3.57 \(\int (a+c x^2)^{3/2} \, dx\)

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

[Out]

(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])

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Rubi [A]  time = 0.0143466, antiderivative size = 65, normalized size of antiderivative = 1., 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 = {195, 217, 206} \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 \sqrt{c}}+\frac{3}{8} a x \sqrt{a+c x^2}+\frac{1}{4} x \left (a+c x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(3/2),x]

[Out]

(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])

Rule 195

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]])

Rule 217

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,
b}, x] &&  !GtQ[a, 0]

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])

Rubi steps

\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 ) \operatorname{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*}

Mathematica [A]  time = 0.0888968, size = 65, normalized size = 1. \[ \frac{1}{8} \sqrt{a+c x^2} \left (\frac{3 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c} \sqrt{\frac{c x^2}{a}+1}}+5 a x+2 c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(3/2),x]

[Out]

(Sqrt[a + c*x^2]*(5*a*x + 2*c*x^3 + (3*a^(3/2)*ArcSinh[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[c]*Sqrt[1 + (c*x^2)/a])))/8

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Maple [A]  time = 0.051, size = 51, normalized size = 0.8 \begin{align*}{\frac{x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,ax}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36263, size = 294, normalized size = 4.52 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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 = 3.8631, size = 70, normalized size = 1.08 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(3/2),x)

[Out]

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
qrt(c))

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Giac [A]  time = 1.22841, size = 66, normalized size = 1.02 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)