3.8.63 \(\int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx\) [763]

Optimal. Leaf size=26 \[ -\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 37} \begin {gather*} -\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x^4,x]

[Out]

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

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)}{x^4} \, dx &=\frac {\sqrt {c x^2} \int \frac {a+b x}{x^3} \, dx}{x}\\ &=-\frac {\sqrt {c x^2} (a+b x)^2}{2 a x^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {c x^2} (a+2 b x)}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[c*x^2]*(a + 2*b*x))/x^3

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Maple [A]
time = 0.02, size = 19, normalized size = 0.73

method	result	size
gosper	\(-\frac {\left (2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}}\)	\(19\)
default	\(-\frac {\left (2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}}\)	\(19\)
risch	\(\frac {\left (-b x -\frac {a}{2}\right ) \sqrt {c \,x^{2}}}{x^{3}}\)	\(20\)
trager	\(\frac {\left (-1+x \right ) \left (a x +2 b x +a \right ) \sqrt {c \,x^{2}}}{2 x^{3}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(c*x^2)^(1/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 0.98, size = 18, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/x^3

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Sympy [A]
time = 0.17, size = 29, normalized size = 1.12 \begin {gather*} - \frac {a \sqrt {c x^{2}}}{2 x^{3}} - \frac {b \sqrt {c x^{2}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x**2)**(1/2)/x**4,x)

[Out]

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

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Giac [A]
time = 2.23, size = 19, normalized size = 0.73 \begin {gather*} -\frac {{\left (2 \, b x \mathrm {sgn}\left (x\right ) + a \mathrm {sgn}\left (x\right )\right )} \sqrt {c}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.14, size = 28, normalized size = 1.08 \begin {gather*} -\frac {a\,\sqrt {c}\,x^2+2\,b\,\sqrt {c}\,x^3}{2\,x\,{\left (x^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(1/2)*(a + b*x))/x^4,x)

[Out]

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

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