Integrand size = 25, antiderivative size = 179 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=-\frac {2 B \sqrt {d+e x}}{c}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \]
-2*B*(e*x+d)^(1/2)/c+arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^ (1/2))*(B*a^(1/2)-A*c^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(1/2)/c^(5/4)/a^(1/2)+ arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(B*a^(1/2)+A*c^ (1/2))*(e*a^(1/2)+d*c^(1/2))^(1/2)/c^(5/4)/a^(1/2)
Time = 0.32 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=\frac {-2 \sqrt {a} B \sqrt {c} \sqrt {d+e x}-\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\left (-\sqrt {a} B+A \sqrt {c}\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e} \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c^{3/2}} \]
(-2*Sqrt[a]*B*Sqrt[c]*Sqrt[d + e*x] - (Sqrt[a]*B + A*Sqrt[c])*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x ])/(Sqrt[c]*d + Sqrt[a]*e)] + (-(Sqrt[a]*B) + A*Sqrt[c])*Sqrt[-(c*d) + Sqr t[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(S qrt[c]*d - Sqrt[a]*e)])/(Sqrt[a]*c^(3/2))
Time = 0.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {653, 25, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx\) |
| \(\Big \downarrow \) 653 |
| \(\displaystyle -\frac {\int -\frac {A c d+a B e+c (B d+A e) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 B \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {A c d+a B e+c (B d+A e) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 B \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {2 \int \frac {B \left (c d^2-a e^2\right )-c (B d+A e) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 B \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {c} \left (\sqrt {a} B-A \sqrt {c}\right ) \left (\sqrt {c} d-\sqrt {a} e\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}-\frac {\sqrt {c} \left (\sqrt {a} B+A \sqrt {c}\right ) \left (\sqrt {a} e+\sqrt {c} d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}\right )}{c}-\frac {2 B \sqrt {d+e x}}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \left (\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\sqrt {c} d-\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \sqrt {\sqrt {a} e+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c}}\right )}{c}-\frac {2 B \sqrt {d+e x}}{c}\) |
(-2*B*Sqrt[d + e*x])/c + (2*(((Sqrt[a]*B - A*Sqrt[c])*Sqrt[Sqrt[c]*d - Sqr t[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(2*S qrt[a]*c^(1/4)) + ((Sqrt[a]*B + A*Sqrt[c])*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*Arc Tanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c^(1 /4))))/c
3.15.49.3.1 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[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x]/(a + c*x^2)), x], x] /; Fr eeQ[{a, c, d, e, f, g}, x] && FractionQ[m] && GtQ[m, 0]
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 = 0.48 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(-\frac {2 B \sqrt {e x +d}}{c}+\frac {\left (A c d e +B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(190\) |
| risch | \(-\frac {2 B \sqrt {e x +d}}{c}+\frac {\left (A c d e +B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(190\) |
| pseudoelliptic | \(-\frac {2 B \sqrt {e x +d}}{c}+\frac {\left (A c d e +B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A c d e -B a \,e^{2}+A \sqrt {a c \,e^{2}}\, e +B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(190\) |
| default | \(-\frac {2 B \sqrt {e x +d}}{c}-\frac {\left (-A c d e -B a \,e^{2}-A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (A c d e +B a \,e^{2}-A \sqrt {a c \,e^{2}}\, e -B \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\) | \(194\) |
-2*B*(e*x+d)^(1/2)/c+(A*c*d*e+B*a*e^2+A*(a*c*e^2)^(1/2)*e+B*(a*c*e^2)^(1/2 )*d)/(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*d*e-B*a*e^2+A*(a*c*e^2)^(1/2)*e+ B*(a*c*e^2)^(1/2)*d)/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arct an(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. 1538 vs. \(2 (129) = 258\).
Time = 0.43 (sec) , antiderivative size = 1538, normalized size of antiderivative = 8.59 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=\text {Too large to display} \]
-1/2*(c*sqrt((2*A*B*a*e + a*c^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)) + (B^2*a + A^2*c)*d)/(a*c^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 - A*a*c^4*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)) + (B^3*a^2*c + A^2*B*a*c^2)*e)*sqrt((2*A*B*a*e + a*c^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)) + (B^2*a + A^2*c)*d)/(a*c^2))) - c*sqrt((2*A*B*a* e + a*c^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)) + (B^2*a + A^2*c)*d)/(a*c^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 - A*a*c^4*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)) + (B^3*a^2*c + A^2*B*a*c^2)*e)*sqrt((2*A*B*a*e + a*c^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 )) + (B^2*a + A^2*c)*d)/(a*c^2))) + c*sqrt((2*A*B*a*e - a*c^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)) + (B^2*a + A^2*c)*d)/(a*c^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 + A* a*c^4*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^...
\[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=- \int \frac {A \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {B x \sqrt {d + e x}}{- a + c x^{2}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=\int { -\frac {{\left (B x + A\right )} \sqrt {e x + d}}{c x^{2} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (129) = 258\).
