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

Optimal. Leaf size=307 \[ \frac{a \sqrt{c} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{3/4}}+\frac{a \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt{2} b^{3/4}}+\frac{(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c} \]

[Out]

((c*x)^(3/2)*(a - b*x^2)^(1/4))/(2*c) - (a*Sqrt[c]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)
^(1/4))])/(4*Sqrt[2]*b^(3/4)) + (a*Sqrt[c]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))]
)/(4*Sqrt[2]*b^(3/4)) + (a*Sqrt[c]*Log[Sqrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[c
*x])/(a - b*x^2)^(1/4)])/(8*Sqrt[2]*b^(3/4)) - (a*Sqrt[c]*Log[Sqrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] +
(Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a - b*x^2)^(1/4)])/(8*Sqrt[2]*b^(3/4))

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Rubi [A]  time = 0.268361, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {279, 329, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{a \sqrt{c} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \log \left (\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}+\sqrt{c}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{3/4}}+\frac{a \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt{2} b^{3/4}}+\frac{(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c} \]

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[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)
^(1/4))])/(4*Sqrt[2]*b^(3/4)) + (a*Sqrt[c]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))]
)/(4*Sqrt[2]*b^(3/4)) + (a*Sqrt[c]*Log[Sqrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] - (Sqrt[2]*b^(1/4)*Sqrt[c
*x])/(a - b*x^2)^(1/4)])/(8*Sqrt[2]*b^(3/4)) - (a*Sqrt[c]*Log[Sqrt[c] + (Sqrt[b]*Sqrt[c]*x)/Sqrt[a - b*x^2] +
(Sqrt[2]*b^(1/4)*Sqrt[c*x])/(a - b*x^2)^(1/4)])/(8*Sqrt[2]*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 297

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

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

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

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 \operatorname{Subst}\left (\int \frac{c-\sqrt{b} x^2}{1+\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{4 \sqrt{b} c}+\frac{a \operatorname{Subst}\left (\int \frac{c+\sqrt{b} x^2}{1+\frac{b x^4}{c^2}} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{4 \sqrt{b} c}\\ &=\frac{(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac{\left (a \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{c}}{\sqrt [4]{b}}+2 x}{-\frac{c}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{\left (a \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{c}}{\sqrt [4]{b}}-2 x}{-\frac{c}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{\frac{c}{\sqrt{b}}-\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 b}+\frac{(a c) \operatorname{Subst}\left (\int \frac{1}{\frac{c}{\sqrt{b}}+\frac{\sqrt{2} \sqrt{c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 b}\\ &=\frac{(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac{a \sqrt{c} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{\left (a \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{3/4}}-\frac{\left (a \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{3/4}}\\ &=\frac{(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}-\frac{a \sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{3/4}}+\frac{a \sqrt{c} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt{c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt{2} b^{3/4}}+\frac{a \sqrt{c} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt{2} b^{3/4}}-\frac{a \sqrt{c} \log \left (\sqrt{c}+\frac{\sqrt{b} \sqrt{c} x}{\sqrt{a-b x^2}}+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt{2} b^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.0110217, size = 57, normalized size = 0.19 \[ \frac{2 x \sqrt{c x} \sqrt [4]{a-b x^2} \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )}{3 \sqrt [4]{1-\frac{b x^2}{a}}} \]

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 = 0.02, 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 = 2.65611, size = 48, normalized size = 0.16 \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^{2 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(2*I*pi)/a)/(2*gamma(7/4))

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Giac [A]  time = 2.24555, size = 459, normalized size = 1.5 \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} \sqrt{{\left | c \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\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 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b^{\frac{3}{4}} c^{2}} - \frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\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 \, b^{\frac{1}{4}} \sqrt{{\left | c \right |}}}\right )}{b^{\frac{3}{4}} c^{2}} - \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\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^{\frac{3}{4}} c^{2}} + \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (-\frac{\sqrt{2}{\left (-b c^{2} x^{2} + a c^{2}\right )}^{\frac{1}{4}} b^{\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^{\frac{3}{4}} 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)*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(a
bs(c))))/(b^(3/4)*c^2) - 2*sqrt(2)*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^(3/4)*c^2) - sqrt(2)*sqrt(abs(c))*log(sq
rt(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^(3/4)*c^2) + sqrt(2)*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^(3/4)*c^2))