∫ x2( c___ a+bx2 )3∕2 dx [243]

Optimal. Leaf size=77 \[ -\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\sqrt {a} c \sqrt {\frac {c}{a+b x^2}} \sqrt {1+\frac {b x^2}{a}} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]

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

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Rubi [A]
time = 0.03, antiderivative size = 77, 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 = {1973, 294, 221} \begin {gather*} \frac {\sqrt {a} c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c}{a+b x^2}} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b} \end {gather*}

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

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^

n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x

], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]

Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Dist[Simp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/

a))^(p*q)], Int[u*(1 + b*(x^n/a))^(p*q), x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]

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

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Mathematica [A]
time = 0.06, size = 64, normalized size = 0.83 \begin {gather*} -\frac {c \sqrt {\frac {c}{a+b x^2}} \left (\sqrt {b} x+\sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{b^{3/2}} \end {gather*}

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

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method	result	size

default	\(\frac {\left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (-x \,b^{\frac {3}{2}}+\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) b \sqrt {b \,x^{2}+a}\right )}{b^{\frac {5}{2}}}\)	\(59\)

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

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.37, size = 141, normalized size = 1.83 \begin {gather*} \left [-\frac {2 \, c x \sqrt {\frac {c}{b x^{2} + a}} - c \sqrt {\frac {c}{b}} \log \left (-2 \, b c x^{2} - a c - 2 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {c}{b x^{2} + a}} \sqrt {\frac {c}{b}}\right )}{2 \, b}, -\frac {c x \sqrt {\frac {c}{b x^{2} + a}} + c \sqrt {-\frac {c}{b}} \arctan \left (\frac {b x \sqrt {\frac {c}{b x^{2} + a}} \sqrt {-\frac {c}{b}}}{c}\right )}{b}\right ] \end {gather*}

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

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

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

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

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Giac [A]
time = 5.72, size = 71, normalized size = 0.92 \begin {gather*} -{\left (\frac {c x \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c x^{2} + a c} b} + \frac {c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b}\right )} c \end {gather*}

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (\frac {c}{b\,x^2+a}\right )}^{3/2} \,d x \end {gather*}

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3.3.43 \(\int x^2 (\frac {c}{a+b x^2})^{3/2} \, dx\) [243]