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

Optimal. Leaf size=33 \[ \frac {b x \log (x)}{c \sqrt {c x^2}}-\frac {a}{c \sqrt {c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {15, 43} \begin {gather*} \frac {b x \log (x)}{c \sqrt {c x^2}}-\frac {a}{c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x))/(c*x^2)^(3/2),x]

[Out]

-(a/(c*Sqrt[c*x^2])) + (b*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 43

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.67 \begin {gather*} \frac {x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(-a + b*x*Log[x]))/(c*x^2)^(3/2)

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IntegrateAlgebraic [A]  time = 0.02, size = 24, normalized size = 0.73 \begin {gather*} \frac {b x^3 \log (x)-a x^2}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.07, size = 23, normalized size = 0.70 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \relax (x) - a\right )}}{c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.22, size = 47, normalized size = 1.42 \begin {gather*} -\frac {b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {2 \, a \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}}{c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 21, normalized size = 0.64 \begin {gather*} \frac {\left (b x \ln \relax (x )-a \right ) x^{2}}{\left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 21, normalized size = 0.64 \begin {gather*} \frac {b \log \relax (x)}{c^{\frac {3}{2}}} - \frac {a}{\sqrt {c x^{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*log(x)/c^(3/2) - a/(sqrt(c*x^2)*c)

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mupad [B]  time = 0.25, size = 28, normalized size = 0.85 \begin {gather*} -\frac {a+b\,x-b\,\ln \left (x+\relax |x|\right )\,\sqrt {x^2}}{c^{3/2}\,\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*x))/(c*x^2)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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