\(\int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx\) [834]

Optimal result

Integrand size = 20, antiderivative size = 26 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {(a+b x)^3}{3 a x^2 \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, 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 = {15, 37} \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {(a+b x)^3}{3 a x^2 \sqrt {c x^2}} \]

[In]

Int[(a + b*x)^2/(x^3*Sqrt[c*x^2]),x]

[Out]

-1/3*(a + b*x)^3/(a*x^2*Sqrt[c*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {c \left (a^2+3 a b x+3 b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15

method	result	size
gosper	\(-\frac {3 b^{2} x^{2}+3 a b x +a^{2}}{3 x^{2} \sqrt {c \,x^{2}}}\)	\(30\)
default	\(-\frac {3 b^{2} x^{2}+3 a b x +a^{2}}{3 x^{2} \sqrt {c \,x^{2}}}\)	\(30\)
risch	\(\frac {-b^{2} x^{2}-a b x -\frac {1}{3} a^{2}}{x^{2} \sqrt {c \,x^{2}}}\)	\(31\)
trager	\(\frac {\left (-1+x \right ) \left (a^{2} x^{2}+3 a b \,x^{2}+3 b^{2} x^{2}+a^{2} x +3 a b x +a^{2}\right ) \sqrt {c \,x^{2}}}{3 c \,x^{4}}\)	\(55\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt {c x^{2}}}{3 \, c x^{4}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=- \frac {a^{2}}{3 x^{2} \sqrt {c x^{2}}} - \frac {a b}{x \sqrt {c x^{2}}} - \frac {b^{2}}{\sqrt {c x^{2}}} \]

[In]

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

[Out]

-a**2/(3*x**2*sqrt(c*x**2)) - a*b/(x*sqrt(c*x**2)) - b**2/sqrt(c*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {b^{2}}{\sqrt {c} x} - \frac {a b}{\sqrt {c} x^{2}} - \frac {a^{2}}{3 \, \sqrt {c} x^{3}} \]

[In]

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

[Out]

-b^2/(sqrt(c)*x) - a*b/(sqrt(c)*x^2) - 1/3*a^2/(sqrt(c)*x^3)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, \sqrt {c} x^{3} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/(sqrt(c)*x^3*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx=-\frac {a^2\,x^2+3\,a\,b\,x^3+3\,b^2\,x^4}{3\,\sqrt {c}\,{\left (x^2\right )}^{5/2}} \]

[In]

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

[Out]

-(a^2*x^2 + 3*b^2*x^4 + 3*a*b*x^3)/(3*c^(1/2)*(x^2)^(5/2))