\(\int \frac {x}{\sqrt {(a+b x)^3}} \, dx\) [137]

Optimal result

Integrand size = 13, antiderivative size = 21 \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2 (2 a+b x)}{b^2 \sqrt {a+b x}} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1973, 45} \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2 (a+b x)^2}{b^2 \sqrt {(a+b x)^3}}+\frac {2 a (a+b x)}{b^2 \sqrt {(a+b x)^3}} \]

[In]

Int[x/Sqrt[(a + b*x)^3],x]

[Out]

(2*a*(a + b*x))/(b^2*Sqrt[(a + b*x)^3]) + (2*(a + b*x)^2)/(b^2*Sqrt[(a + b*x)^3])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1973

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/
a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] &&  !GeQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \frac {x}{\left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{\sqrt {(a+b x)^3}} \\ & = \frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \left (-\frac {a}{b \left (1+\frac {b x}{a}\right )^{3/2}}+\frac {a}{b \sqrt {1+\frac {b x}{a}}}\right ) \, dx}{\sqrt {(a+b x)^3}} \\ & = \frac {2 a (a+b x)}{b^2 \sqrt {(a+b x)^3}}+\frac {2 (a+b x)^2}{b^2 \sqrt {(a+b x)^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2 (a+b x) (2 a+b x)}{b^2 \sqrt {(a+b x)^3}} \]

[In]

Integrate[x/Sqrt[(a + b*x)^3],x]

[Out]

(2*(a + b*x)*(2*a + b*x))/(b^2*Sqrt[(a + b*x)^3])

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29

method	result	size
gosper	\(\frac {2 \left (b x +a \right ) \left (b x +2 a \right )}{b^{2} \sqrt {\left (b x +a \right )^{3}}}\)	\(27\)
default	\(\frac {2 \left (b x +a \right ) \left (b x +2 a \right )}{b^{2} \sqrt {\left (b x +a \right )^{3}}}\)	\(27\)
risch	\(\frac {2 \left (b x +a \right )^{2}}{b^{2} \sqrt {\left (b x +a \right )^{3}}}+\frac {2 a \left (b x +a \right )}{b^{2} \sqrt {\left (b x +a \right )^{3}}}\)	\(43\)
trager	\(\frac {2 \left (b x +2 a \right ) \sqrt {b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}}{\left (b x +a \right )^{2} b^{2}}\)	\(49\)

[In]

int(x/((b*x+a)^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95 \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (b x + 2 \, a\right )}}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}} \]

[In]

integrate(x/((b*x+a)^3)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*(b*x + 2*a)/(b^4*x^2 + 2*a*b^3*x + a^2*b^2)

Sympy [F]

\[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\int \frac {x}{\sqrt {\left (a + b x\right )^{3}}}\, dx \]

[In]

integrate(x/((b*x+a)**3)**(1/2),x)

[Out]

Integral(x/sqrt((a + b*x)**3), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2 \, {\left (b^{2} x^{2} + 3 \, a b x + 2 \, a^{2}\right )}}{{\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \]

[In]

integrate(x/((b*x+a)^3)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2 \, {\left (\frac {\sqrt {b x + a}}{b} + \frac {a}{\sqrt {b x + a} b}\right )}}{b} \]

[In]

integrate(x/((b*x+a)^3)^(1/2),x, algorithm="giac")

[Out]

2*(sqrt(b*x + a)/b + a/(sqrt(b*x + a)*b))/b

Mupad [B] (verification not implemented)

Time = 18.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {(a+b x)^3}} \, dx=\frac {2\,\left (2\,a+b\,x\right )\,\sqrt {{\left (a+b\,x\right )}^3}}{b^2\,{\left (a+b\,x\right )}^2} \]

[In]

int(x/((a + b*x)^3)^(1/2),x)

[Out]

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