\(\int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} (a+c x^2)} \, dx\) [614]

Optimal result

Integrand size = 28, antiderivative size = 625 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {2 (e f-d g) \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {2 \sqrt {e} (e f-d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {g} \left (c f^2+a g^2\right )}-\frac {\sqrt {e} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g} \left (c f^2+a g^2\right )}+\frac {\sqrt {e} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g} \left (c f^2+a g^2\right )}+\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (c f^2+a g^2\right )}-\frac {\sqrt {\sqrt {c} d+\sqrt {-a} e} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c f^2+a g^2\right )} \]

[Out]

-2*(-d*g+e*f)*arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))*e^(1/2)/(a*g^2+c*f^2)/g^(1/2)-arctanh(g^(1/
2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))*(c*d*f+a*e*g-(-d*g+e*f)*(-a)^(1/2)*c^(1/2))*e^(1/2)/(a*g^2+c*f^2)/(-a)
^(1/2)/c^(1/2)/g^(1/2)+arctanh(g^(1/2)*(e*x+d)^(1/2)/e^(1/2)/(g*x+f)^(1/2))*(c*d*f+a*e*g+(-d*g+e*f)*(-a)^(1/2)
*c^(1/2))*e^(1/2)/(a*g^2+c*f^2)/(-a)^(1/2)/c^(1/2)/g^(1/2)+2*(-d*g+e*f)*(e*x+d)^(1/2)/(a*g^2+c*f^2)/(g*x+f)^(1
/2)+arctanh((e*x+d)^(1/2)*(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(-e*(-a)^(1/2)+d*c^(1/2))^(1/2))*(c*d*
f+a*e*g-(-d*g+e*f)*(-a)^(1/2)*c^(1/2))*(-e*(-a)^(1/2)+d*c^(1/2))^(1/2)/(a*g^2+c*f^2)/(-a)^(1/2)/c^(1/2)/(-g*(-
a)^(1/2)+f*c^(1/2))^(1/2)-arctanh((e*x+d)^(1/2)*(g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(e*(-a)^(1/2)+d*c
^(1/2))^(1/2))*(c*d*f+a*e*g+(-d*g+e*f)*(-a)^(1/2)*c^(1/2))*(e*(-a)^(1/2)+d*c^(1/2))^(1/2)/(a*g^2+c*f^2)/(-a)^(
1/2)/c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2))^(1/2)

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {922, 49, 65, 223, 212, 6857, 132, 12, 95, 214} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 \sqrt {e} (e f-d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {g} \left (a g^2+c f^2\right )}-\frac {\sqrt {e} \left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g} \left (a g^2+c f^2\right )}+\frac {\sqrt {e} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g} \left (a g^2+c f^2\right )}+\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (a g^2+c f^2\right )}-\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {-a} g+\sqrt {c} f} \left (a g^2+c f^2\right )}+\frac {2 \sqrt {d+e x} (e f-d g)}{\sqrt {f+g x} \left (a g^2+c f^2\right )} \]

[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 12

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

Rule 49

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 65

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 95

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 132

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

Rule 212

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

Rule 214

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

Rule 223

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 922

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 6857

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]

Rubi steps \begin{align*} \text {integral}& = \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)) \text {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)) \text {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 (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \int \frac {d-\frac {\sqrt {-a} e}{\sqrt {c}}}{\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 (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \int \frac {d+\frac {\sqrt {-a} e}{\sqrt {c}}}{\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 {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 ) \text {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 ) \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 (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {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 ) \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 ) \text {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 (\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 ) \text {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 ) \text {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 (\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 ) \text {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 {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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 8.15 (sec) , antiderivative size = 1049, normalized size of antiderivative = 1.68 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {(e f-d g) \left (2 \sqrt {d+e x}-\frac {1}{2} \sqrt {f+g x} \text {RootSum}\left [c e^4 f^2+a e^4 g^2+4 c e^3 f^2 g \text {$\#$1}^2-8 c d e^2 f g^2 \text {$\#$1}^2-4 a e^3 g^3 \text {$\#$1}^2+6 c e^2 f^2 g^2 \text {$\#$1}^4-16 c d e f g^3 \text {$\#$1}^4+16 c d^2 g^4 \text {$\#$1}^4+6 a e^2 g^4 \text {$\#$1}^4+4 c e f^2 g^3 \text {$\#$1}^6-8 c d f g^4 \text {$\#$1}^6-4 a e g^5 \text {$\#$1}^6+c f^2 g^4 \text {$\#$1}^8+a g^6 \text {$\#$1}^8\&,\frac {-c d e^3 f \log (f+g x)-a e^4 g \log (f+g x)+2 c d e^3 f \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right )+2 a e^4 g \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right )-c d e^2 f g \log (f+g x) \text {$\#$1}^2+2 c d^2 e g^2 \log (f+g x) \text {$\#$1}^2+a e^3 g^2 \log (f+g x) \text {$\#$1}^2+2 c d e^2 f g \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^2-4 c d^2 e g^2 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^2-2 a e^3 g^2 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^2+c d e f g^2 \log (f+g x) \text {$\#$1}^4-2 c d^2 g^3 \log (f+g x) \text {$\#$1}^4-a e^2 g^3 \log (f+g x) \text {$\#$1}^4-2 c d e f g^2 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^4+4 c d^2 g^3 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^4+2 a e^2 g^3 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^4+c d f g^3 \log (f+g x) \text {$\#$1}^6+a e g^4 \log (f+g x) \text {$\#$1}^6-2 c d f g^3 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^6-2 a e g^4 \log \left (\sqrt {d-\frac {e f}{g}}-\sqrt {d+e x}+\sqrt {f+g x} \text {$\#$1}\right ) \text {$\#$1}^6}{c e^3 f^2 \text {$\#$1}-2 c d e^2 f g \text {$\#$1}-a e^3 g^2 \text {$\#$1}+3 c e^2 f^2 g \text {$\#$1}^3-8 c d e f g^2 \text {$\#$1}^3+8 c d^2 g^3 \text {$\#$1}^3+3 a e^2 g^3 \text {$\#$1}^3+3 c e f^2 g^2 \text {$\#$1}^5-6 c d f g^3 \text {$\#$1}^5-3 a e g^4 \text {$\#$1}^5+c f^2 g^3 \text {$\#$1}^7+a g^5 \text {$\#$1}^7}\&\right ]\right )}{\left (c f^2+a g^2\right ) \sqrt {f+g x}} \]

