3.1.78 \(\int \frac {\sqrt {b x+c x^2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=54 \[ -\frac {c \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}-\frac {\sqrt {b x+c x^2}}{x^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {662, 660, 207} \begin {gather*} -\frac {\sqrt {b x+c x^2}}{x^{3/2}}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x + c*x^2]/x^(5/2),x]

[Out]

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

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {\sqrt {b x+c x^2}}{x^{5/2}} \, dx &=-\frac {\sqrt {b x+c x^2}}{x^{3/2}}+\frac {1}{2} c \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=-\frac {\sqrt {b x+c x^2}}{x^{3/2}}+c \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=-\frac {\sqrt {b x+c x^2}}{x^{3/2}}-\frac {c \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 51, normalized size = 0.94 \begin {gather*} -\frac {c x \sqrt {\frac {c x}{b}+1} \tanh ^{-1}\left (\sqrt {\frac {c x}{b}+1}\right )+b+c x}{\sqrt {x} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x + c*x^2]/x^(5/2),x]

[Out]

-((b + c*x + c*x*Sqrt[1 + (c*x)/b]*ArcTanh[Sqrt[1 + (c*x)/b]])/(Sqrt[x]*Sqrt[x*(b + c*x)]))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.15, size = 54, normalized size = 1.00 \begin {gather*} -\frac {c \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{\sqrt {b}}-\frac {\sqrt {b x+c x^2}}{x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b*x + c*x^2]/x^(5/2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 126, normalized size = 2.33 \begin {gather*} \left [\frac {\sqrt {b} c x^{2} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, \sqrt {c x^{2} + b x} b \sqrt {x}}{2 \, b x^{2}}, \frac {\sqrt {-b} c x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - \sqrt {c x^{2} + b x} b \sqrt {x}}{b x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^(5/2),x, algorithm="fricas")

[Out]

[1/2*(sqrt(b)*c*x^2*log(-(c*x^2 + 2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) - 2*sqrt(c*x^2 + b*x)*b*sq
rt(x))/(b*x^2), (sqrt(-b)*c*x^2*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - sqrt(c*x^2 + b*x)*b*sqrt(x))/(b*x
^2)]

________________________________________________________________________________________

giac [A]  time = 0.21, size = 41, normalized size = 0.76 \begin {gather*} \frac {\frac {c^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {\sqrt {c x + b} c}{x}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^(5/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 53, normalized size = 0.98 \begin {gather*} \frac {\left (-c x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-\sqrt {c x +b}\, \sqrt {b}\right ) \sqrt {\left (c x +b \right ) x}}{\sqrt {c x +b}\, \sqrt {b}\, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/x^(5/2),x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^{2} + b x}}{x^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + b*x)/x^(5/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{x^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^(1/2)/x^(5/2),x)

[Out]

int((b*x + c*x^2)^(1/2)/x^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (b + c x\right )}}{x^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(1/2)/x**(5/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/x**(5/2), x)

________________________________________________________________________________________