3.926 \(\int \sqrt{c x} \sqrt [4]{a+b x^2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{a \sqrt{c} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{(c x)^{3/2} \sqrt [4]{a+b x^2}}{2 c} \]

[Out]

((c*x)^(3/2)*(a + b*x^2)^(1/4))/(2*c) - (a*Sqrt[c]*ArcTan[(b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a + b*x^2)^(1/4))])/(4
*b^(3/4)) + (a*Sqrt[c]*ArcTanh[(b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a + b*x^2)^(1/4))])/(4*b^(3/4))

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Rubi [A]  time = 0.0698382, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {279, 329, 331, 298, 205, 208} \[ -\frac{a \sqrt{c} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a+b x^2}}\right )}{4 b^{3/4}}+\frac{(c x)^{3/2} \sqrt [4]{a+b x^2}}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x]*(a + b*x^2)^(1/4),x]

[Out]

((c*x)^(3/2)*(a + b*x^2)^(1/4))/(2*c) - (a*Sqrt[c]*ArcTan[(b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a + b*x^2)^(1/4))])/(4
*b^(3/4)) + (a*Sqrt[c]*ArcTanh[(b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a + b*x^2)^(1/4))])/(4*b^(3/4))

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 205

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

Rule 208

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

Rubi steps

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

Mathematica [C]  time = 0.010098, size = 56, normalized size = 0.48 \[ \frac{2 x \sqrt{c x} \sqrt [4]{a+b x^2} \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )}{3 \sqrt [4]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*x]*(a + b*x^2)^(1/4),x]

[Out]

(2*x*Sqrt[c*x]*(a + b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^(1/4))

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int \sqrt{cx}\sqrt [4]{b{x}^{2}+a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^(1/2)*(b*x^2+a)^(1/4),x)

[Out]

int((c*x)^(1/2)*(b*x^2+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1/2)*(b*x^2+a)^(1/4),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/4)*sqrt(c*x), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1/2)*(b*x^2+a)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [C]  time = 2.50043, size = 46, normalized size = 0.4 \begin{align*} \frac{\sqrt [4]{a} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)**(1/2)*(b*x**2+a)**(1/4),x)

[Out]

a**(1/4)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(7/4))

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Giac [B]  time = 1.64458, size = 498, normalized size = 4.29 \begin{align*} \frac{1}{16} \, a c^{2}{\left (\frac{8 \,{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} x^{2} \sqrt{{\left | c \right |}}}{\sqrt{c x} a c^{2}} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} + \frac{2 \,{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b c^{2}} + \frac{2 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} - \frac{2 \,{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \sqrt{{\left | c \right |}}}{\sqrt{c x}}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b c^{2}} + \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} \log \left (\frac{\sqrt{2}{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{-b}{\left | c \right |} + \frac{\sqrt{b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right )}{b c^{2}} - \frac{\sqrt{2} \left (-b\right )^{\frac{1}{4}} \sqrt{{\left | c \right |}} \log \left (-\frac{\sqrt{2}{\left (b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}{\left | c \right |}}{\sqrt{c x}} + \sqrt{-b}{\left | c \right |} + \frac{\sqrt{b c^{2} x^{2} + a c^{2}}{\left | c \right |}}{c x}\right )}{b c^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(1/2)*(b*x^2+a)^(1/4),x, algorithm="giac")

[Out]

1/16*a*c^2*(8*(b*c^2*x^2 + a*c^2)^(1/4)*x^2*sqrt(abs(c))/(sqrt(c*x)*a*c^2) + 2*sqrt(2)*(-b)^(1/4)*sqrt(abs(c))
*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4)*sqrt(abs(c)) + 2*(b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/((-
b)^(1/4)*sqrt(abs(c))))/(b*c^2) + 2*sqrt(2)*(-b)^(1/4)*sqrt(abs(c))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4)*sq
rt(abs(c)) - 2*(b*c^2*x^2 + a*c^2)^(1/4)*sqrt(abs(c))/sqrt(c*x))/((-b)^(1/4)*sqrt(abs(c))))/(b*c^2) + sqrt(2)*
(-b)^(1/4)*sqrt(abs(c))*log(sqrt(2)*(b*c^2*x^2 + a*c^2)^(1/4)*(-b)^(1/4)*abs(c)/sqrt(c*x) + sqrt(-b)*abs(c) +
sqrt(b*c^2*x^2 + a*c^2)*abs(c)/(c*x))/(b*c^2) - sqrt(2)*(-b)^(1/4)*sqrt(abs(c))*log(-sqrt(2)*(b*c^2*x^2 + a*c^
2)^(1/4)*(-b)^(1/4)*abs(c)/sqrt(c*x) + sqrt(-b)*abs(c) + sqrt(b*c^2*x^2 + a*c^2)*abs(c)/(c*x))/(b*c^2))