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

Optimal. Leaf size=625 \[ \frac{2 \sqrt{d+e x} (e f-d g)}{\sqrt{f+g x} \left (a g^2+c f^2\right )}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (a g^2+c f^2\right )}-\frac{\sqrt{e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{e} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]

[Out]

(2*(e*f - d*g)*Sqrt[d + e*x])/((c*f^2 + a*g^2)*Sqrt[f + g*x]) - (2*Sqrt[e]*(e*f - d*g)*ArcTanh[(Sqrt[g]*Sqrt[d
+ e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[g]*(c*f^2 + a*g^2)) - (Sqrt[e]*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f
- d*g))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2 + a*g^2))
+ (Sqrt[e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*
x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2 + a*g^2)) + (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d*f + a*e*g - Sqrt[-a]*Sqr
t[c]*(e*f - d*g))*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[c]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*(c*f^2 + a*g^2)) - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d*f
+ a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*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[c]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*(c*f^2 + a*g^2))

________________________________________________________________________________________

Rubi [A]  time = 2.53051, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {908, 47, 63, 217, 206, 6725, 105, 93, 208} \[ \frac{2 \sqrt{d+e x} (e f-d g)}{\sqrt{f+g x} \left (a g^2+c f^2\right )}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (a g^2+c f^2\right )}-\frac{\sqrt{e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{e} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (a g^2+c f^2\right )}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (-\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (a g^2+c f^2\right )}-\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} \sqrt{c} (e f-d g)+a e g+c d f\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{\sqrt{-a} g+\sqrt{c} f} \left (a g^2+c f^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(e*f - d*g)*Sqrt[d + e*x])/((c*f^2 + a*g^2)*Sqrt[f + g*x]) - (2*Sqrt[e]*(e*f - d*g)*ArcTanh[(Sqrt[g]*Sqrt[d
+ e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[g]*(c*f^2 + a*g^2)) - (Sqrt[e]*(c*d*f + a*e*g - Sqrt[-a]*Sqrt[c]*(e*f
- d*g))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2 + a*g^2))
+ (Sqrt[e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*
x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]*(c*f^2 + a*g^2)) + (Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d*f + a*e*g - Sqrt[-a]*Sqr
t[c]*(e*f - d*g))*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[c]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*(c*f^2 + a*g^2)) - (Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d*f
+ a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*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[c]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*(c*f^2 + a*g^2))

Rule 908

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -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{(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx &=\frac{\int \frac{\sqrt{d+e x} (c d f+a e g+c (e f-d g) x)}{\sqrt{f+g x} \left (a+c x^2\right )} \, dx}{c f^2+a g^2}-\frac{(g (e f-d g)) \int \frac{\sqrt{d+e x}}{(f+g x)^{3/2}} \, dx}{c f^2+a g^2}\\ &=\frac{2 (e f-d g) \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}+\frac{\int \left (\frac{\left (-a \sqrt{c} (e f-d g)+\sqrt{-a} (c d f+a e g)\right ) \sqrt{d+e x}}{2 a \left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{f+g x}}+\frac{\left (a \sqrt{c} (e f-d g)+\sqrt{-a} (c d f+a e g)\right ) \sqrt{d+e x}}{2 a \left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{f+g x}}\right ) \, dx}{c f^2+a g^2}-\frac{(e (e f-d g)) \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{c f^2+a g^2}\\ &=\frac{2 (e f-d g) \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{(2 (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{d g}{e}+\frac{g x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{c f^2+a g^2}-\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \int \frac{\sqrt{d+e x}}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c f^2+a g^2\right )}-\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \int \frac{\sqrt{d+e x}}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c f^2+a g^2\right )}\\ &=\frac{2 (e f-d g) \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{(2 (e f-d g)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{e}} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{c f^2+a g^2}-\frac{\left (e \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \sqrt{c} \left (c f^2+a g^2\right )}-\frac{\left (\left (d-\frac{\sqrt{-a} e}{\sqrt{c}}\right ) \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \int \frac{1}{\left (\sqrt{-a}+\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c f^2+a g^2\right )}+\frac{\left (e \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \sqrt{c} \left (c f^2+a g^2\right )}-\frac{\left (\left (d+\frac{\sqrt{-a} e}{\sqrt{c}}\right ) \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \int \frac{1}{\left (\sqrt{-a}-\sqrt{c} x\right ) \sqrt{d+e x} \sqrt{f+g x}} \, dx}{2 \sqrt{-a} \left (c f^2+a g^2\right )}\\ &=\frac{2 (e f-d g) \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (c f^2+a g^2\right )}-\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{d g}{e}+\frac{g x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{-a} \sqrt{c} \left (c f^2+a g^2\right )}-\frac{\left (\left (d-\frac{\sqrt{-a} e}{\sqrt{c}}\right ) \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \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} \left (c f^2+a g^2\right )}+\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{f-\frac{d g}{e}+\frac{g x^2}{e}}} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{-a} \sqrt{c} \left (c f^2+a g^2\right )}-\frac{\left (\left (d+\frac{\sqrt{-a} e}{\sqrt{c}}\right ) \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right )\right ) \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} \left (c f^2+a g^2\right )}\\ &=\frac{2 (e f-d g) \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (c f^2+a g^2\right )}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \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{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (c f^2+a g^2\right )}-\frac{\sqrt{\sqrt{c} d+\sqrt{-a} e} \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \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{c} \sqrt{\sqrt{c} f+\sqrt{-a} g} \left (c f^2+a g^2\right )}-\frac{\left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{e}} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \left (c f^2+a g^2\right )}+\frac{\left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{g x^2}{e}} \, dx,x,\frac{\sqrt{d+e x}}{\sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \left (c f^2+a g^2\right )}\\ &=\frac{2 (e f-d g) \sqrt{d+e x}}{\left (c f^2+a g^2\right ) \sqrt{f+g x}}-\frac{2 \sqrt{e} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{g} \left (c f^2+a g^2\right )}-\frac{\sqrt{e} \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (c f^2+a g^2\right )}+\frac{\sqrt{e} \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{\sqrt{-a} \sqrt{c} \sqrt{g} \left (c f^2+a g^2\right )}+\frac{\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (c d f+a e g-\sqrt{-a} \sqrt{c} (e f-d g)\right ) \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{c} \sqrt{\sqrt{c} f-\sqrt{-a} g} \left (c f^2+a g^2\right )}-\frac{\sqrt{\sqrt{c} d+\sqrt{-a} e} \left (c d f+a e g+\sqrt{-a} \sqrt{c} (e f-d g)\right ) \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{c} \sqrt{\sqrt{c} f+\sqrt{-a} g} \left (c f^2+a g^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.891688, size = 340, normalized size = 0.54 \[ -\left (\frac{a d}{(-a)^{3/2}}-\frac{e}{\sqrt{c}}\right ) \left (\frac{\sqrt{d+e x}}{\sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right )}+\frac{\sqrt{\sqrt{-a} e+\sqrt{c} d} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{-\sqrt{-a} g-\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\left (-\sqrt{-a} g-\sqrt{c} f\right )^{3/2}}\right )-\left (\frac{d}{\sqrt{-a}}-\frac{e}{\sqrt{c}}\right ) \left (\frac{\sqrt{d+e x}}{\sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right )}-\frac{\sqrt{\sqrt{-a} e-\sqrt{c} d} \tan ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e-\sqrt{c} d}}\right )}{\left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.508, size = 8264, normalized size = 13.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

[Out]

Timed out

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

[Out]

Timed out