### 3.688 $$\int \frac{\sqrt{d+e x}}{(a+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=298 $\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}$

[Out]

(x*Sqrt[d + e*x])/(a*Sqrt[a + c*x^2]) - (Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*
x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[
c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (d*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/
a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*S
qrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.209321, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {737, 12, 844, 719, 424, 419} $\frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}+\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*Sqrt[d + e*x])/(a*Sqrt[a + c*x^2]) - (Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*
x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[
c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (d*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/
a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*S
qrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 737

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 844

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

Rule 719

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

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

Mathematica [C]  time = 2.05554, size = 408, normalized size = 1.37 $\frac{\sqrt{d+e x} \left (\frac{\sqrt{a} \sqrt{d+e x} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{c} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{e \left (a+c x^2\right )}{c (d+e x)}-\frac{i \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}+x\right )}{a \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[d + e*x]*(x - (e*(a + c*x^2))/(c*(d + e*x)) - (I*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/S
qrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqr
t[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[
a]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x
]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d +
I*Sqrt[a]*e)])/(Sqrt[c]*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(a*Sqrt[a + c*x^2])

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Maple [B]  time = 0.268, size = 649, normalized size = 2.2 \begin{align*}{\frac{1}{ce \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ) a}\sqrt{ex+d}\sqrt{c{x}^{2}+a} \left ({\it EllipticE} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) a{e}^{2}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}+{\it EllipticE} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) c{d}^{2}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}-{\it EllipticF} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) a{e}^{2}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}-{\it EllipticF} \left ( \sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{ \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) de\sqrt{-ac}\sqrt{-{c \left ( ex+d \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}+c{e}^{2}{x}^{2}+cdex \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c
)^(1/2)*e+c*d))^(1/2))*a*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*
d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),
(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*d^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-EllipticF((-(e*x+d)*c
/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*e^2*(-(e*x+d)*c/((-a*c)^(1/
2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d)
)^(1/2)-EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*
d*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+c*e^2*x^2+c*d*e*x)/c/e/(c*e*x^3+c*d*x^2+a*e*x+a*d)/a

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + a)^(3/2), 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} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(d + e*x)/(a + c*x**2)**(3/2), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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