3.441 \(\int \frac{(A+B x) \sqrt{a+c x^2}}{(e x)^{9/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac{2 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}} \]

[Out]

(-4*A*c*Sqrt[a + c*x^2])/(21*a*e^3*(e*x)^(3/2)) - (4*B*c*Sqrt[a + c*x^2])/(5*a*e^4*Sqrt[e*x]) - (2*(5*A + 7*B*
x)*Sqrt[a + c*x^2])/(35*e*(e*x)^(7/2)) + (4*B*c^(3/2)*x*Sqrt[a + c*x^2])/(5*a*e^4*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]
*x)) - (4*B*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan
[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*e^4*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*(21*Sqrt[a]*B - 5*A*Sqrt[c])
*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*S
qrt[x])/a^(1/4)], 1/2])/(105*a^(5/4)*e^4*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.406121, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {811, 835, 842, 840, 1198, 220, 1196} \[ \frac{2 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a + c*x^2])/(e*x)^(9/2),x]

[Out]

(-4*A*c*Sqrt[a + c*x^2])/(21*a*e^3*(e*x)^(3/2)) - (4*B*c*Sqrt[a + c*x^2])/(5*a*e^4*Sqrt[e*x]) - (2*(5*A + 7*B*
x)*Sqrt[a + c*x^2])/(35*e*(e*x)^(7/2)) + (4*B*c^(3/2)*x*Sqrt[a + c*x^2])/(5*a*e^4*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]
*x)) - (4*B*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan
[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*e^4*Sqrt[e*x]*Sqrt[a + c*x^2]) + (2*(21*Sqrt[a]*B - 5*A*Sqrt[c])
*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*S
qrt[x])/a^(1/4)], 1/2])/(105*a^(5/4)*e^4*Sqrt[e*x]*Sqrt[a + c*x^2])

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 842

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[x]/Sqrt[e*x], Int[
(f + g*x)/(Sqrt[x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, e, f, g}, x]

Rule 840

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2, Subst[Int[(f + g*x^2)/Sqrt[
a + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, c, f, g}, x]

Rule 1198

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

Rule 220

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

Rule 1196

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

Rubi steps

\begin{align*} \int \frac{(A+B x) \sqrt{a+c x^2}}{(e x)^{9/2}} \, dx &=-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}-\frac{2 \int \frac{-5 a A c e^2-7 a B c e^2 x}{(e x)^{5/2} \sqrt{a+c x^2}} \, dx}{35 a e^4}\\ &=-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}+\frac{4 \int \frac{\frac{21}{2} a^2 B c e^3-\frac{5}{2} a A c^2 e^3 x}{(e x)^{3/2} \sqrt{a+c x^2}} \, dx}{105 a^2 e^6}\\ &=-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}}-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}-\frac{8 \int \frac{\frac{5}{4} a^2 A c^2 e^4-\frac{21}{4} a^2 B c^2 e^4 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{105 a^3 e^8}\\ &=-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}}-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}-\frac{\left (8 \sqrt{x}\right ) \int \frac{\frac{5}{4} a^2 A c^2 e^4-\frac{21}{4} a^2 B c^2 e^4 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{105 a^3 e^8 \sqrt{e x}}\\ &=-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}}-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}-\frac{\left (16 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{\frac{5}{4} a^2 A c^2 e^4-\frac{21}{4} a^2 B c^2 e^4 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{105 a^3 e^8 \sqrt{e x}}\\ &=-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}}-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}-\frac{\left (4 B c^{3/2} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{5 \sqrt{a} e^4 \sqrt{e x}}+\frac{\left (4 \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) c^{3/2} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{105 a e^4 \sqrt{e x}}\\ &=-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}}-\frac{2 (5 A+7 B x) \sqrt{a+c x^2}}{35 e (e x)^{7/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}+\frac{2 \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0337819, size = 88, normalized size = 0.24 \[ -\frac{2 \sqrt{e x} \sqrt{a+c x^2} \left (5 A \, _2F_1\left (-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{a}\right )+7 B x \, _2F_1\left (-\frac{5}{4},-\frac{1}{2};-\frac{1}{4};-\frac{c x^2}{a}\right )\right )}{35 e^5 x^4 \sqrt{\frac{c x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a + c*x^2])/(e*x)^(9/2),x]

[Out]

(-2*Sqrt[e*x]*Sqrt[a + c*x^2]*(5*A*Hypergeometric2F1[-7/4, -1/2, -3/4, -((c*x^2)/a)] + 7*B*x*Hypergeometric2F1
[-5/4, -1/2, -1/4, -((c*x^2)/a)]))/(35*e^5*x^4*Sqrt[1 + (c*x^2)/a])

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Maple [A]  time = 0.033, size = 340, normalized size = 0.9 \begin{align*} -{\frac{2}{105\,{x}^{3}{e}^{4}a} \left ( 5\,A\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}c-42\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac+21\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac+42\,B{c}^{2}{x}^{5}+10\,A{c}^{2}{x}^{4}+63\,aBc{x}^{3}+25\,aAc{x}^{2}+21\,{a}^{2}Bx+15\,A{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(9/2),x)

[Out]

-2/105/x^3*(5*A*(-a*c)^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2)
)^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*c-42*B*((
c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/
2)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*a*c+21*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2
))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/
2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*a*c+42*B*c^2*x^5+10*A*c^2*x^4+63*a*B*c*x^3+25*a*A*c*x^2+21*a^2*B*x+15
*A*a^2)/(c*x^2+a)^(1/2)/e^4/(e*x)^(1/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)*(B*x + A)/(e*x)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x}}{e^{5} x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(9/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + a)*(B*x + A)*sqrt(e*x)/(e^5*x^5), x)

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Sympy [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((B*x+A)*(c*x**2+a)**(1/2)/(e*x)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}{\left (B x + A\right )}}{\left (e x\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + a)*(B*x + A)/(e*x)^(9/2), x)