3.1100 \(\int \frac{(e x)^{3/2} (c+d x^2)}{(a+b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=139 \[ \frac{\sqrt{a} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (6 b c-5 a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{6 b^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{e \sqrt{e x} \sqrt [4]{a+b x^2} (6 b c-5 a d)}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e} \]

[Out]

((6*b*c - 5*a*d)*e*Sqrt[e*x]*(a + b*x^2)^(1/4))/(6*b^2) + (d*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(3*b*e) + (Sqrt[a]
*(6*b*c - 5*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*b^(3/2)*(a
+ b*x^2)^(3/4))

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Rubi [A]  time = 0.108603, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {459, 321, 329, 237, 335, 275, 231} \[ \frac{e \sqrt{e x} \sqrt [4]{a+b x^2} (6 b c-5 a d)}{6 b^2}+\frac{\sqrt{a} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (6 b c-5 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 b^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*b*c - 5*a*d)*e*Sqrt[e*x]*(a + b*x^2)^(1/4))/(6*b^2) + (d*(e*x)^(5/2)*(a + b*x^2)^(1/4))/(3*b*e) + (Sqrt[a]
*(6*b*c - 5*a*d)*(1 + a/(b*x^2))^(3/4)*(e*x)^(3/2)*EllipticF[ArcCot[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*b^(3/2)*(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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && 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 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{(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}-\frac{\left (-3 b c+\frac{5 a d}{2}\right ) \int \frac{(e x)^{3/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{3 b}\\ &=\frac{(6 b c-5 a d) e \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}-\frac{\left (a (6 b c-5 a d) e^2\right ) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{3/4}} \, dx}{12 b^2}\\ &=\frac{(6 b c-5 a d) e \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}-\frac{(a (6 b c-5 a d) e) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt{e x}\right )}{6 b^2}\\ &=\frac{(6 b c-5 a d) e \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}-\frac{\left (a (6 b c-5 a d) e \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 )}{6 b^2 \left (a+b x^2\right )^{3/4}}\\ &=\frac{(6 b c-5 a d) e \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}+\frac{\left (a (6 b c-5 a d) e \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 )}{6 b^2 \left (a+b x^2\right )^{3/4}}\\ &=\frac{(6 b c-5 a d) e \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}+\frac{\left (a (6 b c-5 a d) e \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 )}{12 b^2 \left (a+b x^2\right )^{3/4}}\\ &=\frac{(6 b c-5 a d) e \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^2}+\frac{d (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 b e}+\frac{\sqrt{a} (6 b c-5 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 )}{6 b^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}

Mathematica [C]  time = 0.103529, size = 97, normalized size = 0.7 \[ \frac{e \sqrt{e x} \left (a \left (\frac{b x^2}{a}+1\right )^{3/4} (5 a d-6 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-\left (a+b x^2\right ) \left (5 a d-2 b \left (3 c+d x^2\right )\right )\right )}{6 b^2 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[e*x]*(-((a + b*x^2)*(5*a*d - 2*b*(3*c + d*x^2))) + a*(-6*b*c + 5*a*d)*(1 + (b*x^2)/a)^(3/4)*Hypergeome
tric2F1[1/4, 3/4, 5/4, -((b*x^2)/a)]))/(6*b^2*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 17.6159, size = 94, normalized size = 0.68 \begin{align*} \frac{c e^{\frac{3}{2}} 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}} \Gamma \left (\frac{9}{4}\right )} + \frac{d e^{\frac{3}{2}} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*e**(3/2)*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)*gamma(9/4)) + d
*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((3/4, 9/4), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/4)*gamma(13/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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