Time = 0.32 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=-\frac {2 \, \sqrt {e x + d} B}{c} - \frac {{\left (\sqrt {a c} A c^{3} d^{2} e - \sqrt {a c} A a c^{2} e^{3} + {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} B {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} A c^{3} d^{2} e - \sqrt {a c} A a c^{2} e^{3} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} B {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \]
-2*sqrt(e*x + d)*B/c - (sqrt(a*c)*A*c^3*d^2*e - sqrt(a*c)*A*a*c^2*e^3 + (a *c^2*d^2 - a^2*c*e^2)*B*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d + sqrt(c^4*d^2 - (c^2*d^2 - a*c*e^2)*c^2))/c^2))/((a*c^3*d - sqrt(a*c)*a*c^ 2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + (sqrt(a*c)*A*c^3*d^2*e - sqrt( a*c)*A*a*c^2*e^3 - (a*c^2*d^2 - a^2*c*e^2)*B*abs(c)*abs(e))*arctan(sqrt(e* x + d)/sqrt(-(c^2*d - sqrt(c^4*d^2 - (c^2*d^2 - a*c*e^2)*c^2))/c^2))/((a*c ^3*d + sqrt(a*c)*a*c^2*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e))
Time = 0.46 (sec) , antiderivative size = 4276, normalized size of antiderivative = 23.89 \[ \int \frac {(A+B x) \sqrt {d+e x}}{a-c x^2} \, dx=\text {Too large to display} \]
- 2*atanh((32*A^2*a*c^2*e^4*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2* c^2) + (A^2*d)/(4*a*c) + (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3 *c^5)^(1/2))/(4*a*c^5) + (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*A ^3*c^2*d^2*e^3 - 16*A^3*a*c*e^5 - 16*A*B^2*a^2*e^5 - (16*A^2*B*e^5*(a^3*c^ 5)^(1/2))/c^2 - (16*B^3*a*e^5*(a^3*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 + (16*B^3*d^2*e^3*(a^3*c^5)^(1/2))/c^2 - 32*A^2*B*a*c*d*e^4 - (32*A*B^2*d* e^4*(a^3*c^5)^(1/2))/c^2 + 16*A*B^2*a*c*d^2*e^3 + (32*A*B^2*d^3*e^2*(a^3*c ^5)^(1/2))/(a*c) + (16*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/(a*c)) + (32*B^2*a^2 *c*e^4*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c ) + (A^2*e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5 ) + (A*B*d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*A^3*c^2*d^2*e^3 - 16*A ^3*a*c*e^5 - 16*A*B^2*a^2*e^5 - (16*A^2*B*e^5*(a^3*c^5)^(1/2))/c^2 - (16*B ^3*a*e^5*(a^3*c^5)^(1/2))/c^3 + 32*A^2*B*c^2*d^3*e^2 + (16*B^3*d^2*e^3*(a^ 3*c^5)^(1/2))/c^2 - 32*A^2*B*a*c*d*e^4 - (32*A*B^2*d*e^4*(a^3*c^5)^(1/2))/ c^2 + 16*A*B^2*a*c*d^2*e^3 + (32*A*B^2*d^3*e^2*(a^3*c^5)^(1/2))/(a*c) + (1 6*A^2*B*d^2*e^3*(a^3*c^5)^(1/2))/(a*c)) + (32*A^2*d*e^3*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((B^2*d)/(4*c^2) + (A*B*e)/(2*c^2) + (A^2*d)/(4*a*c) + (A^2* e*(a^3*c^5)^(1/2))/(4*a^2*c^4) + (B^2*e*(a^3*c^5)^(1/2))/(4*a*c^5) + (A*B* d*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*A*B^2*a^3*e^5 + 16*A^3*a^2*c*e^ 5 + (16*B^3*a^2*e^5*(a^3*c^5)^(1/2))/c^3 - 16*A^3*a*c^2*d^2*e^3 + (16*A...