3.24.59 \(\int \frac {(c+d x)^3}{(a+b (c+d x)^4)^2} \, dx\)

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

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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} \begin {gather*} -\frac {1}{4 b d \left (a+b (c+d x)^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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

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

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

-1/4*1/(b*d*(a + b*(c + d*x)^4))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^3}{\left (a+b (c+d x)^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

IntegrateAlgebraic[(c + d*x)^3/(a + b*(c + d*x)^4)^2, x]

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fricas [B]  time = 0.84, size = 66, normalized size = 2.87 \begin {gather*} -\frac {1}{4 \, {\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x + {\left (b^{2} c^{4} + a b\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 21, normalized size = 0.91 \begin {gather*} -\frac {1}{4 \, {\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/4/(((d*x + c)^4*b + a)*b*d)

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maple [B]  time = 0.00, size = 56, normalized size = 2.43 \begin {gather*} -\frac {1}{4 \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 ) b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+b*(d*x+c)^4)^2,x)

[Out]

-1/4/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)

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maxima [A]  time = 0.46, size = 21, normalized size = 0.91 \begin {gather*} -\frac {1}{4 \, {\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/4/(((d*x + c)^4*b + a)*b*d)

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mupad [B]  time = 0.19, size = 55, normalized size = 2.39 \begin {gather*} -\frac {1}{4\,b\,d\,\left (b\,c^4+4\,b\,c^3\,d\,x+6\,b\,c^2\,d^2\,x^2+4\,b\,c\,d^3\,x^3+b\,d^4\,x^4+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(a + b*(c + d*x)^4)^2,x)

[Out]

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

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sympy [B]  time = 2.02, size = 73, normalized size = 3.17 \begin {gather*} - \frac {1}{4 a b d + 4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+b*(d*x+c)**4)**2,x)

[Out]

-1/(4*a*b*d + 4*b**2*c**4*d + 16*b**2*c**3*d**2*x + 24*b**2*c**2*d**3*x**2 + 16*b**2*c*d**4*x**3 + 4*b**2*d**5
*x**4)

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