3.11.8 \(\int \frac {(\frac {b c}{d}+b x)^4}{(c+d x)^3} \, dx\) [1008]

Optimal. Leaf size=23 \[ \frac {b^4 c x}{d^4}+\frac {b^4 x^2}{2 d^3} \]

[Out]

b^4*c*x/d^4+1/2*b^4*x^2/d^3

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {21} \begin {gather*} \frac {b^4 c x}{d^4}+\frac {b^4 x^2}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b*c)/d + b*x)^4/(c + d*x)^3,x]

[Out]

(b^4*c*x)/d^4 + (b^4*x^2)/(2*d^3)

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])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 19, normalized size = 0.83 \begin {gather*} \frac {b^4 \left (c x+\frac {d x^2}{2}\right )}{d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^4*(c*x + (d*x^2)/2))/d^4

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Mathics [A]
time = 1.60, size = 16, normalized size = 0.70 \begin {gather*} \frac {b^4 x \left (2 c+d x\right )}{2 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b ^ 4 x (2 c + d x) / (2 d ^ 4)

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Maple [A]
time = 0.13, size = 18, normalized size = 0.78

method	result	size
gosper	\(\frac {b^{4} x \left (d x +2 c \right )}{2 d^{4}}\)	\(17\)
default	\(\frac {b^{4} \left (c x +\frac {1}{2} d \,x^{2}\right )}{d^{4}}\)	\(18\)
risch	\(\frac {b^{4} c x}{d^{4}}+\frac {b^{4} x^{2}}{2 d^{3}}\)	\(22\)
norman	\(\frac {\frac {b^{4} d^{2} x^{4}}{2}-\frac {5 c^{4} b^{4}}{2 d^{2}}+2 b^{4} c d \,x^{3}-\frac {4 c^{3} b^{4} x}{d}}{d^{3} \left (d x +c \right )^{2}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*c/d+b*x)^4/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 21, normalized size = 0.91 \begin {gather*} \frac {b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*d*x^2 + 2*b^4*c*x)/d^4

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Fricas [A]
time = 0.29, size = 21, normalized size = 0.91 \begin {gather*} \frac {b^{4} d x^{2} + 2 \, b^{4} c x}{2 \, d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*d*x^2 + 2*b^4*c*x)/d^4

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Sympy [A]
time = 0.06, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} c x}{d^{4}} + \frac {b^{4} x^{2}}{2 d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**4*c*x/d**4 + b**4*x**2/(2*d**3)

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Giac [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\frac {1}{2} x^{2} b^{4} d+x b^{4} c}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*d*x^2 + 2*b^4*c*x)/d^4

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Mupad [B]
time = 0.03, size = 16, normalized size = 0.70 \begin {gather*} \frac {b^4\,x\,\left (2\,c+d\,x\right )}{2\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + (b*c)/d)^4/(c + d*x)^3,x)

[Out]

(b^4*x*(2*c + d*x))/(2*d^4)

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