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

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

[Out]

(b^2*(a + b*x)^3)/(3*d^3)

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Rubi [A]  time = 0.0037963, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ \frac{b^2 (a+b x)^3}{3 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/((a*d)/b + d*x)^3,x]

[Out]

(b^2*(a + b*x)^3)/(3*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])

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

Mathematica [A]  time = 0.0019294, size = 17, normalized size = 1. \[ \frac{b^2 (a+b x)^3}{3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*(a + b*x)^3)/(3*d^3)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \begin{align*}{\frac{{b}^{2} \left ( bx+a \right ) ^{3}}{3\,{d}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.028, size = 42, normalized size = 2.47 \begin{align*} \frac{b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x)/d^3

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Fricas [B]  time = 1.49167, size = 63, normalized size = 3.71 \begin{align*} \frac{b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x)/d^3

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Sympy [B]  time = 0.101171, size = 34, normalized size = 2. \begin{align*} \frac{a^{2} b^{3} x}{d^{3}} + \frac{a b^{4} x^{2}}{d^{3}} + \frac{b^{5} x^{3}}{3 d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.06376, size = 42, normalized size = 2.47 \begin{align*} \frac{b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x)/d^3