Integrand size = 25, antiderivative size = 152 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{c^{3/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \]
arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(B-A*c^(1/2)/a ^(1/2))/c^(3/4)/(-e*a^(1/2)+d*c^(1/2))^(1/2)+arctanh(c^(1/4)*(e*x+d)^(1/2) /(e*a^(1/2)+d*c^(1/2))^(1/2))*(B+A*c^(1/2)/a^(1/2))/c^(3/4)/(e*a^(1/2)+d*c ^(1/2))^(1/2)
Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\frac {\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{\sqrt {a} \sqrt {c}} \]
(((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e] + ((Sqr t[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x] )/(Sqrt[c]*d - Sqrt[a]*e)])/Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])/(Sqrt[a]*Sqr t[c])
Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {654, 1480, 221}
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.
\(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right ) \sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 654 |
\(\displaystyle 2 \int \frac {B d-A e-B (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}\) |
\(\Big \downarrow \) 1480 |
\(\displaystyle 2 \left (-\frac {1}{2} \left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}-\frac {1}{2} \left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 2 \left (\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 c^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\) |
2*(((B - (A*Sqrt[c])/Sqrt[a])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c] *d - Sqrt[a]*e]])/(2*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((B + (A*Sqrt[ c])/Sqrt[a])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]) /(2*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))
3.15.50.3.1 Defintions of rubi rules used
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[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 0.41 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {-\frac {\left (A c e +B \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (A c e -B \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{\sqrt {a c \,e^{2}}}\) | \(133\) |
derivativedivides | \(-2 c \left (-\frac {\left (A c e +B \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (-A c e +B \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(148\) |
default | \(2 c \left (-\frac {\left (-A c e -B \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A c e -B \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, c \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\) | \(150\) |
-1/(a*c*e^2)^(1/2)*(-(A*c*e+B*(a*c*e^2)^(1/2))/((c*d+(a*c*e^2)^(1/2))*c)^( 1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))-(A*c*e-B*(a* c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c *d+(a*c*e^2)^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2385 vs. \(2 (108) = 216\).
Time = 0.46 (sec) , antiderivative size = 2385, normalized size of antiderivative = 15.69 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
1/2*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4 *A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a* c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d ^2 - a^2*c*e^2))*log((2*(A*B^3*a*c - A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e) *sqrt(e*x + d) + (2*A*B^2*a*c^2*d^2 - (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A *B^2*a^2*c + A^3*a*c^2)*e^2 + (A*a*c^4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d *e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)* d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2* e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d + (a*c^2*d^2 - a ^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a ^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^ 3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))) - 1/2*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c )*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3 *B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2* c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))*log((2*(A*B^3*a*c - A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e)*sqrt(e*x + d) - (2*A*B^2*a*c^2*d^2 - (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A*B^2*a^2*c + A^3*a*c^2)*e^2 + (A*a*c^ 4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d*e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^2 *c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4* c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*...
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=- \int \frac {A}{- a \sqrt {d + e x} + c x^{2} \sqrt {d + e x}}\, dx - \int \frac {B x}{- a \sqrt {d + e x} + c x^{2} \sqrt {d + e x}}\, dx \]
-Integral(A/(-a*sqrt(d + e*x) + c*x**2*sqrt(d + e*x)), x) - Integral(B*x/( -a*sqrt(d + e*x) + c*x**2*sqrt(d + e*x)), x)
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\int { -\frac {B x + A}{{\left (c x^{2} - a\right )} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (108) = 216\).
Time = 0.31 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=-\frac {{\left (B a c d {\left | c \right |} {\left | e \right |} - A a c e {\left | c \right |} {\left | e \right |} + \sqrt {a c} A c d e {\left | c \right |} - \sqrt {a c} B a e^{2} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{{\left (a c^{2} d - \sqrt {a c} a c e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} - \frac {{\left (B a c d {\left | c \right |} {\left | e \right |} - A a c e {\left | c \right |} {\left | e \right |} - \sqrt {a c} A c d e {\left | c \right |} + \sqrt {a c} B a e^{2} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{{\left (a c^{2} d + \sqrt {a c} a c e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \]
-(B*a*c*d*abs(c)*abs(e) - A*a*c*e*abs(c)*abs(e) + sqrt(a*c)*A*c*d*e*abs(c) - sqrt(a*c)*B*a*e^2*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c*d + sqrt(c^2*d^ 2 - (c*d^2 - a*e^2)*c))/c))/((a*c^2*d - sqrt(a*c)*a*c*e)*sqrt(-c^2*d - sqr t(a*c)*c*e)*abs(e)) - (B*a*c*d*abs(c)*abs(e) - A*a*c*e*abs(c)*abs(e) - sqr t(a*c)*A*c*d*e*abs(c) + sqrt(a*c)*B*a*e^2*abs(c))*arctan(sqrt(e*x + d)/sqr t(-(c*d - sqrt(c^2*d^2 - (c*d^2 - a*e^2)*c))/c))/((a*c^2*d + sqrt(a*c)*a*c *e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e))
Time = 12.08 (sec) , antiderivative size = 2065, normalized size of antiderivative = 13.59 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\text {Too large to display} \]
atan((a^2*c^5*d^3*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^ 2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/( 4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(3/2)*(d + e*x)^(1/2)*8i + A^2*a^2*c^3*e^2 *((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a ^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4 *a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i - B^2*a^2*c^3*d^2*((B^2*a*e*(a^3*c ^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B* a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(1 /2)*(d + e*x)^(1/2)*2i + B^2*a^3*c^2*e^2*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c *e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B *c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/ 2)*2i - A^2*a*c^4*d^2*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2 ))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i - a^3*c^4*d*e ^2*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2 *a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(3/2)*(d + e*x)^(1/2)*8i)/(A^3*c*e^2*(a^3*c^3)^(1/2) - B^ 3*a^3*c*e^2 - 2*A^2*B*a*c^3*d^2 - B^3*a*d*e*(a^3*c^3)^(1/2) - A^2*B*a^2*c^ 2*e^2 + A*B^2*a*e^2*(a^3*c^3)^(1/2) + 2*A*B^2*c*d^2*(a^3*c^3)^(1/2) + A^3* a*c^3*d*e + 3*A*B^2*a^2*c^2*d*e - 3*A^2*B*c*d*e*(a^3*c^3)^(1/2)))*((B^2...