\(\int \frac {x^2 (a+b x)^2}{(c x^2)^{3/2}} \, dx\) [837]

Optimal result

Integrand size = 20, antiderivative size = 61 \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {2 a b x^2}{c \sqrt {c x^2}}+\frac {b^2 x^3}{2 c \sqrt {c x^2}}+\frac {a^2 x \log (x)}{c \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 61, 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, 45} \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {a^2 x \log (x)}{c \sqrt {c x^2}}+\frac {2 a b x^2}{c \sqrt {c x^2}}+\frac {b^2 x^3}{2 c \sqrt {c x^2}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.57 \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {\frac {1}{2} b x^4 (4 a+b x)+a^2 x^3 \log (x)}{\left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {x^{3} \left (b^{2} x^{2}+2 a^{2} \ln \left (x \right )+4 a b x \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(33\)
risch	\(\frac {x b \left (\frac {1}{2} b \,x^{2}+2 a x \right )}{c \sqrt {c \,x^{2}}}+\frac {a^{2} x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\)	\(43\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.57 \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt {c x^{2}}}{2 \, c^{2} x} \]

[In]

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

[Out]

1/2*(b^2*x^2 + 4*a*b*x + 2*a^2*log(x))*sqrt(c*x^2)/(c^2*x)

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {a^{2} x^{3} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {2 a b x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{5}}{2 \left (c x^{2}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

a**2*x**3*log(x)/(c*x**2)**(3/2) + 2*a*b*x**4/(c*x**2)**(3/2) + b**2*x**5/(2*(c*x**2)**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {b^{2} x^{3}}{2 \, \sqrt {c x^{2}} c} + \frac {2 \, a b x^{2}}{\sqrt {c x^{2}} c} + \frac {a^{2} \log \left (x\right )}{c^{\frac {3}{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.79 \[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {\frac {2 \, a^{2} \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{2} \mathrm {sgn}\left (x\right ) + 4 \, a b c^{\frac {3}{2}} x \mathrm {sgn}\left (x\right )}{c^{2}}}{2 \, c} \]

[In]

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

[Out]

1/2*(2*a^2*log(abs(x))/(sqrt(c)*sgn(x)) + (b^2*c^(3/2)*x^2*sgn(x) + 4*a*b*c^(3/2)*x*sgn(x))/c^2)/c

Mupad [F(-1)]

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

[In]

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

[Out]

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