\(\int \frac {(c x^2)^{3/2} (a+b x)}{x^4} \, dx\) [771]

Optimal result

Integrand size = 18, antiderivative size = 30 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=b c \sqrt {c x^2}+\frac {a c \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, 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} \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=\frac {a c \sqrt {c x^2} \log (x)}{x}+b c \sqrt {c x^2} \]

[In]

Int[((c*x^2)^(3/2)*(a + b*x))/x^4,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=\left (c x^2\right )^{3/2} \left (\frac {b}{x^2}+\frac {a \log (x)}{x^3}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67

method	result	size
default	\(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \ln \left (x \right )\right )}{x^{3}}\)	\(20\)
risch	\(b c \sqrt {c \,x^{2}}+\frac {a c \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(27\)

[In]

int((c*x^2)^(3/2)*(b*x+a)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=\frac {{\left (b c x + a c \log \left (x\right )\right )} \sqrt {c x^{2}}}{x} \]

[In]

integrate((c*x^2)^(3/2)*(b*x+a)/x^4,x, algorithm="fricas")

[Out]

(b*c*x + a*c*log(x))*sqrt(c*x^2)/x

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=\frac {a \left (c x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} + \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c*x^2)^(3/2)*(b*x+a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx={\left (b x \mathrm {sgn}\left (x\right ) + a \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right )\right )} c^{\frac {3}{2}} \]

[In]

integrate((c*x^2)^(3/2)*(b*x+a)/x^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^4} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )}{x^4} \,d x \]

[In]

int(((c*x^2)^(3/2)*(a + b*x))/x^4,x)

[Out]

int(((c*x^2)^(3/2)*(a + b*x))/x^4, x)