\(\int \frac {(a+b x)^5}{(\frac {a d}{b}+d x)^3} \, dx\) [999]

Optimal result

Integrand size = 20, antiderivative size = 17 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^2 (a+b x)^3}{3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^2 (a+b x)^3}{3 d^3} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^2 (a+b x)^3}{3 d^3} \]

[In]

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

[Out]

(b^2*(a + b*x)^3)/(3*d^3)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {b^{2} \left (b x +a \right )^{3}}{3 d^{3}}\)	\(16\)
gosper	\(\frac {b^{3} x \left (b^{2} x^{2}+3 a b x +3 a^{2}\right )}{3 d^{3}}\)	\(28\)
parallelrisch	\(\frac {b^{5} x^{3}+3 a \,b^{4} x^{2}+3 a^{2} b^{3} x}{3 d^{3}}\)	\(32\)
risch	\(\frac {b^{5} x^{3}}{3 d^{3}}+\frac {b^{4} a \,x^{2}}{d^{3}}+\frac {b^{3} a^{2} x}{d^{3}}+\frac {b^{2} a^{3}}{3 d^{3}}\)	\(46\)
norman	\(\frac {\frac {b^{7} x^{5}}{3 d}+\frac {5 a \,b^{6} x^{4}}{3 d}+\frac {10 a^{2} b^{5} x^{3}}{3 d}-\frac {3 a^{5} b^{2}}{d}-\frac {5 a^{4} b^{3} x}{d}}{d^{2} \left (b x +a \right )^{2}}\)	\(73\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).

Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {a^{2} b^{3} x}{d^{3}} + \frac {a b^{4} x^{2}}{d^{3}} + \frac {b^{5} x^{3}}{3 d^{3}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x}{3 \, d^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^5}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3\,x\,\left (3\,a^2+3\,a\,b\,x+b^2\,x^2\right )}{3\,d^3} \]

[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)