Integrand size = 32, antiderivative size = 317 \[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=-\frac {2 A \sqrt {c-d x} \sqrt {a x+b x^2}}{3 a c x^2}+\frac {2 (2 A b c-3 a B c-2 a A d) \sqrt {c-d x} \sqrt {a x+b x^2}}{3 a^2 c^2 x}+\frac {2 \sqrt {d} (2 A b c-3 a B c-2 a A d) \sqrt {1-\frac {d x}{c}} \sqrt {a x+b x^2} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt {x} \sqrt {1+\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {d} (A b c-3 a B c-2 a A d) \sqrt {x} \sqrt {1+\frac {b x}{a}} \sqrt {1-\frac {d x}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {c-d x} \sqrt {a x+b x^2}} \] Output:
-2/3*A*(-d*x+c)^(1/2)*(b*x^2+a*x)^(1/2)/a/c/x^2+2/3*(-2*A*a*d+2*A*b*c-3*B* a*c)*(-d*x+c)^(1/2)*(b*x^2+a*x)^(1/2)/a^2/c^2/x+2/3*d^(1/2)*(-2*A*a*d+2*A* b*c-3*B*a*c)*(1-d*x/c)^(1/2)*(b*x^2+a*x)^(1/2)*EllipticE(d^(1/2)*x^(1/2)/c ^(1/2),(-b*c/a/d)^(1/2))/a^2/c^(3/2)/x^(1/2)/(1+b*x/a)^(1/2)/(-d*x+c)^(1/2 )-2/3*d^(1/2)*(-2*A*a*d+A*b*c-3*B*a*c)*x^(1/2)*(1+b*x/a)^(1/2)*(1-d*x/c)^( 1/2)*EllipticF(d^(1/2)*x^(1/2)/c^(1/2),(-b*c/a/d)^(1/2))/a/c^(3/2)/(-d*x+c )^(1/2)/(b*x^2+a*x)^(1/2)
Result contains complex when optimal does not.
Time = 19.62 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=-\frac {2 \left (a A c (a+b x) (c-d x)+i \sqrt {\frac {a}{b}} b d (-2 A b c+3 a B c+2 a A d) \sqrt {1+\frac {a}{b x}} \sqrt {1-\frac {c}{d x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|-\frac {b c}{a d}\right )-i \sqrt {\frac {a}{b}} b d (-A b c+3 a B c+2 a A d) \sqrt {1+\frac {a}{b x}} \sqrt {1-\frac {c}{d x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),-\frac {b c}{a d}\right )\right )}{3 a^2 c^2 x \sqrt {x (a+b x)} \sqrt {c-d x}} \] Input:
Integrate[(A + B*x)/(x^2*Sqrt[c - d*x]*Sqrt[a*x + b*x^2]),x]
Output:
(-2*(a*A*c*(a + b*x)*(c - d*x) + I*Sqrt[a/b]*b*d*(-2*A*b*c + 3*a*B*c + 2*a *A*d)*Sqrt[1 + a/(b*x)]*Sqrt[1 - c/(d*x)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt [a/b]/Sqrt[x]], -((b*c)/(a*d))] - I*Sqrt[a/b]*b*d*(-(A*b*c) + 3*a*B*c + 2* a*A*d)*Sqrt[1 + a/(b*x)]*Sqrt[1 - c/(d*x)]*x^(5/2)*EllipticF[I*ArcSinh[Sqr t[a/b]/Sqrt[x]], -((b*c)/(a*d))]))/(3*a^2*c^2*x*Sqrt[x*(a + b*x)]*Sqrt[c - d*x])
Time = 1.40 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.55, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2154, 27, 1261, 115, 27, 124, 123, 169, 27, 176, 122, 120, 127, 126}
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.
