3.10.100 \(\int \frac {(a+b x)^4}{(\frac {a d}{b}+d x)^3} \, dx\) [1000]

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

[Out]

a*b^3*x/d^3+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 {a b^3 x}{d^3}+\frac {b^4 x^2}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/((a*d)/b + d*x)^3,x]

[Out]

(a*b^3*x)/d^3 + (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 {(a+b x)^4}{\left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int (a+b x) \, dx}{d^3}\\ &=\frac {a b^3 x}{d^3}+\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^3 \left (a x+\frac {b x^2}{2}\right )}{d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*(a*x + (b*x^2)/2))/d^3

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Maple [A]
time = 0.14, size = 18, normalized size = 0.78

method	result	size
gosper	\(\frac {b^{3} x \left (b x +2 a \right )}{2 d^{3}}\)	\(17\)
default	\(\frac {b^{3} \left (\frac {1}{2} x^{2} b +a x \right )}{d^{3}}\)	\(18\)
risch	\(\frac {a \,b^{3} x}{d^{3}}+\frac {b^{4} x^{2}}{2 d^{3}}\)	\(22\)
norman	\(\frac {\frac {b^{6} x^{4}}{2 d}+\frac {2 a \,b^{5} x^{3}}{d}-\frac {5 a^{4} b^{2}}{2 d}-\frac {4 a^{3} b^{3} x}{d}}{d^{2} \left (b x +a \right )^{2}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*x^2 + 2*a*b^3*x)/d^3

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Fricas [A]
time = 0.45, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*x^2 + 2*a*b^3*x)/d^3

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Sympy [A]
time = 0.03, size = 20, normalized size = 0.87 \begin {gather*} \frac {a b^{3} x}{d^{3}} + \frac {b^{4} x^{2}}{2 d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b**3*x/d**3 + b**4*x**2/(2*d**3)

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Giac [A]
time = 1.31, size = 20, normalized size = 0.87 \begin {gather*} \frac {b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*x^2 + 2*a*b^3*x)/d^3

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Mupad [B]
time = 0.03, size = 16, normalized size = 0.70 \begin {gather*} \frac {b^3\,x\,\left (2\,a+b\,x\right )}{2\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/(d*x + (a*d)/b)^3,x)

[Out]

(b^3*x*(2*a + b*x))/(2*d^3)

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