\(\int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} (a+c x^2)} \, dx\) [615]

Optimal result

Integrand size = 28, antiderivative size = 351 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {\left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (c f^2+a g^2\right )}-\frac {\left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c f^2+a g^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {922, 37, 6857, 95, 214} \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {\left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (a g^2+c f^2\right )}-\frac {\left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f} \left (a g^2+c f^2\right )}-\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )} \]

[In]

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

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.27 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {i \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \arctan \left (\frac {\sqrt {c f^2+a g^2} \sqrt {d+e x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {f+g x}}\right )}{\sqrt {a} \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {c f^2+a g^2}}-\frac {i \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \arctan \left (\frac {\sqrt {c f^2+a g^2} \sqrt {d+e x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {f+g x}}\right )}{\sqrt {a} \left (\sqrt {c} f-i \sqrt {a} g\right ) \sqrt {c f^2+a g^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5382\) vs. \(2(279)=558\).

Time = 0.46 (sec) , antiderivative size = 5383, normalized size of antiderivative = 15.34

method	result	size
default	\(\text {Expression too large to display}\)	\(5383\)

[In]

int((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. 5844 vs. \(2 (279) = 558\).

Time = 46.42 (sec) , antiderivative size = 5844, normalized size of antiderivative = 16.65 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[In]

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

[Out]

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