| \(\displaystyle \int \frac {A+B x}{x^2 \sqrt {a x+b x^2} \sqrt {c-d x}} \, dx\) |
| \(\Big \downarrow \) 2154 |
| \(\displaystyle A \int \frac {1}{x^2 \sqrt {c-d x} \sqrt {b x^2+a x}}dx+\int \frac {B}{x \sqrt {c-d x} \sqrt {b x^2+a x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle A \int \frac {1}{x^2 \sqrt {c-d x} \sqrt {b x^2+a x}}dx+B \int \frac {1}{x \sqrt {c-d x} \sqrt {b x^2+a x}}dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \int \frac {1}{x^{5/2} \sqrt {a+b x} \sqrt {c-d x}}dx}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \int \frac {1}{x^{3/2} \sqrt {a+b x} \sqrt {c-d x}}dx}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \int \frac {2 b c-2 a d-b d x}{2 x^{3/2} \sqrt {a+b x} \sqrt {c-d x}}dx}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \int \frac {b d \sqrt {x}}{2 \sqrt {a+b x} \sqrt {c-d x}}dx}{a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\int \frac {2 (b c-a d)-b d x}{x^{3/2} \sqrt {a+b x} \sqrt {c-d x}}dx}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {b d \int \frac {\sqrt {x}}{\sqrt {a+b x} \sqrt {c-d x}}dx}{a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\int \frac {2 (b c-a d)-b d x}{x^{3/2} \sqrt {a+b x} \sqrt {c-d x}}dx}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {b d \sqrt {x} \sqrt {\frac {b (c-d x)}{a d+b c}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c+a d}-\frac {b d x}{b c+a d}}}dx}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\int \frac {2 (b c-a d)-b d x}{x^{3/2} \sqrt {a+b x} \sqrt {c-d x}}dx}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {2 \int \frac {b d (a c+2 (b c-a d) x)}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c-d x}}dx}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {b d \int \frac {a c+2 (b c-a d) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c-d x}}dx}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {b d \left (\frac {c (2 b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c-d x}}dx}{d}-\frac {2 (b c-a d) \int \frac {\sqrt {c-d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {b d \left (\frac {c (2 b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c-d x}}dx}{d}-\frac {2 \sqrt {\frac {b x}{a}+1} \sqrt {c-d x} (b c-a d) \int \frac {\sqrt {1-\frac {d x}{c}}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {1-\frac {d x}{c}}}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {b d \left (\frac {c (2 b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c-d x}}dx}{d}-\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c-d x} (b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {1-\frac {d x}{c}}}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {b d \left (\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {1-\frac {d x}{c}} (2 b c-a d) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {1-\frac {d x}{c}}}dx}{d \sqrt {a+b x} \sqrt {c-d x}}-\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c-d x} (b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {1-\frac {d x}{c}}}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {b d \left (\frac {2 c^{3/2} \sqrt {\frac {b x}{a}+1} \sqrt {1-\frac {d x}{c}} (2 b c-a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{d^{3/2} \sqrt {a+b x} \sqrt {c-d x}}-\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c-d x} (b c-a d) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {1-\frac {d x}{c}}}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c-d x} (b c-a d)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (-\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d+b c} \sqrt {\frac {b (c-d x)}{a d+b c}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c+a d}}\right )|\frac {b c}{a d}+1\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c-d x}}-\frac {2 \sqrt {a+b x} \sqrt {c-d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
Input:
Int[(A + B*x)/(x^2*Sqrt[c - d*x]*Sqrt[a*x + b*x^2]),x]
Output:
(B*Sqrt[x]*Sqrt[a + b*x]*((-2*Sqrt[a + b*x]*Sqrt[c - d*x])/(a*c*Sqrt[x]) - (2*Sqrt[d]*Sqrt[b*c + a*d]*Sqrt[x]*Sqrt[(b*(c - d*x))/(b*c + a*d)]*Ellipt icE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c + a*d]], 1 + (b*c)/(a*d)])/(a* c*Sqrt[-((b*x)/a)]*Sqrt[c - d*x])))/Sqrt[a*x + b*x^2] + (A*Sqrt[x]*Sqrt[a + b*x]*((-2*Sqrt[a + b*x]*Sqrt[c - d*x])/(3*a*c*x^(3/2)) - ((-4*(b*c - a*d )*Sqrt[a + b*x]*Sqrt[c - d*x])/(a*c*Sqrt[x]) - (b*d*((-4*Sqrt[-a]*(b*c - a *d)*Sqrt[1 + (b*x)/a]*Sqrt[c - d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[x])/Sqr t[-a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[1 - (d*x)/c]) + (2* c^(3/2)*(2*b*c - a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 - (d*x)/c]*EllipticF[ArcSin [(Sqrt[d]*Sqrt[x])/Sqrt[c]], -((b*c)/(a*d))])/(d^(3/2)*Sqrt[a + b*x]*Sqrt[ c - d*x])))/(a*c))/(3*a*c)))/Sqrt[a*x + b*x^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn omialRemainder[Px, d + e*x, x] Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x ^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x ] && LtQ[m, 0] && !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(534\) vs. \(2(263)=526\).
