\(\int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx\) [811]

Optimal result

Integrand size = 20, antiderivative size = 54 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=-\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=-\frac {a^2 \sqrt {c x^2}}{2 x^3}-\frac {2 a b \sqrt {c x^2}}{x^2}+\frac {b^2 \sqrt {c x^2} \log (x)}{x} \]

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=\sqrt {c x^2} \left (-\frac {a (a+4 b x)}{2 x^3}+\frac {b^2 \log (x)}{x}\right ) \]

[In]

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

[Out]

Sqrt[c*x^2]*(-1/2*(a*(a + 4*b*x))/x^3 + (b^2*Log[x])/x)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.63

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} \ln \left (x \right ) x^{2}-4 a b x -a^{2}\right )}{2 x^{3}}\)	\(34\)
risch	\(\frac {\sqrt {c \,x^{2}}\, \left (-\frac {1}{2} a^{2}-2 a b x \right )}{x^{3}}+\frac {b^{2} \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=\frac {{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, x^{3}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=- \frac {a^{2} \sqrt {c x^{2}}}{2 x^{3}} - \frac {2 a b \sqrt {c x^{2}}}{x^{2}} + \frac {b^{2} \sqrt {c x^{2}} \log {\left (x \right )}}{x} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((b*x+a)^2*(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 = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=\frac {1}{2} \, {\left (2 \, b^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {4 \, a b x \mathrm {sgn}\left (x\right ) + a^{2} \mathrm {sgn}\left (x\right )}{x^{2}}\right )} \sqrt {c} \]

[In]

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

[Out]

1/2*(2*b^2*log(abs(x))*sgn(x) - (4*a*b*x*sgn(x) + a^2*sgn(x))/x^2)*sqrt(c)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^4} \, dx=\int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \]

[In]

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

[Out]

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