3.1370 \(\int \frac{a+b x}{(c+d x)^8} \, dx\)

Optimal. Leaf size=38 \[ \frac{b c-a d}{7 d^2 (c+d x)^7}-\frac{b}{6 d^2 (c+d x)^6} \]

[Out]

(b*c - a*d)/(7*d^2*(c + d*x)^7) - b/(6*d^2*(c + d*x)^6)

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Rubi [A]  time = 0.0518894, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b c-a d}{7 d^2 (c+d x)^7}-\frac{b}{6 d^2 (c+d x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(c + d*x)^8,x]

[Out]

(b*c - a*d)/(7*d^2*(c + d*x)^7) - b/(6*d^2*(c + d*x)^6)

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Rubi in Sympy [A]  time = 9.8356, size = 32, normalized size = 0.84 \[ - \frac{b}{6 d^{2} \left (c + d x\right )^{6}} - \frac{a d - b c}{7 d^{2} \left (c + d x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/(d*x+c)**8,x)

[Out]

-b/(6*d**2*(c + d*x)**6) - (a*d - b*c)/(7*d**2*(c + d*x)**7)

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Mathematica [A]  time = 0.0144264, size = 27, normalized size = 0.71 \[ -\frac{6 a d+b (c+7 d x)}{42 d^2 (c+d x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(c + d*x)^8,x]

[Out]

-(6*a*d + b*(c + 7*d*x))/(42*d^2*(c + d*x)^7)

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Maple [A]  time = 0.009, size = 35, normalized size = 0.9 \[ -{\frac{b}{6\,{d}^{2} \left ( dx+c \right ) ^{6}}}-{\frac{ad-bc}{7\,{d}^{2} \left ( dx+c \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)^8,x)

[Out]

-1/6*b/d^2/(d*x+c)^6-1/7*(a*d-b*c)/d^2/(d*x+c)^7

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Maxima [A]  time = 1.33712, size = 127, normalized size = 3.34 \[ -\frac{7 \, b d x + b c + 6 \, a d}{42 \,{\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(d*x + c)^8,x, algorithm="maxima")

[Out]

-1/42*(7*b*d*x + b*c + 6*a*d)/(d^9*x^7 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d
^6*x^4 + 35*c^4*d^5*x^3 + 21*c^5*d^4*x^2 + 7*c^6*d^3*x + c^7*d^2)

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Fricas [A]  time = 0.199788, size = 127, normalized size = 3.34 \[ -\frac{7 \, b d x + b c + 6 \, a d}{42 \,{\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(d*x + c)^8,x, algorithm="fricas")

[Out]

-1/42*(7*b*d*x + b*c + 6*a*d)/(d^9*x^7 + 7*c*d^8*x^6 + 21*c^2*d^7*x^5 + 35*c^3*d
^6*x^4 + 35*c^4*d^5*x^3 + 21*c^5*d^4*x^2 + 7*c^6*d^3*x + c^7*d^2)

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Sympy [A]  time = 1.64463, size = 100, normalized size = 2.63 \[ - \frac{6 a d + b c + 7 b d x}{42 c^{7} d^{2} + 294 c^{6} d^{3} x + 882 c^{5} d^{4} x^{2} + 1470 c^{4} d^{5} x^{3} + 1470 c^{3} d^{6} x^{4} + 882 c^{2} d^{7} x^{5} + 294 c d^{8} x^{6} + 42 d^{9} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(d*x+c)**8,x)

[Out]

-(6*a*d + b*c + 7*b*d*x)/(42*c**7*d**2 + 294*c**6*d**3*x + 882*c**5*d**4*x**2 +
1470*c**4*d**5*x**3 + 1470*c**3*d**6*x**4 + 882*c**2*d**7*x**5 + 294*c*d**8*x**6
+ 42*d**9*x**7)

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GIAC/XCAS [A]  time = 0.21714, size = 34, normalized size = 0.89 \[ -\frac{7 \, b d x + b c + 6 \, a d}{42 \,{\left (d x + c\right )}^{7} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(d*x + c)^8,x, algorithm="giac")

[Out]

-1/42*(7*b*d*x + b*c + 6*a*d)/((d*x + c)^7*d^2)