\(\int \frac {x^2}{(c x^2)^{3/2} (a+b x)^2} \, dx\) [920]

Optimal result

Integrand size = 20, antiderivative size = 68 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {x}{a c \sqrt {c x^2} (a+b x)}+\frac {x \log (x)}{a^2 c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a^2 c \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, 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} \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {x \log (a+b x)}{a^2 c \sqrt {c x^2}}+\frac {x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {x}{a c \sqrt {c x^2} (a+b x)} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {\frac {x^3}{a (a+b x)}+\frac {x^3 \log (x)}{a^2}-\frac {x^3 \log (a+b x)}{a^2}}{\left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {x^{3} \left (b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a \ln \left (x \right )-a \ln \left (b x +a \right )+a \right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} a^{2} \left (b x +a \right )}\)	\(52\)
risch	\(\frac {x}{a c \left (b x +a \right ) \sqrt {c \,x^{2}}}+\frac {x \ln \left (-x \right )}{c \sqrt {c \,x^{2}}\, a^{2}}-\frac {x \ln \left (b x +a \right )}{a^{2} c \sqrt {c \,x^{2}}}\)	\(65\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (\frac {x}{b x + a}\right ) + a\right )}}{a^{2} b c^{2} x^{2} + a^{3} c^{2} x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\int \frac {x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {1}{\sqrt {c x^{2}} b^{2} c x + \sqrt {c x^{2}} a b c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{2} c^{\frac {3}{2}}} + \frac {1}{\sqrt {c x^{2}} a b c} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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