\(\int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx\) [948]

Optimal result

Integrand size = 20, antiderivative size = 59 \[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=-\frac {a x (a+b x)^{1+n}}{b^2 (1+n) \sqrt {c x^2}}+\frac {x (a+b x)^{2+n}}{b^2 (2+n) \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 59, 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} \[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{n+2}}{b^2 (n+2) \sqrt {c x^2}}-\frac {a x (a+b x)^{n+1}}{b^2 (n+1) \sqrt {c x^2}} \]

[In]

Int[(x^2*(a + b*x)^n)/Sqrt[c*x^2],x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{1+n} (-a+b (1+n) x)}{b^2 (1+n) (2+n) \sqrt {c x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75

method	result	size
gosper	\(-\frac {x \left (b x +a \right )^{1+n} \left (-b n x -b x +a \right )}{b^{2} \sqrt {c \,x^{2}}\, \left (n^{2}+3 n +2\right )}\)	\(44\)
risch	\(-\frac {x \left (-b^{2} n \,x^{2}-a b n x -b^{2} x^{2}+a^{2}\right ) \left (b x +a \right )^{n}}{\sqrt {c \,x^{2}}\, b^{2} \left (2+n \right ) \left (1+n \right )}\)	\(58\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (a b n x + {\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} c n^{2} + 3 \, b^{2} c n + 2 \, b^{2} c\right )} x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=\begin {cases} \frac {a^{n} x^{3}}{2 \sqrt {c x^{2}}} & \text {for}\: b = 0 \\\int \frac {x^{2}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{2}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} x \left (a + b x\right )^{n}}{b^{2} n^{2} \sqrt {c x^{2}} + 3 b^{2} n \sqrt {c x^{2}} + 2 b^{2} \sqrt {c x^{2}}} + \frac {a b n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} \sqrt {c x^{2}} + 3 b^{2} n \sqrt {c x^{2}} + 2 b^{2} \sqrt {c x^{2}}} + \frac {b^{2} n x^{3} \left (a + b x\right )^{n}}{b^{2} n^{2} \sqrt {c x^{2}} + 3 b^{2} n \sqrt {c x^{2}} + 2 b^{2} \sqrt {c x^{2}}} + \frac {b^{2} x^{3} \left (a + b x\right )^{n}}{b^{2} n^{2} \sqrt {c x^{2}} + 3 b^{2} n \sqrt {c x^{2}} + 2 b^{2} \sqrt {c x^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**n*x**3/(2*sqrt(c*x**2)), Eq(b, 0)), (Integral(x**2/(sqrt(c*x**2)*(a + b*x)**2), x), Eq(n, -2)),
(Integral(x**2/(sqrt(c*x**2)*(a + b*x)), x), Eq(n, -1)), (-a**2*x*(a + b*x)**n/(b**2*n**2*sqrt(c*x**2) + 3*b**
2*n*sqrt(c*x**2) + 2*b**2*sqrt(c*x**2)) + a*b*n*x**2*(a + b*x)**n/(b**2*n**2*sqrt(c*x**2) + 3*b**2*n*sqrt(c*x*
*2) + 2*b**2*sqrt(c*x**2)) + b**2*n*x**3*(a + b*x)**n/(b**2*n**2*sqrt(c*x**2) + 3*b**2*n*sqrt(c*x**2) + 2*b**2
*sqrt(c*x**2)) + b**2*x**3*(a + b*x)**n/(b**2*n**2*sqrt(c*x**2) + 3*b**2*n*sqrt(c*x**2) + 2*b**2*sqrt(c*x**2))
, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} \sqrt {c}} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate(x^2*(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,[0,1,1,0,0]%%%} / %%%{1,[0,0,0,1,1]%%%} Error: Bad Arg
ument Value

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.20 \[ \int \frac {x^2 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^3\,\left (n+1\right )}{n^2+3\,n+2}-\frac {a^2\,x}{b^2\,\left (n^2+3\,n+2\right )}+\frac {a\,n\,x^2}{b\,\left (n^2+3\,n+2\right )}\right )}{\sqrt {c\,x^2}} \]

[In]

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

[Out]

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