3.11.14 \(\int \frac {1}{(\frac {b c}{d}+b x)^3 (c+d x)^3} \, dx\) [1014]

Optimal. Leaf size=17 \[ -\frac {d^2}{5 b^3 (c+d x)^5} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \begin {gather*} -\frac {d^2}{5 b^3 (c+d x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(((b*c)/d + b*x)^3*(c + d*x)^3),x]

[Out]

-1/5*d^2/(b^3*(c + d*x)^5)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (\frac {b c}{d}+b x\right )^3 (c+d x)^3} \, dx &=\frac {d^3 \int \frac {1}{(c+d x)^6} \, dx}{b^3}\\ &=-\frac {d^2}{5 b^3 (c+d x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {d^2}{5 b^3 (c+d x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(((b*c)/d + b*x)^3*(c + d*x)^3),x]

[Out]

-1/5*d^2/(b^3*(c + d*x)^5)

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(17)=34\).
time = 2.20, size = 59, normalized size = 3.47 \begin {gather*} -\frac {d^2}{5 b^3 \left (c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/((b*c/d + b*x)^3*(c + d*x)^3),x]')

[Out]

-d ^ 2 / (5 b ^ 3 (c ^ 5 + 5 c ^ 4 d x + 10 c ^ 3 d ^ 2 x ^ 2 + 10 c ^ 2 d ^ 3 x ^ 3 + 5 c d ^ 4 x ^ 4 + d ^ 5
x ^ 5))

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Maple [A]
time = 0.16, size = 16, normalized size = 0.94

method	result	size
gosper	\(-\frac {d^{2}}{5 b^{3} \left (d x +c \right )^{5}}\)	\(16\)
default	\(-\frac {d^{2}}{5 b^{3} \left (d x +c \right )^{5}}\)	\(16\)
norman	\(-\frac {d^{2}}{5 b^{3} \left (d x +c \right )^{5}}\)	\(16\)
risch	\(-\frac {d^{2}}{5 b^{3} \left (d x +c \right )^{5}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*c/d+b*x)^3/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (15) = 30\).
time = 0.25, size = 75, normalized size = 4.41 \begin {gather*} -\frac {d^{2}}{5 \, {\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*c/d+b*x)^3/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (15) = 30\).
time = 0.29, size = 75, normalized size = 4.41 \begin {gather*} -\frac {d^{2}}{5 \, {\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*c/d+b*x)^3/(d*x+c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (15) = 30\).
time = 0.21, size = 83, normalized size = 4.88 \begin {gather*} - \frac {d^{3}}{5 b^{3} c^{5} d + 25 b^{3} c^{4} d^{2} x + 50 b^{3} c^{3} d^{3} x^{2} + 50 b^{3} c^{2} d^{4} x^{3} + 25 b^{3} c d^{5} x^{4} + 5 b^{3} d^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*c/d+b*x)**3/(d*x+c)**3,x)

[Out]

-d**3/(5*b**3*c**5*d + 25*b**3*c**4*d**2*x + 50*b**3*c**3*d**3*x**2 + 50*b**3*c**2*d**4*x**3 + 25*b**3*c*d**5*
x**4 + 5*b**3*d**6*x**5)

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Giac [A]
time = 0.00, size = 18, normalized size = 1.06 \begin {gather*} -\frac {d^{2}}{5 b^{3} \left (x d+c\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*c/d+b*x)^3/(d*x+c)^3,x)

[Out]

-1/5*d^2/((d*x + c)^5*b^3)

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Mupad [B]
time = 0.17, size = 77, normalized size = 4.53 \begin {gather*} -\frac {d^2}{5\,\left (b^3\,c^5+5\,b^3\,c^4\,d\,x+10\,b^3\,c^3\,d^2\,x^2+10\,b^3\,c^2\,d^3\,x^3+5\,b^3\,c\,d^4\,x^4+b^3\,d^5\,x^5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + (b*c)/d)^3*(c + d*x)^3),x)

[Out]

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

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