Integrand size = 28, antiderivative size = 625 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\frac {2 (e f-d g) \sqrt {d+e x}}{\left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {2 \sqrt {e} (e f-d g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {g} \left (c f^2+a g^2\right )}-\frac {\sqrt {e} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g} \left (c f^2+a g^2\right )}+\frac {\sqrt {e} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g} \left (c f^2+a g^2\right )}+\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g} \left (c f^2+a g^2\right )}-\frac {\sqrt {\sqrt {c} d+\sqrt {-a} e} \left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (c f^2+a g^2\right )} \]
-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)
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}} \]
((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*Log[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] + Sqrt[f + g*x]*#1]*#1^2 - 4*c*d^2*e*g^2*Log[Sqrt[d - (e*f)/g] - Sqrt[d + e*x] + Sq rt[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] + Sqrt[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...
Time = 1.30 (sec) , antiderivative size = 574, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {660, 57, 66, 221, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right ) (f+g x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 660 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (c d f+a e g+c (e f-d g) x)}{\sqrt {f+g x} \left (c x^2+a\right )}dx}{a g^2+c f^2}-\frac {g (e f-d g) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2}}dx}{a g^2+c f^2}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (c d f+a e g+c (e f-d g) x)}{\sqrt {f+g x} \left (c x^2+a\right )}dx}{a g^2+c f^2}-\frac {g (e f-d g) \left (\frac {e \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x}}dx}{g}-\frac {2 \sqrt {d+e x}}{g \sqrt {f+g x}}\right )}{a g^2+c f^2}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (c d f+a e g+c (e f-d g) x)}{\sqrt {f+g x} \left (c x^2+a\right )}dx}{a g^2+c f^2}-\frac {g (e f-d g) \left (\frac {2 e \int \frac {1}{e-\frac {g (d+e x)}{f+g x}}d\frac {\sqrt {d+e x}}{\sqrt {f+g x}}}{g}-\frac {2 \sqrt {d+e x}}{g \sqrt {f+g x}}\right )}{a g^2+c f^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\int \frac {\sqrt {d+e x} (c d f+a e g+c (e f-d g) x)}{\sqrt {f+g x} \left (c x^2+a\right )}dx}{a g^2+c f^2}-\frac {g (e f-d g) \left (\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{g^{3/2}}-\frac {2 \sqrt {d+e x}}{g \sqrt {f+g x}}\right )}{a g^2+c f^2}\) |
| \(\Big \downarrow \) 2348 |
| \(\displaystyle \frac {\int \left (\frac {\sqrt {d+e x} \left (\sqrt {-a} (c d f+a e g)-a \sqrt {c} (e f-d g)\right )}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {f+g x}}+\frac {\left (a \sqrt {c} (e f-d g)+\sqrt {-a} (c d f+a e g)\right ) \sqrt {d+e x}}{2 a \left (\sqrt {c} x+\sqrt {-a}\right ) \sqrt {f+g x}}\right )dx}{a g^2+c f^2}-\frac {g (e f-d g) \left (\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{g^{3/2}}-\frac {2 \sqrt {d+e x}}{g \sqrt {f+g x}}\right )}{a g^2+c f^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt {e} \left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g}}+\frac {\sqrt {e} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {g}}+\frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (-\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} f-\sqrt {-a} g}}-\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (\sqrt {-a} \sqrt {c} (e f-d g)+a e g+c d f\right ) \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {-a} g+\sqrt {c} f}}}{a g^2+c f^2}-\frac {g (e f-d g) \left (\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {d+e x}}{\sqrt {e} \sqrt {f+g x}}\right )}{g^{3/2}}-\frac {2 \sqrt {d+e x}}{g \sqrt {f+g x}}\right )}{a g^2+c f^2}\) |
-((g*(e*f - d*g)*((-2*Sqrt[d + e*x])/(g*Sqrt[f + g*x]) + (2*Sqrt[e]*ArcTan h[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/g^(3/2)))/(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 ])) + (Sqrt[e]*(c*d*f + a*e*g + Sqrt[-a]*Sqrt[c]*(e*f - d*g))*ArcTanh[(Sqr t[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[g]) + (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]) - (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)
3.7.14.3.1 Defintions of rubi rules used
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] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & 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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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]
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. \(8263\) vs. \(2(497)=994\).
Time = 0.48 (sec) , antiderivative size = 8264, normalized size of antiderivative = 13.22
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (a + c x^{2}\right ) \left (f + g x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{3/2} \left (a+c x^2\right )} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )} \,d x \]