\(\int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx\) [949]

Optimal result

Integrand size = 18, antiderivative size = 28 \[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{1+n}}{b (1+n) \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, 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, 32} \[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{n+1}}{b (n+1) \sqrt {c x^2}} \]

[In]

Int[(x*(a + b*x)^n)/Sqrt[c*x^2],x]

[Out]

(x*(a + b*x)^(1 + n))/(b*(1 + n)*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x \int (a+b x)^n \, dx}{\sqrt {c x^2}} \\ & = \frac {x (a+b x)^{1+n}}{b (1+n) \sqrt {c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{1+n}}{b (1+n) \sqrt {c x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96

method	result	size
gosper	\(\frac {x \left (b x +a \right )^{1+n}}{b \left (1+n \right ) \sqrt {c \,x^{2}}}\)	\(27\)
risch	\(\frac {\left (b x +a \right ) x \left (b x +a \right )^{n}}{b \left (1+n \right ) \sqrt {c \,x^{2}}}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {\sqrt {c x^{2}} {\left (b x + a\right )} {\left (b x + a\right )}^{n}}{{\left (b c n + b c\right )} x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\begin {cases} \frac {x^{2}}{a \sqrt {c x^{2}}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {a^{n} x^{2}}{\sqrt {c x^{2}}} & \text {for}\: b = 0 \\\int \frac {x}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {a x \left (a + b x\right )^{n}}{b n \sqrt {c x^{2}} + b \sqrt {c x^{2}}} + \frac {b x^{2} \left (a + b x\right )^{n}}{b n \sqrt {c x^{2}} + b \sqrt {c x^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**2/(a*sqrt(c*x**2)), Eq(b, 0) & Eq(n, -1)), (a**n*x**2/sqrt(c*x**2), Eq(b, 0)), (Integral(x/(sqrt
(c*x**2)*(a + b*x)), x), Eq(n, -1)), (a*x*(a + b*x)**n/(b*n*sqrt(c*x**2) + b*sqrt(c*x**2)) + b*x**2*(a + b*x)*
*n/(b*n*sqrt(c*x**2) + b*sqrt(c*x**2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (b \sqrt {c} x + a \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{b c {\left (n + 1\right )}} \]

[In]

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

[Out]

(b*sqrt(c)*x + a*sqrt(c))*(b*x + a)^n/(b*c*(n + 1))

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,1,0,0]%%%} / %%%{1,[0,0,1,1]%%%} Error: Bad Argumen
t Value

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {x (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {\left (\frac {x^2}{n+1}+\frac {a\,x}{b\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{\sqrt {c\,x^2}} \]

[In]

int((x*(a + b*x)^n)/(c*x^2)^(1/2),x)

[Out]

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