\(\int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx\) [809]

Optimal result

Integrand size = 20, antiderivative size = 49 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx=2 a b \sqrt {c x^2}+\frac {1}{2} b^2 x \sqrt {c x^2}+\frac {a^2 \sqrt {c x^2} \log (x)}{x} \]

[Out]

2*a*b*(c*x^2)^(1/2)+1/2*b^2*x*(c*x^2)^(1/2)+a^2*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} \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx=\frac {a^2 \sqrt {c x^2} \log (x)}{x}+2 a b \sqrt {c x^2}+\frac {1}{2} b^2 x \sqrt {c x^2} \]

[In]

Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^2,x]

[Out]

2*a*b*Sqrt[c*x^2] + (b^2*x*Sqrt[c*x^2])/2 + (a^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} \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{x} \\ & = 2 a b \sqrt {c x^2}+\frac {1}{2} b^2 x \sqrt {c x^2}+\frac {a^2 \sqrt {c x^2} \log (x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx=\frac {c x \left (b x (4 a+b x)+2 a^2 \log (x)\right )}{2 \sqrt {c x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (b^{2} x^{2}+2 a^{2} \ln \left (x \right )+4 a b x \right )}{2 x}\)	\(33\)
risch	\(\frac {\sqrt {c \,x^{2}}\, b \left (\frac {1}{2} b \,x^{2}+2 a x \right )}{x}+\frac {a^{2} \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(41\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx=\frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt {c x^{2}}}{2 \, x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx=\frac {a^{2} \sqrt {c x^{2}} \log {\left (x \right )}}{x} + 2 a b \sqrt {c x^{2}} + b^{2} \left (\begin {cases} \frac {x \sqrt {c x^{2}}}{2} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a**2*sqrt(c*x**2)*log(x)/x + 2*a*b*sqrt(c*x**2) + b**2*Piecewise((x*sqrt(c*x**2)/2, Ne(c, 0)), (0, True))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {c x^2} (a+b x)^2}{x^2} \, dx=\frac {1}{2} \, {\left (b^{2} x^{2} \mathrm {sgn}\left (x\right ) + 4 \, a b x \mathrm {sgn}\left (x\right ) + 2 \, a^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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