\(\int \frac {x^2}{(c x^2)^{3/2} (a+b x)} \, dx\) [889]

Optimal result

Integrand size = 20, antiderivative size = 44 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}} \]

[In]

Int[x^2/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

(x*Log[x])/(a*c*Sqrt[c*x^2]) - (x*Log[a + b*x])/(a*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^3 (\log (x)-\log (a (a+b x)))}{a \left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59

method	result	size
default	\(\frac {x^{3} \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} a}\)	\(26\)
risch	\(\frac {x \ln \left (-x \right )}{c \sqrt {c \,x^{2}}\, a}-\frac {x \ln \left (b x +a \right )}{a c \sqrt {c \,x^{2}}}\)	\(43\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.59 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a c^{2} x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a c^{2}}\right ] \]

[In]

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

[Out]

[sqrt(c*x^2)*log(x/(b*x + a))/(a*c^2*x), 2*sqrt(-c)*arctan(sqrt(c*x^2)*(2*b*x + a)*sqrt(-c)/(a*c*x))/(a*c^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=-\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a c^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int(x^2/((c*x^2)^(3/2)*(a + b*x)),x)

[Out]

int(x^2/((c*x^2)^(3/2)*(a + b*x)), x)