3.1101 \(\int \frac{c+d x^2}{\sqrt{e x} (a+b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=102 \[ \frac{d \sqrt{e x} \sqrt [4]{a+b x^2}}{b e}-\frac{(e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{\sqrt{a} \sqrt{b} e^2 \left (a+b x^2\right )^{3/4}} \]

[Out]

(d*Sqrt[e*x]*(a + b*x^2)^(1/4))/(b*e) - ((2*b*c - a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcCot[(Sqr
t[b]*x)/Sqrt[a]]/2, 2])/(Sqrt[a]*Sqrt[b]*e^2*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.0879124, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {459, 329, 237, 335, 275, 231} \[ \frac{d \sqrt{e x} \sqrt [4]{a+b x^2}}{b e}-\frac{(e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} e^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(3/4)),x]

[Out]

(d*Sqrt[e*x]*(a + b*x^2)^(1/4))/(b*e) - ((2*b*c - a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcCot[(Sqr
t[b]*x)/Sqrt[a]]/2, 2])/(Sqrt[a]*Sqrt[b]*e^2*(a + b*x^2)^(3/4))

Rule 459

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

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 237

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 335

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

Rule 275

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

Rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rubi steps

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

Mathematica [C]  time = 0.0616701, size = 77, normalized size = 0.75 \[ \frac{x \left (\frac{b x^2}{a}+1\right )^{3/4} (2 b c-a d) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+d x \left (a+b x^2\right )}{b \sqrt{e x} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(Sqrt[e*x]*(a + b*x^2)^(3/4)),x]

[Out]

(d*x*(a + b*x^2) + (2*b*c - a*d)*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)])/(b*Sq
rt[e*x]*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c){\frac{1}{\sqrt{ex}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x)

[Out]

int((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*sqrt(e*x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}{\left (d x^{2} + c\right )} \sqrt{e x}}{b e x^{3} + a e x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/4)*(d*x^2 + c)*sqrt(e*x)/(b*e*x^3 + a*e*x), x)

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Sympy [C]  time = 5.36119, size = 78, normalized size = 0.76 \begin{align*} - \frac{c{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac{3}{4}} \sqrt{e} x} + \frac{d x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \sqrt{e} \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/(e*x)**(1/2)/(b*x**2+a)**(3/4),x)

[Out]

-c*hyper((1/2, 3/4), (3/2,), a*exp_polar(I*pi)/(b*x**2))/(b**(3/4)*sqrt(e)*x) + d*x**(5/2)*gamma(5/4)*hyper((3
/4, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/4)*sqrt(e)*gamma(9/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*sqrt(e*x)), x)