∫ (a + cx2)3∕2 dx [57]

Optimal. Leaf size=65 \[ \frac {3}{8} a x \sqrt {a+c x^2}+\frac {1}{4} x \left (a+c x^2\right )^{3/2}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}} \]

1/4*x*(c*x^2+a)^(3/2)+3/8*a^2*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))/c^(1/2)+3/8*a*x*(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {201, 223, 212} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c}}+\frac {3}{8} a x \sqrt {a+c x^2}+\frac {1}{4} x \left (a+c x^2\right )^{3/2} \end {gather*}

(3*a*x*Sqrt[a + c*x^2])/8 + (x*(a + c*x^2)^(3/2))/4 + (3*a^2*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c])

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

Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2

] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

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

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

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Mathematica [A]
time = 0.06, size = 60, normalized size = 0.92 \begin {gather*} \frac {1}{8} x \sqrt {a+c x^2} \left (5 a+2 c x^2\right )-\frac {3 a^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{8 \sqrt {c}} \end {gather*}

(x*Sqrt[a + c*x^2]*(5*a + 2*c*x^2))/8 - (3*a^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/(8*Sqrt[c])

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method	result	size

risch	\(\frac {x \left (2 c \,x^{2}+5 a \right ) \sqrt {c \,x^{2}+a}}{8}+\frac {3 a^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{8 \sqrt {c}}\)	\(48\)
default	\(\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\)	\(52\)

1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(c^(1/2)*x+(c*x^2+a)^(1/2)))

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Maxima [A]
time = 0.30, size = 43, normalized size = 0.66 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a x + \frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} \end {gather*}

1/4*(c*x^2 + a)^(3/2)*x + 3/8*sqrt(c*x^2 + a)*a*x + 3/8*a^2*arcsinh(c*x/sqrt(a*c))/sqrt(c)

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Fricas [A]
time = 1.59, size = 124, normalized size = 1.91 \begin {gather*} \left [\frac {3 \, a^{2} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (2 \, c^{2} x^{3} + 5 \, a c x\right )} \sqrt {c x^{2} + a}}{16 \, c}, -\frac {3 \, a^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} x^{3} + 5 \, a c x\right )} \sqrt {c x^{2} + a}}{8 \, c}\right ] \end {gather*}

[1/16*(3*a^2*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(2*c^2*x^3 + 5*a*c*x)*sqrt(c*x^2 + a)

)/c, -1/8*(3*a^2*sqrt(-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (2*c^2*x^3 + 5*a*c*x)*sqrt(c*x^2 + a))/c]

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Sympy [A]
time = 1.56, size = 70, normalized size = 1.08 \begin {gather*} \frac {5 a^{\frac {3}{2}} x \sqrt {1 + \frac {c x^{2}}{a}}}{8} + \frac {\sqrt {a} c x^{3} \sqrt {1 + \frac {c x^{2}}{a}}}{4} + \frac {3 a^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} \end {gather*}

5*a**(3/2)*x*sqrt(1 + c*x**2/a)/8 + sqrt(a)*c*x**3*sqrt(1 + c*x**2/a)/4 + 3*a**2*asinh(sqrt(c)*x/sqrt(a))/(8*s

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Giac [A]
time = 1.21, size = 49, normalized size = 0.75 \begin {gather*} \frac {1}{8} \, {\left (2 \, c x^{2} + 5 \, a\right )} \sqrt {c x^{2} + a} x - \frac {3 \, a^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{8 \, \sqrt {c}} \end {gather*}

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

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Mupad [B]
time = 0.16, size = 37, normalized size = 0.57 \begin {gather*} \frac {x\,{\left (c\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {c\,x^2}{a}\right )}{{\left (\frac {c\,x^2}{a}+1\right )}^{3/2}} \end {gather*}

(x*(a + c*x^2)^(3/2)*hypergeom([-3/2, 1/2], 3/2, -(c*x^2)/a))/((c*x^2)/a + 1)^(3/2)

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3.1.57 \(\int (a+c x^2)^{3/2} \, dx\) [57]