Integrand size = 28, antiderivative size = 351 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {\left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (c f^2+a g^2\right )}-\frac {\left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c f^2+a g^2\right )} \] Output:
-2*g*(e*x+d)^(1/2)/(a*g^2+c*f^2)/(g*x+f)^(1/2)+(c*d*f+a*e*g-(-a)^(1/2)*c^( 1/2)*(-d*g+e*f))*arctanh((c^(1/2)*f-(-a)^(1/2)*g)^(1/2)*(e*x+d)^(1/2)/(c^( 1/2)*d-(-a)^(1/2)*e)^(1/2)/(g*x+f)^(1/2))/(-a)^(1/2)/(c^(1/2)*d-(-a)^(1/2) *e)^(1/2)/(c^(1/2)*f-(-a)^(1/2)*g)^(1/2)/(a*g^2+c*f^2)-(c*d*f+a*e*g+(-a)^( 1/2)*c^(1/2)*(-d*g+e*f))*arctanh((c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(e*x+d)^(1 /2)/(c^(1/2)*d+(-a)^(1/2)*e)^(1/2)/(g*x+f)^(1/2))/(-a)^(1/2)/(c^(1/2)*d+(- a)^(1/2)*e)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)/(a*g^2+c*f^2)
Result contains complex when optimal does not.
Time = 2.03 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 g \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {i \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \arctan \left (\frac {\sqrt {c f^2+a g^2} \sqrt {d+e x}}{\sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )} \sqrt {f+g x}}\right )}{\sqrt {a} \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {c f^2+a g^2}}-\frac {i \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \arctan \left (\frac {\sqrt {c f^2+a g^2} \sqrt {d+e x}}{\sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )} \sqrt {f+g x}}\right )}{\sqrt {a} \left (\sqrt {c} f-i \sqrt {a} g\right ) \sqrt {c f^2+a g^2}} \] Input:
Integrate[Sqrt[d + e*x]/((f + g*x)^(3/2)*(a + c*x^2)),x]
Output:
(-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]*S qrt[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)*(Sqr t[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])
Time = 0.83 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {660, 48, 2348, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right ) (f+g x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 660 |
\(\displaystyle \frac {\int \frac {c d f+a e g+c (e f-d g) x}{\sqrt {d+e x} \sqrt {f+g x} \left (c x^2+a\right )}dx}{a g^2+c f^2}-\frac {g (e f-d g) \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2}}dx}{a g^2+c f^2}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {\int \frac {c d f+a e g+c (e f-d g) x}{\sqrt {d+e x} \sqrt {f+g x} \left (c x^2+a\right )}dx}{a g^2+c f^2}-\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )}\) |
\(\Big \downarrow \) 2348 |
\(\displaystyle \frac {\int \left (\frac {\sqrt {-a} (c d f+a e g)-a \sqrt {c} (e f-d g)}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {a \sqrt {c} (e f-d g)+\sqrt {-a} (c d f+a e g)}{2 a \left (\sqrt {c} x+\sqrt {-a}\right ) \sqrt {d+e x} \sqrt {f+g x}}\right )dx}{a g^2+c f^2}-\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {\sqrt {-a} e+\sqrt {c} d} \sqrt {\sqrt {-a} g+\sqrt {c} f}}}{a g^2+c f^2}-\frac {2 g \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right )}\) |
Input:
Int[Sqrt[d + e*x]/((f + g*x)^(3/2)*(a + c*x^2)),x]
Output:
(-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[Sqr t[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*f - Sqrt[-a]*g]) - ((c*d*f + a*e*g + Sqr t[-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[Sqrt[c ]*d + Sqrt[-a]*e]*Sqrt[Sqrt[c]*f + Sqrt[-a]*g]))/(c*f^2 + a*g^2)
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_ )^2), x_Symbol] :> Simp[(-g)*((e*f - d*g)/(c*f^2 + a*g^2)) Int[(d + e*x)^ (m - 1)*(f + g*x)^n, x], x] + Simp[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] && !IntegerQ[m] && !IntegerQ[n] & & GtQ[m, 0] && LtQ[n, -1]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(5382\) vs. \(2(279)=558\).
Time = 1.64 (sec) , antiderivative size = 5383, normalized size of antiderivative = 15.34
Input:
int((e*x+d)^(1/2)/(g*x+f)^(3/2)/(c*x^2+a),x,method=_RETURNVERBOSE)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 5844 vs. \(2 (279) = 558\).
Time = 37.10 (sec) , antiderivative size = 5844, normalized size of antiderivative = 16.65 \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)^(3/2)/(c*x^2+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {\sqrt {d + e x}}{\left (a + c x^{2}\right ) \left (f + g x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x+d)**(1/2)/(g*x+f)**(3/2)/(c*x**2+a),x)
Output:
Integral(sqrt(d + e*x)/((a + c*x**2)*(f + g*x)**(3/2)), x)
\[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + a\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)^(3/2)/(c*x^2+a),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)/((c*x^2 + a)*(g*x + f)^(3/2)), x)
Timed out. \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)^(3/2)/(c*x^2+a),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )} \,d x \] Input:
int((d + e*x)^(1/2)/((f + g*x)^(3/2)*(a + c*x^2)),x)
Output:
int((d + e*x)^(1/2)/((f + g*x)^(3/2)*(a + c*x^2)), x)
\[ \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {\sqrt {e x +d}}{\sqrt {g x +f}\, a f +\sqrt {g x +f}\, a g x +\sqrt {g x +f}\, c f \,x^{2}+\sqrt {g x +f}\, c g \,x^{3}}d x \] Input:
int((e*x+d)^(1/2)/(g*x+f)^(3/2)/(c*x^2+a),x)
Output:
int(sqrt(d + e*x)/(sqrt(f + g*x)*a*f + sqrt(f + g*x)*a*g*x + sqrt(f + g*x) *c*f*x**2 + sqrt(f + g*x)*c*g*x**3),x)