[In]

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

[Out]

((e*f - d*g)*(2*Sqrt[d + e*x] - (Sqrt[f + g*x]*RootSum[c*e^4*f^2 + a*e^4*g^2 + 4*c*e^3*f^2*g*#1^2 - 8*c*d*e^2*
f*g^2*#1^2 - 4*a*e^3*g^3*#1^2 + 6*c*e^2*f^2*g^2*#1^4 - 16*c*d*e*f*g^3*#1^4 + 16*c*d^2*g^4*#1^4 + 6*a*e^2*g^4*#
1^4 + 4*c*e*f^2*g^3*#1^6 - 8*c*d*f*g^4*#1^6 - 4*a*e*g^5*#1^6 + c*f^2*g^4*#1^8 + a*g^6*#1^8 & , (-(c*d*e^3*f*Lo
g[f + g*x]) - a*e^4*g*Log[f + g*x] + 2*c*d*e^3*f*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1] + 2
*a*e^4*g*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1] - c*d*e^2*f*g*Log[f + g*x]*#1^2 + 2*c*d^2*e
*g^2*Log[f + g*x]*#1^2 + a*e^3*g^2*Log[f + g*x]*#1^2 + 2*c*d*e^2*f*g*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + S
qrt[f + g*x]*#1]*#1^2 - 4*c*d^2*e*g^2*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1]*#1^2 - 2*a*e^3
*g^2*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1]*#1^2 + c*d*e*f*g^2*Log[f + g*x]*#1^4 - 2*c*d^2*
g^3*Log[f + g*x]*#1^4 - a*e^2*g^3*Log[f + g*x]*#1^4 - 2*c*d*e*f*g^2*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sq
rt[f + g*x]*#1]*#1^4 + 4*c*d^2*g^3*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1]*#1^4 + 2*a*e^2*g^
3*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1]*#1^4 + c*d*f*g^3*Log[f + g*x]*#1^6 + a*e*g^4*Log[f
+ g*x]*#1^6 - 2*c*d*f*g^3*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1]*#1^6 - 2*a*e*g^4*Log[Sqrt
[d - (e*f)/g] - Sqrt[d + e*x] + Sqrt[f + g*x]*#1]*#1^6)/(c*e^3*f^2*#1 - 2*c*d*e^2*f*g*#1 - a*e^3*g^2*#1 + 3*c*
e^2*f^2*g*#1^3 - 8*c*d*e*f*g^2*#1^3 + 8*c*d^2*g^3*#1^3 + 3*a*e^2*g^3*#1^3 + 3*c*e*f^2*g^2*#1^5 - 6*c*d*f*g^3*#
1^5 - 3*a*e*g^4*#1^5 + c*f^2*g^3*#1^7 + a*g^5*#1^7) & ])/2))/((c*f^2 + a*g^2)*Sqrt[f + g*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(8263\) vs. \(2(497)=994\).

Time = 0.48 (sec) , antiderivative size = 8264, normalized size of antiderivative = 13.22

method	result	size
default	\(\text {Expression too large to display}\)	\(8264\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (a + c x^{2}\right ) \left (f + g x\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )} \,d x \]

[In]

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

[Out]

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