\(\int \frac {a+b x}{(\frac {a d}{b}+d x)^3} \, dx\) [1003]

Optimal result

Integrand size = 18, antiderivative size = 15 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^2}{d^3 (a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^2}{d^3 (a+b x)} \]

[In]

Int[(a + b*x)/((a*d)/b + d*x)^3,x]

[Out]

-(b^2/(d^3*(a + b*x)))

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 \frac {1}{(a+b x)^2} \, dx}{d^3} \\ & = -\frac {b^2}{d^3 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^2}{d^3 (a+b x)} \]

[In]

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

[Out]

-(b^2/(d^3*(a + b*x)))

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
gosper	\(-\frac {b^{2}}{d^{3} \left (b x +a \right )}\)	\(16\)
default	\(-\frac {b^{2}}{d^{3} \left (b x +a \right )}\)	\(16\)
risch	\(-\frac {b^{2}}{d^{3} \left (b x +a \right )}\)	\(16\)
parallelrisch	\(\frac {b^{3} x}{a \,d^{3} \left (b x +a \right )}\)	\(19\)
norman	\(\frac {-\frac {a \,b^{2}}{d}-\frac {b^{3} x}{d}}{d^{2} \left (b x +a \right )^{2}}\)	\(31\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^{2}}{b d^{3} x + a d^{3}} \]

[In]

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

[Out]

-b^2/(b*d^3*x + a*d^3)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=- \frac {b^{3}}{a b d^{3} + b^{2} d^{3} x} \]

[In]

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

[Out]

-b**3/(a*b*d**3 + b**2*d**3*x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^{2}}{b d^{3} x + a d^{3}} \]

[In]

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

[Out]

-b^2/(b*d^3*x + a*d^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^{2}}{{\left (b x + a\right )} d^{3}} \]

[In]

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

[Out]

-b^2/((b*x + a)*d^3)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx=-\frac {b^2}{d^3\,\left (a+b\,x\right )} \]

[In]

int((a + b*x)/(d*x + (a*d)/b)^3,x)

[Out]

-b^2/(d^3*(a + b*x))