\(\int \frac {x^2 (a+b x)}{(c x^2)^{3/2}} \, dx\) [789]

Optimal result

Integrand size = 18, antiderivative size = 35 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {b x^2}{c \sqrt {c x^2}}+\frac {a x \log (x)}{c \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {a x \log (x)}{c \sqrt {c x^2}}+\frac {b x^2}{c \sqrt {c x^2}} \]

[In]

Int[(x^2*(a + b*x))/(c*x^2)^(3/2),x]

[Out]

(b*x^2)/(c*Sqrt[c*x^2]) + (a*x*Log[x])/(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 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {b x^4+a x^3 \log (x)}{\left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

(b*x^4 + a*x^3*Log[x])/(c*x^2)^(3/2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57

method	result	size
default	\(\frac {x^{3} \left (b x +a \ln \left (x \right )\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(20\)
risch	\(\frac {b \,x^{2}}{c \sqrt {c \,x^{2}}}+\frac {a x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c x^{2}} {\left (b x + a \log \left (x\right )\right )}}{c^{2} x} \]

[In]

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c^2*x)

Sympy [A] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {a x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {b x^{2}}{\sqrt {c x^{2}} c} + \frac {a \log \left (x\right )}{c^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {\frac {b x}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {a \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \]

[In]

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

[Out]

(b*x/(sqrt(c)*sgn(x)) + a*log(abs(x))/(sqrt(c)*sgn(x)))/c

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx=\frac {b\,\left |x\right |}{c^{3/2}}+\frac {a\,\ln \left (x+\left |x\right |\right )}{c^{3/2}}-\frac {a\,x}{c^{3/2}\,\sqrt {x^2}} \]

[In]

int((x^2*(a + b*x))/(c*x^2)^(3/2),x)

[Out]

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