\(\int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx\) [900]

Optimal result

Integrand size = 20, antiderivative size = 112 \[ \int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx=-\frac {\sqrt {c x^2}}{2 a^2 x^3}+\frac {2 b \sqrt {c x^2}}{a^3 x^2}+\frac {b^2 \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {3 b^2 \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 \sqrt {c x^2} \log (a+b x)}{a^4 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 112, 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, 46} \[ \int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx=\frac {3 b^2 \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 \sqrt {c x^2} \log (a+b x)}{a^4 x}+\frac {b^2 \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {2 b \sqrt {c x^2}}{a^3 x^2}-\frac {\sqrt {c x^2}}{2 a^2 x^3} \]

[In]

Int[Sqrt[c*x^2]/(x^4*(a + b*x)^2),x]

[Out]

-1/2*Sqrt[c*x^2]/(a^2*x^3) + (2*b*Sqrt[c*x^2])/(a^3*x^2) + (b^2*Sqrt[c*x^2])/(a^3*x*(a + b*x)) + (3*b^2*Sqrt[c
*x^2]*Log[x])/(a^4*x) - (3*b^2*Sqrt[c*x^2]*Log[a + b*x])/(a^4*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 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int \frac {1}{x^3 (a+b x)^2} \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx}{x} \\ & = -\frac {\sqrt {c x^2}}{2 a^2 x^3}+\frac {2 b \sqrt {c x^2}}{a^3 x^2}+\frac {b^2 \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {3 b^2 \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 \sqrt {c x^2} \log (a+b x)}{a^4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx=\frac {\sqrt {c x^2} \left (\frac {a \left (-a^2+3 a b x+6 b^2 x^2\right )}{a+b x}+6 b^2 x^2 \log (x)-6 b^2 x^2 \log (a+b x)\right )}{2 a^4 x^3} \]

[In]

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

[Out]

(Sqrt[c*x^2]*((a*(-a^2 + 3*a*b*x + 6*b^2*x^2))/(a + b*x) + 6*b^2*x^2*Log[x] - 6*b^2*x^2*Log[a + b*x]))/(2*a^4*
x^3)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {\sqrt {c \,x^{2}}\, \left (\frac {3 b^{2} x^{2}}{a^{3}}+\frac {3 b x}{2 a^{2}}-\frac {1}{2 a}\right )}{x^{3} \left (b x +a \right )}+\frac {3 \sqrt {c \,x^{2}}\, b^{2} \ln \left (-x \right )}{x \,a^{4}}-\frac {3 b^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a^{4} x}\)	\(90\)
default	\(\frac {\sqrt {c \,x^{2}}\, \left (6 b^{3} \ln \left (x \right ) x^{3}-6 b^{3} \ln \left (b x +a \right ) x^{3}+6 a \,b^{2} \ln \left (x \right ) x^{2}-6 \ln \left (b x +a \right ) x^{2} a \,b^{2}+6 a \,b^{2} x^{2}+3 a^{2} b x -a^{3}\right )}{2 x^{3} a^{4} \left (b x +a \right )}\)	\(95\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx=\frac {{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac {x}{b x + a}\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \]

[In]

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

[Out]

1/2*(6*a*b^2*x^2 + 3*a^2*b*x - a^3 + 6*(b^3*x^3 + a*b^2*x^2)*log(x/(b*x + a)))*sqrt(c*x^2)/(a^4*b*x^4 + a^5*x^
3)

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(c*x**2)/(x**4*(a + b*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx=\frac {6 \, b^{2} \sqrt {c} x^{2} + 3 \, a b \sqrt {c} x - a^{2} \sqrt {c}}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac {3 \, b^{2} \sqrt {c} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, b^{2} \sqrt {c} \log \left (x\right )}{a^{4}} \]

[In]

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

[Out]

1/2*(6*b^2*sqrt(c)*x^2 + 3*a*b*sqrt(c)*x - a^2*sqrt(c))/(a^3*b*x^3 + a^4*x^2) - 3*b^2*sqrt(c)*log(b*x + a)/a^4
+ 3*b^2*sqrt(c)*log(x)/a^4

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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