\(\int \frac {(a+b x)^2}{x^{5/2}} \, dx\) [441]

Optimal result

Integrand size = 13, antiderivative size = 32 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=-\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 b^2 \sqrt {x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=-\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 b^2 \sqrt {x} \]

[In]

Int[(a + b*x)^2/x^(5/2),x]

[Out]

(-2*a^2)/(3*x^(3/2)) - (4*a*b)/Sqrt[x] + 2*b^2*Sqrt[x]

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])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2}{x^{5/2}}+\frac {2 a b}{x^{3/2}}+\frac {b^2}{\sqrt {x}}\right ) \, dx \\ & = -\frac {2 a^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 b^2 \sqrt {x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=-\frac {2 \left (a^2+6 a b x-3 b^2 x^2\right )}{3 x^{3/2}} \]

[In]

Integrate[(a + b*x)^2/x^(5/2),x]

[Out]

(-2*(a^2 + 6*a*b*x - 3*b^2*x^2))/(3*x^(3/2))

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72

method	result	size
gosper	\(-\frac {2 \left (-3 b^{2} x^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}}\)	\(23\)
trager	\(-\frac {2 \left (-3 b^{2} x^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}}\)	\(23\)
risch	\(-\frac {2 \left (-3 b^{2} x^{2}+6 a b x +a^{2}\right )}{3 x^{\frac {3}{2}}}\)	\(23\)
derivativedivides	\(-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 b^{2} \sqrt {x}\)	\(25\)
default	\(-\frac {2 a^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 b^{2} \sqrt {x}\)	\(25\)

[In]

int((b*x+a)^2/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(-3*b^2*x^2+6*a*b*x+a^2)/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x+a)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/3*(3*b^2*x^2 - 6*a*b*x - a^2)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=- \frac {2 a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 a b}{\sqrt {x}} + 2 b^{2} \sqrt {x} \]

[In]

integrate((b*x+a)**2/x**(5/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x+a)^2/x^(5/2),x, algorithm="maxima")

[Out]

2*b^2*sqrt(x) - 2/3*(6*a*b*x + a^2)/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=2 \, b^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x+a)^2/x^(5/2),x, algorithm="giac")

[Out]

2*b^2*sqrt(x) - 2/3*(6*a*b*x + a^2)/x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^2}{x^{5/2}} \, dx=-\frac {2\,a^2+12\,a\,b\,x-6\,b^2\,x^2}{3\,x^{3/2}} \]

[In]

int((a + b*x)^2/x^(5/2),x)

[Out]

-(2*a^2 - 6*b^2*x^2 + 12*a*b*x)/(3*x^(3/2))