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]

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

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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} \[ \frac {b^2 (a+b x)^3}{3 d^3} \]

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {b^2 (a+b x)^3}{3 d^3} \]

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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fricas [B]  time = 0.40, size = 31, normalized size = 1.82 \[ \frac {b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

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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giac [B]  time = 1.09, size = 31, normalized size = 1.82 \[ \frac {b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

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

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \frac {\left (b x +a \right )^{3} b^{2}}{3 d^{3}} \]

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 = 1.37, size = 31, normalized size = 1.82 \[ \frac {b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

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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mupad [B]  time = 0.05, size = 27, normalized size = 1.59 \[ \frac {b^3\,x\,\left (3\,a^2+3\,a\,b\,x+b^2\,x^2\right )}{3\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(d*x + (a*d)/b)^3,x)

[Out]

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

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sympy [B]  time = 0.12, size = 34, normalized size = 2.00 \[ \frac {a^{2} b^{3} x}{d^{3}} + \frac {a b^{4} x^{2}}{d^{3}} + \frac {b^{5} x^{3}}{3 d^{3}} \]

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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