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

Optimal. Leaf size=76 \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c} \]

[Out]

3/4*b^2*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(5/2)-3/4*b*(c*x^2+b*x)^(1/2)/c^2+1/2*x*(c*x^2+b*x)^(1/2)/c

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Rubi [A]  time = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {670, 640, 620, 206} \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[b*x + c*x^2],x]

[Out]

(-3*b*Sqrt[b*x + c*x^2])/(4*c^2) + (x*Sqrt[b*x + c*x^2])/(2*c) + (3*b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]]
)/(4*c^(5/2))

Rule 206

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

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 640

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

Rule 670

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

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx &=\frac {x \sqrt {b x+c x^2}}{2 c}-\frac {(3 b) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{4 c}\\ &=-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 c^2}\\ &=-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 c^2}\\ &=-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 88, normalized size = 1.16 \[ \frac {3 b^{5/2} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {c} x \left (-3 b^2-b c x+2 c^2 x^2\right )}{4 c^{5/2} \sqrt {x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[c]*x*(-3*b^2 - b*c*x + 2*c^2*x^2) + 3*b^(5/2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b
]])/(4*c^(5/2)*Sqrt[x*(b + c*x)])

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fricas [A]  time = 0.79, size = 126, normalized size = 1.66 \[ \left [\frac {3 \, b^{2} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, c^{2} x - 3 \, b c\right )} \sqrt {c x^{2} + b x}}{8 \, c^{3}}, -\frac {3 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (2 \, c^{2} x - 3 \, b c\right )} \sqrt {c x^{2} + b x}}{4 \, c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*b^2*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(2*c^2*x - 3*b*c)*sqrt(c*x^2 + b*x))/c^3,
-1/4*(3*b^2*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (2*c^2*x - 3*b*c)*sqrt(c*x^2 + b*x))/c^3]

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giac [A]  time = 0.23, size = 65, normalized size = 0.86 \[ \frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (\frac {2 \, x}{c} - \frac {3 \, b}{c^{2}}\right )} - \frac {3 \, b^{2} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(c*x^2 + b*x)*(2*x/c - 3*b/c^2) - 3/8*b^2*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(
5/2)

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maple [A]  time = 0.05, size = 68, normalized size = 0.89 \[ \frac {3 b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{2}+b x}\, x}{2 c}-\frac {3 \sqrt {c \,x^{2}+b x}\, b}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*(c*x^2+b*x)^(1/2)/c-3/4*b*(c*x^2+b*x)^(1/2)/c^2+3/8*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x)^(1/2)
)

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maxima [A]  time = 1.34, size = 66, normalized size = 0.87 \[ \frac {\sqrt {c x^{2} + b x} x}{2 \, c} + \frac {3 \, b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^2 + b*x)*x/c + 3/8*b^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 3/4*sqrt(c*x^2 + b*
x)*b/c^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {c\,x^2+b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x + c*x^2)^(1/2),x)

[Out]

int(x^2/(b*x + c*x^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2/sqrt(x*(b + c*x)), x)

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