### 3.99 $$\int \frac{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=56 $\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0220336, antiderivative size = 56, 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 = {672, 660, 207} $\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x^(3/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), I
nt[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 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{1}{x^{3/2} \sqrt{b x+c x^2}} \, dx &=-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}-\frac{c \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{2 b}\\ &=-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}-\frac{c \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{b}\\ &=-\frac{\sqrt{b x+c x^2}}{b x^{3/2}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0583343, size = 63, normalized size = 1.12 $\frac{2 c \sqrt{x (b+c x)} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{c x}{b}+1}\right )}{2 \sqrt{\frac{c x}{b}+1}}-\frac{b}{2 c x}\right )}{b^2 \sqrt{x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(x^(3/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.196, size = 52, normalized size = 0.9 \begin{align*}{\sqrt{x \left ( cx+b \right ) } \left ({\it Artanh} \left ({\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ) xc-\sqrt{cx+b}\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cx+b}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + b x} x^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*x^(3/2)), x)

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Fricas [A]  time = 1.93033, size = 315, normalized size = 5.62 \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^{2} 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^{2} x^{2}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(c*x^2+b*x)^(1/2),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^2*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^2*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{3}{2}} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(x**(3/2)*sqrt(x*(b + c*x))), x)

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Giac [A]  time = 1.25034, size = 59, normalized size = 1.05 \begin{align*} -c{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{\sqrt{c x + b}}{b c x}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

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