3.616 \(\int \frac{1}{\sqrt{d+e x} (f+g x)^{3/2} (a+c x^2)} \, dx\)

Optimal. Leaf size=354 \[ \frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]

[Out]

(g*Sqrt[d + e*x])/(Sqrt[-a]*(Sqrt[c]*f - Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) - (g*Sqrt[d + e*x])/(Sqrt[-a]*
(Sqrt[c]*f + Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) + (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d +
e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(Sqrt[c]*f - Sqrt[
-a]*g)^(3/2)) - (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sq
rt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.762631, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {912, 96, 93, 208} \[ \frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(f + g*x)^(3/2)*(a + c*x^2)),x]

[Out]

(g*Sqrt[d + e*x])/(Sqrt[-a]*(Sqrt[c]*f - Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) - (g*Sqrt[d + e*x])/(Sqrt[-a]*
(Sqrt[c]*f + Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]) + (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d +
e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(Sqrt[c]*f - Sqrt[
-a]*g)^(3/2)) - (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sq
rt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))

Rule 912

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.843464, size = 291, normalized size = 0.82 \[ \frac{\frac{2 \sqrt{-a} g^2 \sqrt{d+e x}}{\sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (-\sqrt{-a} g-\sqrt{c} f\right )^{3/2}}-\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\sqrt{\sqrt{-a} e-\sqrt{c} d} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}}{\sqrt{-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(f + g*x)^(3/2)*(a + c*x^2)),x]

[Out]

((2*Sqrt[-a]*g^2*Sqrt[d + e*x])/((e*f - d*g)*(c*f^2 + a*g^2)*Sqrt[f + g*x]) + (Sqrt[c]*ArcTan[(Sqrt[-(Sqrt[c]*
f) - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(
-(Sqrt[c]*f) - Sqrt[-a]*g)^(3/2)) - (Sqrt[c]*ArcTan[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[-(Sqrt[
c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*(Sqrt[c]*f - Sqrt[-a]*g)^(3/2)))/Sqrt[-a
]

________________________________________________________________________________________

Maple [B]  time = 0.572, size = 10977, normalized size = 31. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )} \sqrt{e x + d}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*x^2 + a)*sqrt(e*x + d)*(g*x + f)^(3/2)), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right ) \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + c*x**2)*sqrt(d + e*x)*(f + g*x)**(3/2)), x)

________________________________________________________________________________________

Giac [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(1/(e*x+d)^(1/2)/(g*x+f)^(3/2)/(c*x^2+a),x, algorithm="giac")

[Out]

Timed out