3.1368 \(\int \frac{(a+b x)^3}{(c+d x)^8} \, dx\)

Optimal. Leaf size=92 \[ \frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b^3}{4 d^4 (c+d x)^4} \]

[Out]

(b*c - a*d)^3/(7*d^4*(c + d*x)^7) - (b*(b*c - a*d)^2)/(2*d^4*(c + d*x)^6) + (3*b
^2*(b*c - a*d))/(5*d^4*(c + d*x)^5) - b^3/(4*d^4*(c + d*x)^4)

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Rubi [A]  time = 0.139578, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{3 b^2 (b c-a d)}{5 d^4 (c+d x)^5}-\frac{b (b c-a d)^2}{2 d^4 (c+d x)^6}+\frac{(b c-a d)^3}{7 d^4 (c+d x)^7}-\frac{b^3}{4 d^4 (c+d x)^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3/(c + d*x)^8,x]

[Out]

(b*c - a*d)^3/(7*d^4*(c + d*x)^7) - (b*(b*c - a*d)^2)/(2*d^4*(c + d*x)^6) + (3*b
^2*(b*c - a*d))/(5*d^4*(c + d*x)^5) - b^3/(4*d^4*(c + d*x)^4)

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Rubi in Sympy [A]  time = 27.5598, size = 82, normalized size = 0.89 \[ - \frac{b^{3}}{4 d^{4} \left (c + d x\right )^{4}} - \frac{3 b^{2} \left (a d - b c\right )}{5 d^{4} \left (c + d x\right )^{5}} - \frac{b \left (a d - b c\right )^{2}}{2 d^{4} \left (c + d x\right )^{6}} - \frac{\left (a d - b c\right )^{3}}{7 d^{4} \left (c + d x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3/(d*x+c)**8,x)

[Out]

-b**3/(4*d**4*(c + d*x)**4) - 3*b**2*(a*d - b*c)/(5*d**4*(c + d*x)**5) - b*(a*d
- b*c)**2/(2*d**4*(c + d*x)**6) - (a*d - b*c)**3/(7*d**4*(c + d*x)**7)

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Mathematica [A]  time = 0.0519185, size = 94, normalized size = 1.02 \[ -\frac{20 a^3 d^3+10 a^2 b d^2 (c+7 d x)+4 a b^2 d \left (c^2+7 c d x+21 d^2 x^2\right )+b^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )}{140 d^4 (c+d x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3/(c + d*x)^8,x]

[Out]

-(20*a^3*d^3 + 10*a^2*b*d^2*(c + 7*d*x) + 4*a*b^2*d*(c^2 + 7*c*d*x + 21*d^2*x^2)
+ b^3*(c^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3))/(140*d^4*(c + d*x)^7)

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Maple [A]  time = 0.009, size = 122, normalized size = 1.3 \[ -{\frac{{b}^{3}}{4\,{d}^{4} \left ( dx+c \right ) ^{4}}}-{\frac{{a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3}}{7\,{d}^{4} \left ( dx+c \right ) ^{7}}}-{\frac{b \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{2\,{d}^{4} \left ( dx+c \right ) ^{6}}}-{\frac{3\,{b}^{2} \left ( ad-bc \right ) }{5\,{d}^{4} \left ( dx+c \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3/(d*x+c)^8,x)

[Out]

-1/4*b^3/d^4/(d*x+c)^4-1/7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/d^4/(d*
x+c)^7-1/2*b*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d^4/(d*x+c)^6-3/5*b^2*(a*d-b*c)/d^4/(d*
x+c)^5

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Maxima [A]  time = 1.35947, size = 246, normalized size = 2.67 \[ -\frac{35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \,{\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \,{\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(d*x + c)^8,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*d^3*x^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3 +
21*(b^3*c*d^2 + 4*a*b^2*d^3)*x^2 + 7*(b^3*c^2*d + 4*a*b^2*c*d^2 + 10*a^2*b*d^3)
*x)/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3*d^8*x^4 + 35*c^4*d^7*x^3
+ 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4)

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Fricas [A]  time = 0.20106, size = 246, normalized size = 2.67 \[ -\frac{35 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3} + 21 \,{\left (b^{3} c d^{2} + 4 \, a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + 10 \, a^{2} b d^{3}\right )} x}{140 \,{\left (d^{11} x^{7} + 7 \, c d^{10} x^{6} + 21 \, c^{2} d^{9} x^{5} + 35 \, c^{3} d^{8} x^{4} + 35 \, c^{4} d^{7} x^{3} + 21 \, c^{5} d^{6} x^{2} + 7 \, c^{6} d^{5} x + c^{7} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(d*x + c)^8,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*d^3*x^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20*a^3*d^3 +
21*(b^3*c*d^2 + 4*a*b^2*d^3)*x^2 + 7*(b^3*c^2*d + 4*a*b^2*c*d^2 + 10*a^2*b*d^3)
*x)/(d^11*x^7 + 7*c*d^10*x^6 + 21*c^2*d^9*x^5 + 35*c^3*d^8*x^4 + 35*c^4*d^7*x^3
+ 21*c^5*d^6*x^2 + 7*c^6*d^5*x + c^7*d^4)

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Sympy [A]  time = 6.08197, size = 194, normalized size = 2.11 \[ - \frac{20 a^{3} d^{3} + 10 a^{2} b c d^{2} + 4 a b^{2} c^{2} d + b^{3} c^{3} + 35 b^{3} d^{3} x^{3} + x^{2} \left (84 a b^{2} d^{3} + 21 b^{3} c d^{2}\right ) + x \left (70 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 7 b^{3} c^{2} d\right )}{140 c^{7} d^{4} + 980 c^{6} d^{5} x + 2940 c^{5} d^{6} x^{2} + 4900 c^{4} d^{7} x^{3} + 4900 c^{3} d^{8} x^{4} + 2940 c^{2} d^{9} x^{5} + 980 c d^{10} x^{6} + 140 d^{11} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3/(d*x+c)**8,x)

[Out]

-(20*a**3*d**3 + 10*a**2*b*c*d**2 + 4*a*b**2*c**2*d + b**3*c**3 + 35*b**3*d**3*x
**3 + x**2*(84*a*b**2*d**3 + 21*b**3*c*d**2) + x*(70*a**2*b*d**3 + 28*a*b**2*c*d
**2 + 7*b**3*c**2*d))/(140*c**7*d**4 + 980*c**6*d**5*x + 2940*c**5*d**6*x**2 + 4
900*c**4*d**7*x**3 + 4900*c**3*d**8*x**4 + 2940*c**2*d**9*x**5 + 980*c*d**10*x**
6 + 140*d**11*x**7)

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GIAC/XCAS [A]  time = 0.218683, size = 154, normalized size = 1.67 \[ -\frac{35 \, b^{3} d^{3} x^{3} + 21 \, b^{3} c d^{2} x^{2} + 84 \, a b^{2} d^{3} x^{2} + 7 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 70 \, a^{2} b d^{3} x + b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 10 \, a^{2} b c d^{2} + 20 \, a^{3} d^{3}}{140 \,{\left (d x + c\right )}^{7} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3/(d*x + c)^8,x, algorithm="giac")

[Out]

-1/140*(35*b^3*d^3*x^3 + 21*b^3*c*d^2*x^2 + 84*a*b^2*d^3*x^2 + 7*b^3*c^2*d*x + 2
8*a*b^2*c*d^2*x + 70*a^2*b*d^3*x + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2 + 20
*a^3*d^3)/((d*x + c)^7*d^4)