3.2.18 \(\int \frac {(a+b x)^3 (A+B x)}{x^{10}} \, dx\) [118]

Optimal. Leaf size=75 \[ -\frac {a^3 A}{9 x^9}-\frac {a^2 (3 A b+a B)}{8 x^8}-\frac {3 a b (A b+a B)}{7 x^7}-\frac {b^2 (A b+3 a B)}{6 x^6}-\frac {b^3 B}{5 x^5} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \begin {gather*} -\frac {a^3 A}{9 x^9}-\frac {a^2 (a B+3 A b)}{8 x^8}-\frac {b^2 (3 a B+A b)}{6 x^6}-\frac {3 a b (a B+A b)}{7 x^7}-\frac {b^3 B}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/x^10,x]

[Out]

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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 69, normalized size = 0.92 \begin {gather*} -\frac {84 b^3 x^3 (5 A+6 B x)+180 a b^2 x^2 (6 A+7 B x)+135 a^2 b x (7 A+8 B x)+35 a^3 (8 A+9 B x)}{2520 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2520*(84*b^3*x^3*(5*A + 6*B*x) + 180*a*b^2*x^2*(6*A + 7*B*x) + 135*a^2*b*x*(7*A + 8*B*x) + 35*a^3*(8*A + 9*
B*x))/x^9

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Maple [A]
time = 0.06, size = 66, normalized size = 0.88

method	result	size
default	\(-\frac {a^{3} A}{9 x^{9}}-\frac {a^{2} \left (3 A b +B a \right )}{8 x^{8}}-\frac {3 a b \left (A b +B a \right )}{7 x^{7}}-\frac {b^{2} \left (A b +3 B a \right )}{6 x^{6}}-\frac {b^{3} B}{5 x^{5}}\)	\(66\)
norman	\(\frac {-\frac {b^{3} B \,x^{4}}{5}+\left (-\frac {1}{6} b^{3} A -\frac {1}{2} a \,b^{2} B \right ) x^{3}+\left (-\frac {3}{7} a \,b^{2} A -\frac {3}{7} a^{2} b B \right ) x^{2}+\left (-\frac {3}{8} a^{2} b A -\frac {1}{8} a^{3} B \right ) x -\frac {a^{3} A}{9}}{x^{9}}\)	\(74\)
risch	\(\frac {-\frac {b^{3} B \,x^{4}}{5}+\left (-\frac {1}{6} b^{3} A -\frac {1}{2} a \,b^{2} B \right ) x^{3}+\left (-\frac {3}{7} a \,b^{2} A -\frac {3}{7} a^{2} b B \right ) x^{2}+\left (-\frac {3}{8} a^{2} b A -\frac {1}{8} a^{3} B \right ) x -\frac {a^{3} A}{9}}{x^{9}}\)	\(74\)
gosper	\(-\frac {504 b^{3} B \,x^{4}+420 A \,b^{3} x^{3}+1260 B a \,b^{2} x^{3}+1080 a A \,b^{2} x^{2}+1080 B \,a^{2} b \,x^{2}+945 a^{2} A b x +315 a^{3} B x +280 a^{3} A}{2520 x^{9}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/x^10,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.33, size = 73, normalized size = 0.97 \begin {gather*} -\frac {504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2}\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)^3*(B*x+A)/x^10,x, algorithm="maxima")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b + A*a*b^2)*x^2 + 315*(B*a^3 +
3*A*a^2*b)*x)/x^9

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Fricas [A]
time = 1.17, size = 73, normalized size = 0.97 \begin {gather*} -\frac {504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2}\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)^3*(B*x+A)/x^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b + A*a*b^2)*x^2 + 315*(B*a^3 +
3*A*a^2*b)*x)/x^9

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Sympy [A]
time = 2.21, size = 82, normalized size = 1.09 \begin {gather*} \frac {- 280 A a^{3} - 504 B b^{3} x^{4} + x^{3} \left (- 420 A b^{3} - 1260 B a b^{2}\right ) + x^{2} \left (- 1080 A a b^{2} - 1080 B a^{2} b\right ) + x \left (- 945 A a^{2} b - 315 B a^{3}\right )}{2520 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/x**10,x)

[Out]

(-280*A*a**3 - 504*B*b**3*x**4 + x**3*(-420*A*b**3 - 1260*B*a*b**2) + x**2*(-1080*A*a*b**2 - 1080*B*a**2*b) +
x*(-945*A*a**2*b - 315*B*a**3))/(2520*x**9)

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Giac [A]
time = 0.74, size = 75, normalized size = 1.00 \begin {gather*} -\frac {504 \, B b^{3} x^{4} + 1260 \, B a b^{2} x^{3} + 420 \, A b^{3} x^{3} + 1080 \, B a^{2} b x^{2} + 1080 \, A a b^{2} 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)^3*(B*x+A)/x^10,x, algorithm="giac")

[Out]

-1/2520*(504*B*b^3*x^4 + 1260*B*a*b^2*x^3 + 420*A*b^3*x^3 + 1080*B*a^2*b*x^2 + 1080*A*a*b^2*x^2 + 315*B*a^3*x
+ 945*A*a^2*b*x + 280*A*a^3)/x^9

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Mupad [B]
time = 0.05, size = 74, normalized size = 0.99 \begin {gather*} -\frac {x^2\,\left (\frac {3\,B\,a^2\,b}{7}+\frac {3\,A\,a\,b^2}{7}\right )+x\,\left (\frac {B\,a^3}{8}+\frac {3\,A\,b\,a^2}{8}\right )+\frac {A\,a^3}{9}+x^3\,\left (\frac {A\,b^3}{6}+\frac {B\,a\,b^2}{2}\right )+\frac {B\,b^3\,x^4}{5}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/x^10,x)

[Out]

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

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