\(\int \frac {x^3}{(c x^2)^{3/2} (a+b x)^2} \, dx\) [919]

Optimal result

Integrand size = 20, antiderivative size = 25 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {x}{b c \sqrt {c x^2} (a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {x}{b c \sqrt {c x^2} (a+b x)} \]

[In]

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

[Out]

-(x/(b*c*Sqrt[c*x^2]*(a + b*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{(a+b x)^2} \, dx}{c \sqrt {c x^2}} \\ & = -\frac {x}{b c \sqrt {c x^2} (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {x^3}{b \left (c x^2\right )^{3/2} (a+b x)} \]

[In]

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

[Out]

-(x^3/(b*(c*x^2)^(3/2)*(a + b*x)))

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

method	result	size
gosper	\(-\frac {x^{3}}{\left (b x +a \right ) b \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(23\)
default	\(-\frac {x^{3}}{\left (b x +a \right ) b \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(23\)
risch	\(-\frac {x}{b c \left (b x +a \right ) \sqrt {c \,x^{2}}}\)	\(24\)
trager	\(\frac {\left (-1+x \right ) \sqrt {c \,x^{2}}}{c^{2} \left (b x +a \right ) \left (a +b \right ) x}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {\sqrt {c x^{2}}}{b^{2} c^{2} x^{2} + a b c^{2} x} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).

Time = 0.52 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\begin {cases} - \frac {x^{3}}{a b \left (c x^{2}\right )^{\frac {3}{2}} + b^{2} x \left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{a^{2} \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-x**3/(a*b*(c*x**2)**(3/2) + b**2*x*(c*x**2)**(3/2)), Ne(b, 0)), (x**4/(a**2*(c*x**2)**(3/2)), True
))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).

Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {a}{\sqrt {c x^{2}} b^{3} c x + \sqrt {c x^{2}} a b^{2} c} - \frac {1}{\sqrt {c x^{2}} b^{2} c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {\frac {\mathrm {sgn}\left (x\right )}{a b \sqrt {c}} - \frac {1}{{\left (b x + a\right )} b \sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \]

[In]

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

[Out]

(sgn(x)/(a*b*sqrt(c)) - 1/((b*x + a)*b*sqrt(c)*sgn(x)))/c

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {\sqrt {c\,x^2}}{b\,c^2\,x\,\left (a+b\,x\right )} \]

[In]

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

[Out]

-(c*x^2)^(1/2)/(b*c^2*x*(a + b*x))