3.9.90 \(\int \frac {x}{(c x^2)^{3/2} (a+b x)} \, dx\) [890]

Optimal. Leaf size=63 \[ -\frac {1}{a c \sqrt {c x^2}}-\frac {b x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 c \sqrt {c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 46} \begin {gather*} -\frac {b x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 c \sqrt {c x^2}}-\frac {1}{a c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 35, normalized size = 0.56 \begin {gather*} \frac {x^2 (-a-b x \log (x)+b x \log (a+b x))}{a^2 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 33, normalized size = 0.52

method	result	size
default	\(-\frac {x^{2} \left (b x \ln \left (x \right )-b \ln \left (b x +a \right ) x +a \right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} a^{2}}\)	\(33\)
risch	\(-\frac {1}{a c \sqrt {c \,x^{2}}}+\frac {x b \ln \left (-b x -a \right )}{c \sqrt {c \,x^{2}}\, a^{2}}-\frac {b x \ln \left (x \right )}{a^{2} c \sqrt {c \,x^{2}}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 51, normalized size = 0.81 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} 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 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.43, size = 34, normalized size = 0.54 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(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:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((c*x^2)^(3/2)*(a + b*x)),x)

[Out]

int(x/((c*x^2)^(3/2)*(a + b*x)), x)

________________________________________________________________________________________