3.84 \(\int \frac{(a+b x)^3 (A+B x)}{x^7} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^3 A}{6 x^6}-\frac{a^2 (a B+3 A b)}{5 x^5}-\frac{b^2 (3 a B+A b)}{3 x^3}-\frac{3 a b (a B+A b)}{4 x^4}-\frac{b^3 B}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.104703, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^3 A}{6 x^6}-\frac{a^2 (a B+3 A b)}{5 x^5}-\frac{b^2 (3 a B+A b)}{3 x^3}-\frac{3 a b (a B+A b)}{4 x^4}-\frac{b^3 B}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/x^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 21.875, size = 71, normalized size = 0.95 \[ - \frac{A a^{3}}{6 x^{6}} - \frac{B b^{3}}{2 x^{2}} - \frac{a^{2} \left (3 A b + B a\right )}{5 x^{5}} - \frac{3 a b \left (A b + B a\right )}{4 x^{4}} - \frac{b^{2} \left (A b + 3 B a\right )}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0315647, size = 69, normalized size = 0.92 \[ -\frac{2 a^3 (5 A+6 B x)+9 a^2 b x (4 A+5 B x)+15 a b^2 x^2 (3 A+4 B x)+10 b^3 x^3 (2 A+3 B x)}{60 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/x^7,x]

[Out]

-(10*b^3*x^3*(2*A + 3*B*x) + 15*a*b^2*x^2*(3*A + 4*B*x) + 9*a^2*b*x*(4*A + 5*B*x
) + 2*a^3*(5*A + 6*B*x))/(60*x^6)

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Maple [A]  time = 0.009, size = 66, normalized size = 0.9 \[ -{\frac{A{a}^{3}}{6\,{x}^{6}}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{3\,ab \left ( Ab+Ba \right ) }{4\,{x}^{4}}}-{\frac{{b}^{2} \left ( Ab+3\,Ba \right ) }{3\,{x}^{3}}}-{\frac{B{b}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.34828, size = 99, normalized size = 1.32 \[ -\frac{30 \, B b^{3} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^7,x, algorithm="maxima")

[Out]

-1/60*(30*B*b^3*x^4 + 10*A*a^3 + 20*(3*B*a*b^2 + A*b^3)*x^3 + 45*(B*a^2*b + A*a*
b^2)*x^2 + 12*(B*a^3 + 3*A*a^2*b)*x)/x^6

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Fricas [A]  time = 0.200175, size = 99, normalized size = 1.32 \[ -\frac{30 \, B b^{3} x^{4} + 10 \, A a^{3} + 20 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 12 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^7,x, algorithm="fricas")

[Out]

-1/60*(30*B*b^3*x^4 + 10*A*a^3 + 20*(3*B*a*b^2 + A*b^3)*x^3 + 45*(B*a^2*b + A*a*
b^2)*x^2 + 12*(B*a^3 + 3*A*a^2*b)*x)/x^6

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Sympy [A]  time = 7.48343, size = 78, normalized size = 1.04 \[ - \frac{10 A a^{3} + 30 B b^{3} x^{4} + x^{3} \left (20 A b^{3} + 60 B a b^{2}\right ) + x^{2} \left (45 A a b^{2} + 45 B a^{2} b\right ) + x \left (36 A a^{2} b + 12 B a^{3}\right )}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/x**7,x)

[Out]

-(10*A*a**3 + 30*B*b**3*x**4 + x**3*(20*A*b**3 + 60*B*a*b**2) + x**2*(45*A*a*b**
2 + 45*B*a**2*b) + x*(36*A*a**2*b + 12*B*a**3))/(60*x**6)

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GIAC/XCAS [A]  time = 0.295586, size = 101, normalized size = 1.35 \[ -\frac{30 \, B b^{3} x^{4} + 60 \, B a b^{2} x^{3} + 20 \, A b^{3} x^{3} + 45 \, B a^{2} b x^{2} + 45 \, A a b^{2} x^{2} + 12 \, B a^{3} x + 36 \, A a^{2} b x + 10 \, A a^{3}}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^7,x, algorithm="giac")

[Out]

-1/60*(30*B*b^3*x^4 + 60*B*a*b^2*x^3 + 20*A*b^3*x^3 + 45*B*a^2*b*x^2 + 45*A*a*b^
2*x^2 + 12*B*a^3*x + 36*A*a^2*b*x + 10*A*a^3)/x^6