\(\int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} (a+c x^2)} \, dx\) [616]

Optimal result

Integrand size = 28, antiderivative size = 354 \[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}-\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \left (\sqrt {c} f-\sqrt {-a} g\right )^{3/2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d+\sqrt {-a} e} \left (\sqrt {c} f+\sqrt {-a} g\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {926, 98, 95, 214} \[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \left (\sqrt {c} f-\sqrt {-a} g\right )^{3/2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {\sqrt {-a} e+\sqrt {c} d} \left (\sqrt {-a} g+\sqrt {c} f\right )^{3/2}}+\frac {g \sqrt {d+e x}}{\sqrt {-a} \sqrt {f+g x} \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g)}-\frac {g \sqrt {d+e x}}{\sqrt {-a} \sqrt {f+g x} \left (\sqrt {-a} g+\sqrt {c} f\right ) (e f-d g)} \]

[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 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 98

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 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 926

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]

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

Result contains complex when optimal does not.

Time = 0.93 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {2 g^2 \sqrt {d+e x}}{(e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {i \sqrt {c} \left (\sqrt {c} f-i \sqrt {a} g\right )^2 \arctan \left (\frac {\sqrt {c f^2+a g^2} \sqrt {d+e x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {f+g x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \left (c f^2+a g^2\right )^{3/2}}+\frac {i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right )^2 \arctan \left (\frac {\sqrt {c f^2+a g^2} \sqrt {d+e x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {f+g x}}\right )}{\sqrt {a} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \left (c f^2+a g^2\right )^{3/2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(10976\) vs. \(2(270)=540\).

Time = 0.46 (sec) , antiderivative size = 10977, normalized size of antiderivative = 31.01

method	result	size
default	\(\text {Expression too large to display}\)	\(10977\)

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12028 vs. \(2 (270) = 540\).

Time = 90.51 (sec) , antiderivative size = 12028, normalized size of antiderivative = 33.98 \[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \sqrt {d + e x} \left (f + g x\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} \sqrt {e x + d} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )\,\sqrt {d+e\,x}} \,d x \]

[In]

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

[Out]

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