\(\int \frac {(c x^2)^{3/2}}{x^6 (a+b x)^2} \, dx\) [908]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=-\frac {c \sqrt {c x^2}}{2 a^2 x^3}+\frac {2 b c \sqrt {c x^2}}{a^3 x^2}+\frac {b^2 c \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {3 b^2 c \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 c \sqrt {c x^2} \log (a+b x)}{a^4 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 117, 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} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=\frac {3 b^2 c \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 c \sqrt {c x^2} \log (a+b x)}{a^4 x}+\frac {b^2 c \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {2 b c \sqrt {c x^2}}{a^3 x^2}-\frac {c \sqrt {c x^2}}{2 a^2 x^3} \]

[In]

Int[(c*x^2)^(3/2)/(x^6*(a + b*x)^2),x]

[Out]

-1/2*(c*Sqrt[c*x^2])/(a^2*x^3) + (2*b*c*Sqrt[c*x^2])/(a^3*x^2) + (b^2*c*Sqrt[c*x^2])/(a^3*x*(a + b*x)) + (3*b^
2*c*Sqrt[c*x^2]*Log[x])/(a^4*x) - (3*b^2*c*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 {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^3 (a+b x)^2} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \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 {c \sqrt {c x^2}}{2 a^2 x^3}+\frac {2 b c \sqrt {c x^2}}{a^3 x^2}+\frac {b^2 c \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {3 b^2 c \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 c \sqrt {c x^2} \log (a+b x)}{a^4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=\frac {\left (c x^2\right )^{3/2} \left (\frac {a \left (-a^2+3 a b x+6 b^2 x^2\right )}{a+b x}+6 b^2 x^2 \log (x)-6 b^2 x^2 \log (a+b x)\right )}{2 a^4 x^5} \]

[In]

Integrate[(c*x^2)^(3/2)/(x^6*(a + b*x)^2),x]

[Out]

((c*x^2)^(3/2)*((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^5)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {c \sqrt {c \,x^{2}}\, \left (\frac {3 b^{2} x^{2}}{a^{3}}+\frac {3 b x}{2 a^{2}}-\frac {1}{2 a}\right )}{x^{3} \left (b x +a \right )}+\frac {3 c \sqrt {c \,x^{2}}\, b^{2} \ln \left (-x \right )}{x \,a^{4}}-\frac {3 b^{2} c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a^{4} x}\)	\(93\)
default	\(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (6 b^{3} \ln \left (x \right ) x^{3}-6 b^{3} \ln \left (b x +a \right ) x^{3}+6 a \,b^{2} \ln \left (x \right ) x^{2}-6 \ln \left (b x +a \right ) x^{2} a \,b^{2}+6 a \,b^{2} x^{2}+3 a^{2} b x -a^{3}\right )}{2 x^{5} a^{4} \left (b x +a \right )}\)	\(95\)

[In]

int((c*x^2)^(3/2)/x^6/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=\frac {{\left (6 \, a b^{2} c x^{2} + 3 \, a^{2} b c x - a^{3} c + 6 \, {\left (b^{3} c x^{3} + a b^{2} c x^{2}\right )} \log \left (\frac {x}{b x + a}\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \]

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{6} \left (a + b x\right )^{2}}\, dx \]

[In]

integrate((c*x**2)**(3/2)/x**6/(b*x+a)**2,x)

[Out]

Integral((c*x**2)**(3/2)/(x**6*(a + b*x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=-\frac {3 \, b^{2} c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, b^{2} c^{\frac {3}{2}} \log \left (x\right )}{a^{4}} + \frac {6 \, b^{2} c^{\frac {3}{2}} x^{2} + 3 \, a b c^{\frac {3}{2}} x - a^{2} c^{\frac {3}{2}}}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \]

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c*x^2)^(3/2)/x^6/(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. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)^2} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^6\,{\left (a+b\,x\right )}^2} \,d x \]

[In]

int((c*x^2)^(3/2)/(x^6*(a + b*x)^2),x)

[Out]

int((c*x^2)^(3/2)/(x^6*(a + b*x)^2), x)