3.534 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^7} \, dx\)

Optimal. Leaf size=44 \[ \frac{(a+b x)^5 (A b-6 a B)}{30 a^2 x^5}-\frac{A (a+b x)^5}{6 a x^6} \]

[Out]

-(A*(a + b*x)^5)/(6*a*x^6) + ((A*b - 6*a*B)*(a + b*x)^5)/(30*a^2*x^5)

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Rubi [A]  time = 0.056326, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^5 (A b-6 a B)}{30 a^2 x^5}-\frac{A (a+b x)^5}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^7,x]

[Out]

-(A*(a + b*x)^5)/(6*a*x^6) + ((A*b - 6*a*B)*(a + b*x)^5)/(30*a^2*x^5)

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Rubi in Sympy [A]  time = 15.2461, size = 37, normalized size = 0.84 \[ - \frac{A \left (a + b x\right )^{5}}{6 a x^{6}} + \frac{\left (a + b x\right )^{5} \left (A b - 6 B a\right )}{30 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**7,x)

[Out]

-A*(a + b*x)**5/(6*a*x**6) + (a + b*x)**5*(A*b - 6*B*a)/(30*a**2*x**5)

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Mathematica [A]  time = 0.0501948, size = 85, normalized size = 1.93 \[ -\frac{a^4 (5 A+6 B x)+6 a^3 b x (4 A+5 B x)+15 a^2 b^2 x^2 (3 A+4 B x)+20 a b^3 x^3 (2 A+3 B x)+15 b^4 x^4 (A+2 B x)}{30 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^7,x]

[Out]

-(15*b^4*x^4*(A + 2*B*x) + 20*a*b^3*x^3*(2*A + 3*B*x) + 15*a^2*b^2*x^2*(3*A + 4*
B*x) + 6*a^3*b*x*(4*A + 5*B*x) + a^4*(5*A + 6*B*x))/(30*x^6)

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Maple [B]  time = 0.008, size = 88, normalized size = 2. \[ -{\frac{A{a}^{4}}{6\,{x}^{6}}}-{\frac{{a}^{2}b \left ( 3\,Ab+2\,Ba \right ) }{2\,{x}^{4}}}-{\frac{2\,a{b}^{2} \left ( 2\,Ab+3\,Ba \right ) }{3\,{x}^{3}}}-{\frac{{b}^{3} \left ( Ab+4\,Ba \right ) }{2\,{x}^{2}}}-{\frac{{a}^{3} \left ( 4\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{{b}^{4}B}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 0.676984, size = 134, normalized size = 3.05 \[ -\frac{30 \, B b^{4} x^{5} + 5 \, A a^{4} + 15 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 20 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 15 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^7,x, algorithm="maxima")

[Out]

-1/30*(30*B*b^4*x^5 + 5*A*a^4 + 15*(4*B*a*b^3 + A*b^4)*x^4 + 20*(3*B*a^2*b^2 + 2
*A*a*b^3)*x^3 + 15*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 6*(B*a^4 + 4*A*a^3*b)*x)/x^6

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Fricas [A]  time = 0.279205, size = 134, normalized size = 3.05 \[ -\frac{30 \, B b^{4} x^{5} + 5 \, A a^{4} + 15 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 20 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 15 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^7,x, algorithm="fricas")

[Out]

-1/30*(30*B*b^4*x^5 + 5*A*a^4 + 15*(4*B*a*b^3 + A*b^4)*x^4 + 20*(3*B*a^2*b^2 + 2
*A*a*b^3)*x^3 + 15*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 + 6*(B*a^4 + 4*A*a^3*b)*x)/x^6

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Sympy [A]  time = 9.37471, size = 102, normalized size = 2.32 \[ - \frac{5 A a^{4} + 30 B b^{4} x^{5} + x^{4} \left (15 A b^{4} + 60 B a b^{3}\right ) + x^{3} \left (40 A a b^{3} + 60 B a^{2} b^{2}\right ) + x^{2} \left (45 A a^{2} b^{2} + 30 B a^{3} b\right ) + x \left (24 A a^{3} b + 6 B a^{4}\right )}{30 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**7,x)

[Out]

-(5*A*a**4 + 30*B*b**4*x**5 + x**4*(15*A*b**4 + 60*B*a*b**3) + x**3*(40*A*a*b**3
+ 60*B*a**2*b**2) + x**2*(45*A*a**2*b**2 + 30*B*a**3*b) + x*(24*A*a**3*b + 6*B*
a**4))/(30*x**6)

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GIAC/XCAS [A]  time = 0.267176, size = 134, normalized size = 3.05 \[ -\frac{30 \, B b^{4} x^{5} + 60 \, B a b^{3} x^{4} + 15 \, A b^{4} x^{4} + 60 \, B a^{2} b^{2} x^{3} + 40 \, A a b^{3} x^{3} + 30 \, B a^{3} b x^{2} + 45 \, A a^{2} b^{2} x^{2} + 6 \, B a^{4} x + 24 \, A a^{3} b x + 5 \, A a^{4}}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/x^7,x, algorithm="giac")

[Out]

-1/30*(30*B*b^4*x^5 + 60*B*a*b^3*x^4 + 15*A*b^4*x^4 + 60*B*a^2*b^2*x^3 + 40*A*a*
b^3*x^3 + 30*B*a^3*b*x^2 + 45*A*a^2*b^2*x^2 + 6*B*a^4*x + 24*A*a^3*b*x + 5*A*a^4
)/x^6