3*b*c*x + 3*c^2*x^2)) + 10*x^3*(A*(b^3 + 6*b^2*c*x - 2*c^3*x^3) - B*x*(-2*b^3 + 6*b*c^2*x^2 + c^3*x^
3)) - 60*c*(b^2*B + A*b*c + a*B*c)*x^5*Log[x])/x^5

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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^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^6,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/x^6, x]

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fricas [A]  time = 0.40, size = 168, normalized size = 1.09 \begin {gather*} \frac {10 \, B c^{3} x^{7} + 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 60 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} \log \relax (x) - 20 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 4 \, A a^{3} - 10 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 20 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \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^6,x, algorithm="fricas")

[Out]

1/20*(10*B*c^3*x^7 + 20*(3*B*b*c^2 + A*c^3)*x^6 + 60*(B*b^2*c + (B*a + A*b)*c^2)*x^5*log(x) - 20*(B*b^3 + 3*A*
a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 4*A*a^3 - 10*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 - 20*(B*a^2*
b + A*a*b^2 + A*a^2*c)*x^2 - 5*(B*a^3 + 3*A*a^2*b)*x)/x^5

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giac [A]  time = 0.16, size = 162, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, B c^{3} x^{2} + 3 \, B b c^{2} x + A c^{3} x + 3 \, {\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac {20 \, {\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 4 \, A a^{3} + 10 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 20 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \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^6,x, algorithm="giac")

[Out]

1/2*B*c^3*x^2 + 3*B*b*c^2*x + A*c^3*x + 3*(B*b^2*c + B*a*c^2 + A*b*c^2)*log(abs(x)) - 1/20*(20*(B*b^3 + 6*B*a*
b*c + 3*A*b^2*c + 3*A*a*c^2)*x^4 + 4*A*a^3 + 10*(3*B*a*b^2 + A*b^3 + 3*B*a^2*c + 6*A*a*b*c)*x^3 + 20*(B*a^2*b
+ A*a*b^2 + A*a^2*c)*x^2 + 5*(B*a^3 + 3*A*a^2*b)*x)/x^5

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maple [A]  time = 0.05, size = 186, normalized size = 1.21 \begin {gather*} \frac {B \,c^{3} x^{2}}{2}+3 A b \,c^{2} \ln \relax (x )+A \,c^{3} x +3 B a \,c^{2} \ln \relax (x )+3 B \,b^{2} c \ln \relax (x )+3 B b \,c^{2} x -\frac {3 A a \,c^{2}}{x}-\frac {3 A \,b^{2} c}{x}-\frac {6 B a b c}{x}-\frac {B \,b^{3}}{x}-\frac {3 A a b c}{x^{2}}-\frac {A \,b^{3}}{2 x^{2}}-\frac {3 B \,a^{2} c}{2 x^{2}}-\frac {3 B a \,b^{2}}{2 x^{2}}-\frac {A \,a^{2} c}{x^{3}}-\frac {A a \,b^{2}}{x^{3}}-\frac {B \,a^{2} b}{x^{3}}-\frac {3 A \,a^{2} b}{4 x^{4}}-\frac {B \,a^{3}}{4 x^{4}}-\frac {A \,a^{3}}{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^6,x)

[Out]

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

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maxima [A]  time = 0.62, size = 163, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, B c^{3} x^{2} + {\left (3 \, B b c^{2} + A c^{3}\right )} x + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} \log \relax (x) - \frac {20 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 4 \, A a^{3} + 10 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 20 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \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^6,x, algorithm="maxima")

[Out]

1/2*B*c^3*x^2 + (3*B*b*c^2 + A*c^3)*x + 3*(B*b^2*c + (B*a + A*b)*c^2)*log(x) - 1/20*(20*(B*b^3 + 3*A*a*c^2 + 3
*(2*B*a*b + A*b^2)*c)*x^4 + 4*A*a^3 + 10*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 20*(B*a^2*b + A*a*b
^2 + A*a^2*c)*x^2 + 5*(B*a^3 + 3*A*a^2*b)*x)/x^5

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mupad [B]  time = 1.18, size = 162, normalized size = 1.05 \begin {gather*} x\,\left (A\,c^3+3\,B\,b\,c^2\right )-\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{2}+\frac {3\,B\,a\,b^2}{2}+3\,A\,c\,a\,b+\frac {A\,b^3}{2}\right )+x^4\,\left (B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2\right )+x\,\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )+\frac {A\,a^3}{5}+x^2\,\left (B\,a^2\,b+A\,c\,a^2+A\,a\,b^2\right )}{x^5}+\ln \relax (x)\,\left (3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2\right )+\frac {B\,c^3\,x^2}{2} \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^6,x)

[Out]

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

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sympy [A]  time = 11.62, size = 182, normalized size = 1.18 \begin {gather*} \frac {B c^{3} x^{2}}{2} + 3 c \left (A b c + B a c + B b^{2}\right ) \log {\relax (x )} + x \left (A c^{3} + 3 B b c^{2}\right ) + \frac {- 4 A a^{3} + x^{4} \left (- 60 A a c^{2} - 60 A b^{2} c - 120 B a b c - 20 B b^{3}\right ) + x^{3} \left (- 60 A a b c - 10 A b^{3} - 30 B a^{2} c - 30 B a b^{2}\right ) + x^{2} \left (- 20 A a^{2} c - 20 A a b^{2} - 20 B a^{2} b\right ) + x \left (- 15 A a^{2} b - 5 B a^{3}\right )}{20 x^{5}} \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**6,x)

[Out]

B*c**3*x**2/2 + 3*c*(A*b*c + B*a*c + B*b**2)*log(x) + x*(A*c**3 + 3*B*b*c**2) + (-4*A*a**3 + x**4*(-60*A*a*c**
2 - 60*A*b**2*c - 120*B*a*b*c - 20*B*b**3) + x**3*(-60*A*a*b*c - 10*A*b**3 - 30*B*a**2*c - 30*B*a*b**2) + x**2
*(-20*A*a**2*c - 20*A*a*b**2 - 20*B*a**2*b) + x*(-15*A*a**2*b - 5*B*a**3))/(20*x**5)

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