3.1269 \(\int \frac{(c+d x)^3}{(a+b x)^6} \, dx\)

Optimal. Leaf size=58 \[ \frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)} \]

[Out]

-(c + d*x)^4/(5*(b*c - a*d)*(a + b*x)^5) + (d*(c + d*x)^4)/(20*(b*c - a*d)^2*(a
+ b*x)^4)

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Rubi [A]  time = 0.0392504, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3/(a + b*x)^6,x]

[Out]

-(c + d*x)^4/(5*(b*c - a*d)*(a + b*x)^5) + (d*(c + d*x)^4)/(20*(b*c - a*d)^2*(a
+ b*x)^4)

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Rubi in Sympy [A]  time = 7.71727, size = 46, normalized size = 0.79 \[ \frac{d \left (c + d x\right )^{4}}{20 \left (a + b x\right )^{4} \left (a d - b c\right )^{2}} + \frac{\left (c + d x\right )^{4}}{5 \left (a + b x\right )^{5} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3/(b*x+a)**6,x)

[Out]

d*(c + d*x)**4/(20*(a + b*x)**4*(a*d - b*c)**2) + (c + d*x)**4/(5*(a + b*x)**5*(
a*d - b*c))

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Mathematica [A]  time = 0.0640932, size = 97, normalized size = 1.67 \[ -\frac{a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )}{20 b^4 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^3/(a + b*x)^6,x]

[Out]

-(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a*b^2*d*(3*c^2 + 10*c*d*x + 10*d^2*x^2) +
b^3*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3))/(20*b^4*(a + b*x)^5)

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Maple [B]  time = 0.007, size = 121, normalized size = 2.1 \[ -{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}bc{d}^{2}-3\,a{b}^{2}{c}^{2}d+{b}^{3}{c}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}+{\frac{{d}^{2} \left ( ad-bc \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/(b*x+a)^6,x)

[Out]

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

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Maxima [A]  time = 1.36538, size = 216, normalized size = 3.72 \[ -\frac{10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \,{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/(b*x + a)^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224366, size = 216, normalized size = 3.72 \[ -\frac{10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \,{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/(b*x + a)^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.94557, size = 170, normalized size = 2.93 \[ - \frac{a^{3} d^{3} + 2 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 4 b^{3} c^{3} + 10 b^{3} d^{3} x^{3} + x^{2} \left (10 a b^{2} d^{3} + 20 b^{3} c d^{2}\right ) + x \left (5 a^{2} b d^{3} + 10 a b^{2} c d^{2} + 15 b^{3} c^{2} d\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3/(b*x+a)**6,x)

[Out]

-(a**3*d**3 + 2*a**2*b*c*d**2 + 3*a*b**2*c**2*d + 4*b**3*c**3 + 10*b**3*d**3*x**
3 + x**2*(10*a*b**2*d**3 + 20*b**3*c*d**2) + x*(5*a**2*b*d**3 + 10*a*b**2*c*d**2
+ 15*b**3*c**2*d))/(20*a**5*b**4 + 100*a**4*b**5*x + 200*a**3*b**6*x**2 + 200*a
**2*b**7*x**3 + 100*a*b**8*x**4 + 20*b**9*x**5)

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GIAC/XCAS [A]  time = 0.220012, size = 154, normalized size = 2.66 \[ -\frac{10 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c d^{2} x^{2} + 10 \, a b^{2} d^{3} x^{2} + 15 \, b^{3} c^{2} d x + 10 \, a b^{2} c d^{2} x + 5 \, a^{2} b d^{3} x + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}}{20 \,{\left (b x + a\right )}^{5} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/(b*x + a)^6,x, algorithm="giac")

[Out]

-1/20*(10*b^3*d^3*x^3 + 20*b^3*c*d^2*x^2 + 10*a*b^2*d^3*x^2 + 15*b^3*c^2*d*x + 1
0*a*b^2*c*d^2*x + 5*a^2*b*d^3*x + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^
3*d^3)/((b*x + a)^5*b^4)