### 3.78 $$\int \frac{\sqrt{b x+c x^2}}{x^{5/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0207981, antiderivative size = 54, normalized size of antiderivative = 1., 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} $-\frac{\sqrt{b x+c x^2}}{x^{3/2}}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}$

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 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]

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 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])

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 = 0.0408061, size = 51, normalized size = 0.94 $-\frac{c x \sqrt{\frac{c x}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )+b+c x}{\sqrt{x} \sqrt{x (b+c x)}}$

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)]))

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Maple [A]  time = 0.183, size = 53, normalized size = 1. \begin{align*}{ \left ( -{\it Artanh} \left ({\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) xc-\sqrt{cx+b}\sqrt{b} \right ) \sqrt{x \left ( cx+b \right ) }{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \end{align*}

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

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

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)

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Fricas [A]  time = 2.02776, size = 308, normalized size = 5.7 \begin{align*} \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{align*}

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)]

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

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)

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Giac [A]  time = 1.21015, size = 51, normalized size = 0.94 \begin{align*} c{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{c x + b}}{c x}\right )} \end{align*}

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