Time = 2.76 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.69
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-d x +c \right ) x \left (b x +a \right )}\, \left (-\frac {2 A \sqrt {-b d \,x^{3}-a d \,x^{2}+b c \,x^{2}+a c x}}{3 a c \,x^{2}}-\frac {2 \left (-b d \,x^{2}-a d x +c b x +a c \right ) \left (2 A a d -2 A b c +3 a B c \right )}{3 a^{2} c^{2} \sqrt {x \left (-b d \,x^{2}-a d x +c b x +a c \right )}}+\frac {2 \left (\frac {A b d}{3 a c}+\frac {\left (a d -b c \right ) \left (2 A a d -2 A b c +3 a B c \right )}{3 a^{2} c^{2}}+\frac {\left (-a d +b c \right ) \left (2 A a d -2 A b c +3 a B c \right )}{3 a^{2} c^{2}}\right ) a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {c}{d}-\frac {a}{b}}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \sqrt {-\frac {a}{b \left (-\frac {c}{d}-\frac {a}{b}\right )}}\right )}{b \sqrt {-b d \,x^{3}-a d \,x^{2}+b c \,x^{2}+a c x}}-\frac {2 d \left (2 A a d -2 A b c +3 a B c \right ) \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {c}{d}-\frac {a}{b}}}\, \sqrt {-\frac {x b}{a}}\, \left (\left (-\frac {c}{d}-\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \sqrt {-\frac {a}{b \left (-\frac {c}{d}-\frac {a}{b}\right )}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \sqrt {-\frac {a}{b \left (-\frac {c}{d}-\frac {a}{b}\right )}}\right )}{d}\right )}{3 a \,c^{2} \sqrt {-b d \,x^{3}-a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {-d x +c}}\) | \(535\) |
| default | \(-\frac {2 \left (A \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a^{2} b c d x -2 A \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a \,b^{2} c^{2} x -2 A \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a^{3} d^{2} x +2 A \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a \,b^{2} c^{2} x +3 B \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a^{2} b \,c^{2} x -3 B \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a^{3} c d x -3 B \sqrt {\frac {b x +a}{a}}\, \sqrt {\frac {\left (-d x +c \right ) b}{a d +b c}}\, \sqrt {-\frac {x b}{a}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +a}{a}}, \sqrt {\frac {d a}{a d +b c}}\right ) a^{2} b \,c^{2} x -2 A a \,b^{2} d^{2} x^{3}+2 A \,b^{3} c d \,x^{3}-3 B a \,b^{2} c d \,x^{3}-2 A \,a^{2} b \,d^{2} x^{2}+3 A a \,b^{2} c d \,x^{2}-2 A \,b^{3} c^{2} x^{2}-3 B \,a^{2} b c d \,x^{2}+3 B a \,b^{2} c^{2} x^{2}+A \,a^{2} b c d x -A a \,b^{2} c^{2} x +3 B \,a^{2} b \,c^{2} x +A \,a^{2} b \,c^{2}\right ) \sqrt {x \left (b x +a \right )}\, \sqrt {-d x +c}}{3 x^{2} c^{2} a^{2} b \left (-b d \,x^{2}-a d x +c b x +a c \right )}\) | \(720\) |
Input:
int((B*x+A)/x^2/(-d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
