\(\int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx\) [763]

Optimal result

Integrand size = 18, antiderivative size = 26 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=-\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 37} \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=-\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3} \]

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x^4,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=\frac {\sqrt {c x^2} (-a-2 b x)}{2 x^3} \]

[In]

Integrate[(Sqrt[c*x^2]*(a + b*x))/x^4,x]

[Out]

(Sqrt[c*x^2]*(-a - 2*b*x))/(2*x^3)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73

method	result	size
gosper	\(-\frac {\left (2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}}\)	\(19\)
default	\(-\frac {\left (2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}}\)	\(19\)
risch	\(\frac {\left (-b x -\frac {a}{2}\right ) \sqrt {c \,x^{2}}}{x^{3}}\)	\(20\)
trager	\(\frac {\left (-1+x \right ) \left (a x +2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}}\)	\(25\)

[In]

int((b*x+a)*(c*x^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=-\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, x^{3}} \]

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/x^3

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=- \frac {a \sqrt {c x^{2}}}{2 x^{3}} - \frac {b \sqrt {c x^{2}}}{x^{2}} \]

[In]

integrate((b*x+a)*(c*x**2)**(1/2)/x**4,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=-\frac {{\left (2 \, b x \mathrm {sgn}\left (x\right ) + a \mathrm {sgn}\left (x\right )\right )} \sqrt {c}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx=-\frac {a\,\sqrt {c}\,x^2+2\,b\,\sqrt {c}\,x^3}{2\,x\,{\left (x^2\right )}^{3/2}} \]

[In]

int(((c*x^2)^(1/2)*(a + b*x))/x^4,x)

[Out]

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