\(\int \frac {(c x^2)^{3/2}}{x^4 (a+b x)^2} \, dx\) [906]

Optimal result

Integrand size = 20, antiderivative size = 68 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\frac {c \sqrt {c x^2}}{a x (a+b x)}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (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} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=-\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {c \sqrt {c x^2}}{a x (a+b x)} \]

[In]

Int[(c*x^2)^(3/2)/(x^4*(a + b*x)^2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\left (c x^2\right )^{3/2} \left (\frac {1}{a x^3 (a+b x)}+\frac {\log (x)}{a^2 x^3}-\frac {\log (a+b x)}{a^2 x^3}\right ) \]

[In]

Integrate[(c*x^2)^(3/2)/(x^4*(a + b*x)^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76

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

[In]

int((c*x^2)^(3/2)/x^4/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} {\left (a c + {\left (b c x + a c\right )} \log \left (\frac {x}{b x + a}\right )\right )}}{a^{2} b x^{2} + a^{3} x} \]

[In]

integrate((c*x^2)^(3/2)/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{4} \left (a + b x\right )^{2}}\, dx \]

[In]

integrate((c*x**2)**(3/2)/x**4/(b*x+a)**2,x)

[Out]

Integral((c*x**2)**(3/2)/(x**4*(a + b*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.56 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\frac {c^{\frac {3}{2}}}{a b x + a^{2}} - \frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{2}} + \frac {c^{\frac {3}{2}} \log \left (x\right )}{a^{2}} \]

[In]

integrate((c*x^2)^(3/2)/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c*x^2)^(3/2)/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. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^4\,{\left (a+b\,x\right )}^2} \,d x \]

[In]

int((c*x^2)^(3/2)/(x^4*(a + b*x)^2),x)

[Out]

int((c*x^2)^(3/2)/(x^4*(a + b*x)^2), x)