3.2916 \(\int \frac {(c+d x)^3}{(a+b (c+d x)^4)^3} \, dx\)

Optimal. Leaf size=23 \[ -\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2} \]

[Out]

-1/8/b/d/(a+b*(d*x+c)^4)^2

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {372, 261} \[ -\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*(c + d*x)^4)^3,x]

[Out]

-1/(8*b*d*(a + b*(c + d*x)^4)^2)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rubi steps

\begin {align*} \int \frac {(c+d x)^3}{\left (a+b (c+d x)^4\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x^4\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ -\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + b*(c + d*x)^4)^3,x]

[Out]

-1/8*1/(b*d*(a + b*(c + d*x)^4)^2)

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fricas [B]  time = 1.01, size = 172, normalized size = 7.48 \[ -\frac {1}{8 \, {\left (b^{3} d^{9} x^{8} + 8 \, b^{3} c d^{8} x^{7} + 28 \, b^{3} c^{2} d^{7} x^{6} + 56 \, b^{3} c^{3} d^{6} x^{5} + 2 \, {\left (35 \, b^{3} c^{4} + a b^{2}\right )} d^{5} x^{4} + 8 \, {\left (7 \, b^{3} c^{5} + a b^{2} c\right )} d^{4} x^{3} + 4 \, {\left (7 \, b^{3} c^{6} + 3 \, a b^{2} c^{2}\right )} d^{3} x^{2} + 8 \, {\left (b^{3} c^{7} + a b^{2} c^{3}\right )} d^{2} x + {\left (b^{3} c^{8} + 2 \, a b^{2} c^{4} + a^{2} b\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/8/(b^3*d^9*x^8 + 8*b^3*c*d^8*x^7 + 28*b^3*c^2*d^7*x^6 + 56*b^3*c^3*d^6*x^5 + 2*(35*b^3*c^4 + a*b^2)*d^5*x^4
+ 8*(7*b^3*c^5 + a*b^2*c)*d^4*x^3 + 4*(7*b^3*c^6 + 3*a*b^2*c^2)*d^3*x^2 + 8*(b^3*c^7 + a*b^2*c^3)*d^2*x + (b^
3*c^8 + 2*a*b^2*c^4 + a^2*b)*d)

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giac [A]  time = 0.25, size = 21, normalized size = 0.91 \[ -\frac {1}{8 \, {\left ({\left (d x + c\right )}^{4} b + a\right )}^{2} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/8/(((d*x + c)^4*b + a)^2*b*d)

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maple [B]  time = 0.00, size = 56, normalized size = 2.43 \[ -\frac {1}{8 \left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )^{2} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+b*(d*x+c)^4)^3,x)

[Out]

-1/8/b/d/(b*d^4*x^4+4*b*c*d^3*x^3+6*b*c^2*d^2*x^2+4*b*c^3*d*x+b*c^4+a)^2

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maxima [A]  time = 0.56, size = 21, normalized size = 0.91 \[ -\frac {1}{8 \, {\left ({\left (d x + c\right )}^{4} b + a\right )}^{2} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8/(((d*x + c)^4*b + a)^2*b*d)

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mupad [B]  time = 2.35, size = 171, normalized size = 7.43 \[ -\frac {1}{8\,b\,d\,\left (x^4\,\left (70\,b^2\,c^4\,d^4+2\,a\,b\,d^4\right )+x^3\,\left (56\,b^2\,c^5\,d^3+8\,a\,b\,c\,d^3\right )+a^2+x\,\left (8\,d\,b^2\,c^7+8\,a\,d\,b\,c^3\right )+b^2\,c^8+x^2\,\left (28\,b^2\,c^6\,d^2+12\,a\,b\,c^2\,d^2\right )+b^2\,d^8\,x^8+2\,a\,b\,c^4+8\,b^2\,c\,d^7\,x^7+56\,b^2\,c^3\,d^5\,x^5+28\,b^2\,c^2\,d^6\,x^6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + b*(c + d*x)^4)^3,x)

[Out]

-1/(8*b*d*(x^4*(70*b^2*c^4*d^4 + 2*a*b*d^4) + x^3*(56*b^2*c^5*d^3 + 8*a*b*c*d^3) + a^2 + x*(8*b^2*c^7*d + 8*a*
b*c^3*d) + b^2*c^8 + x^2*(28*b^2*c^6*d^2 + 12*a*b*c^2*d^2) + b^2*d^8*x^8 + 2*a*b*c^4 + 8*b^2*c*d^7*x^7 + 56*b^
2*c^3*d^5*x^5 + 28*b^2*c^2*d^6*x^6))

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sympy [B]  time = 5.42, size = 197, normalized size = 8.57 \[ - \frac {1}{8 a^{2} b d + 16 a b^{2} c^{4} d + 8 b^{3} c^{8} d + 448 b^{3} c^{3} d^{6} x^{5} + 224 b^{3} c^{2} d^{7} x^{6} + 64 b^{3} c d^{8} x^{7} + 8 b^{3} d^{9} x^{8} + x^{4} \left (16 a b^{2} d^{5} + 560 b^{3} c^{4} d^{5}\right ) + x^{3} \left (64 a b^{2} c d^{4} + 448 b^{3} c^{5} d^{4}\right ) + x^{2} \left (96 a b^{2} c^{2} d^{3} + 224 b^{3} c^{6} d^{3}\right ) + x \left (64 a b^{2} c^{3} d^{2} + 64 b^{3} c^{7} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+b*(d*x+c)**4)**3,x)

[Out]

-1/(8*a**2*b*d + 16*a*b**2*c**4*d + 8*b**3*c**8*d + 448*b**3*c**3*d**6*x**5 + 224*b**3*c**2*d**7*x**6 + 64*b**
3*c*d**8*x**7 + 8*b**3*d**9*x**8 + x**4*(16*a*b**2*d**5 + 560*b**3*c**4*d**5) + x**3*(64*a*b**2*c*d**4 + 448*b
**3*c**5*d**4) + x**2*(96*a*b**2*c**2*d**3 + 224*b**3*c**6*d**3) + x*(64*a*b**2*c**3*d**2 + 64*b**3*c**7*d**2)
)

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