3.1000 \(\int \frac{(a+b x)^4}{(\frac{a d}{b}+d x)^3} \, dx\)

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 + (b^4*x^2)/(2*d^3)

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Rubi [A]  time = 0.0057754, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {21} \[ \frac{a b^3 x}{d^3}+\frac{b^4 x^2}{2 d^3} \]

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*}

Mathematica [A]  time = 0.0009898, size = 19, normalized size = 0.83 \[ \frac{b^3 \left (a x+\frac{b x^2}{2}\right )}{d^3} \]

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., size = 18, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}}{{d}^{3}} \left ( ax+{\frac{b{x}^{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01743, size = 27, normalized size = 1.17 \begin{align*} \frac{b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end{align*}

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 = 1.56128, size = 42, normalized size = 1.83 \begin{align*} \frac{b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end{align*}

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 = 0.096131, size = 20, normalized size = 0.87 \begin{align*} \frac{a b^{3} x}{d^{3}} + \frac{b^{4} x^{2}}{2 d^{3}} \end{align*}

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 = 1.05981, size = 27, normalized size = 1.17 \begin{align*} \frac{b^{4} x^{2} + 2 \, a b^{3} x}{2 \, d^{3}} \end{align*}

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