\(\int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx\) [810]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=b^2 \sqrt {c x^2}-\frac {a^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, 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} \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=-\frac {a^2 \sqrt {c x^2}}{x^2}+\frac {2 a b \sqrt {c x^2} \log (x)}{x}+b^2 \sqrt {c x^2} \]

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=\frac {c \left (-a^2+b^2 x^2+2 a b x \log (x)\right )}{\sqrt {c x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (2 a b \ln \left (x \right ) x +b^{2} x^{2}-a^{2}\right )}{x^{2}}\)	\(32\)
risch	\(b^{2} \sqrt {c \,x^{2}}-\frac {a^{2} \sqrt {c \,x^{2}}}{x^{2}}+\frac {2 a b \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(44\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt {c x^{2}}}{x^{2}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=- \frac {a^{2} \sqrt {c x^{2}}}{x^{2}} + \frac {2 a b \sqrt {c x^{2}} \log {\left (x \right )}}{x} + b^{2} \sqrt {c x^{2}} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((b*x+a)^2*(c*x^2)^(1/2)/x^3,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 = 31, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx={\left (b^{2} x \mathrm {sgn}\left (x\right ) + 2 \, a b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {a^{2} \mathrm {sgn}\left (x\right )}{x}\right )} \sqrt {c} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^3} \, dx=\int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2}{x^3} \,d x \]

[In]

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

[Out]

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