3.10.38 \(\int \frac {(\frac {b c}{d}+b x)^5}{(c+d x)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {b^5 (c+d x)^3}{3 d^6} \]

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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 {b^5 (c+d x)^3}{3 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b*c)/d + b*x)^5/(c + d*x)^3,x]

[Out]

(b^5*(c + d*x)^3)/(3*d^6)

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b^5 (c+d x)^3}{3 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b*c)/d + b*x)^5/(c + d*x)^3,x]

[Out]

(b^5*(c + d*x)^3)/(3*d^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {b c}{d}+b x\right )^5}{(c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((b*c)/d + b*x)^5/(c + d*x)^3,x]

[Out]

IntegrateAlgebraic[((b*c)/d + b*x)^5/(c + d*x)^3, x]

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fricas [B]  time = 1.25, size = 35, normalized size = 2.06 \begin {gather*} \frac {b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^5/(d*x+c)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 1.01, size = 35, normalized size = 2.06 \begin {gather*} \frac {b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^5/(d*x+c)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (d x +c \right )^{3} b^{5}}{3 d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*c/d+b*x)^5/(d*x+c)^3,x)

[Out]

1/3*b^5*(d*x+c)^3/d^6

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maxima [B]  time = 1.37, size = 35, normalized size = 2.06 \begin {gather*} \frac {b^{5} d^{2} x^{3} + 3 \, b^{5} c d x^{2} + 3 \, b^{5} c^{2} x}{3 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)^5/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.16, size = 27, normalized size = 1.59 \begin {gather*} \frac {b^5\,x\,\left (3\,c^2+3\,c\,d\,x+d^2\,x^2\right )}{3\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + (b*c)/d)^5/(c + d*x)^3,x)

[Out]

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

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sympy [B]  time = 0.13, size = 34, normalized size = 2.00 \begin {gather*} \frac {b^{5} c^{2} x}{d^{5}} + \frac {b^{5} c x^{2}}{d^{4}} + \frac {b^{5} x^{3}}{3 d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*c/d+b*x)**5/(d*x+c)**3,x)

[Out]

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

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