((-d*x+c)*x*(b*x+a))^(1/2)/(x*(b*x+a))^(1/2)/(-d*x+c)^(1/2)*(-2/3*A/a/c*(- b*d*x^3-a*d*x^2+b*c*x^2+a*c*x)^(1/2)/x^2-2/3*(-b*d*x^2-a*d*x+b*c*x+a*c)/a^ 2/c^2*(2*A*a*d-2*A*b*c+3*B*a*c)/(x*(-b*d*x^2-a*d*x+b*c*x+a*c))^(1/2)+2*(1/ 3*A*b*d/a/c+1/3*(a*d-b*c)*(2*A*a*d-2*A*b*c+3*B*a*c)/a^2/c^2+1/3*(-a*d+b*c) /a^2/c^2*(2*A*a*d-2*A*b*c+3*B*a*c))*a/b*((x+a/b)/a*b)^(1/2)*((x-c/d)/(-c/d -a/b))^(1/2)*(-1/a*x*b)^(1/2)/(-b*d*x^3-a*d*x^2+b*c*x^2+a*c*x)^(1/2)*Ellip ticF(((x+a/b)/a*b)^(1/2),(-a/b/(-c/d-a/b))^(1/2))-2/3*d*(2*A*a*d-2*A*b*c+3 *B*a*c)/a/c^2*((x+a/b)/a*b)^(1/2)*((x-c/d)/(-c/d-a/b))^(1/2)*(-1/a*x*b)^(1 /2)/(-b*d*x^3-a*d*x^2+b*c*x^2+a*c*x)^(1/2)*((-c/d-a/b)*EllipticE(((x+a/b)/ a*b)^(1/2),(-a/b/(-c/d-a/b))^(1/2))+c/d*EllipticF(((x+a/b)/a*b)^(1/2),(-a/ b/(-c/d-a/b))^(1/2))))
Time = 0.09 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.33 \[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=-\frac {2 \, {\left ({\left (2 \, A a^{2} d^{2} - {\left (3 \, B a b - 2 \, A b^{2}\right )} c^{2} + {\left (3 \, B a^{2} - A a b\right )} c d\right )} \sqrt {-b d} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, \frac {4 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x - b c + a d}{3 \, b d}\right ) + 3 \, {\left (2 \, A a b d^{2} + {\left (3 \, B a b - 2 \, A b^{2}\right )} c d\right )} \sqrt {-b d} x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, \frac {4 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, \frac {4 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x - b c + a d}{3 \, b d}\right )\right ) + 3 \, {\left (A a b c d + {\left (2 \, A a b d^{2} + {\left (3 \, B a b - 2 \, A b^{2}\right )} c d\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {-d x + c}\right )}}{9 \, a^{2} b c^{2} d x^{2}} \] Input:
integrate((B*x+A)/x^2/(-d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x, algorithm="frica s")
Output:
-2/9*((2*A*a^2*d^2 - (3*B*a*b - 2*A*b^2)*c^2 + (3*B*a^2 - A*a*b)*c*d)*sqrt (-b*d)*x^2*weierstrassPInverse(4/3*(b^2*c^2 + a*b*c*d + a^2*d^2)/(b^2*d^2) , 4/27*(2*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x - b*c + a*d)/(b*d)) + 3*(2*A*a*b*d^2 + (3*B*a*b - 2*A*b^2)*c* d)*sqrt(-b*d)*x^2*weierstrassZeta(4/3*(b^2*c^2 + a*b*c*d + a^2*d^2)/(b^2*d ^2), 4/27*(2*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 2*a^3*d^3)/(b^3*d^3 ), weierstrassPInverse(4/3*(b^2*c^2 + a*b*c*d + a^2*d^2)/(b^2*d^2), 4/27*( 2*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b *d*x - b*c + a*d)/(b*d))) + 3*(A*a*b*c*d + (2*A*a*b*d^2 + (3*B*a*b - 2*A*b ^2)*c*d)*x)*sqrt(b*x^2 + a*x)*sqrt(-d*x + c))/(a^2*b*c^2*d*x^2)
\[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=\int \frac {A + B x}{x^{2} \sqrt {x \left (a + b x\right )} \sqrt {c - d x}}\, dx \] Input:
integrate((B*x+A)/x**2/(-d*x+c)**(1/2)/(b*x**2+a*x)**(1/2),x)
Output:
Integral((A + B*x)/(x**2*sqrt(x*(a + b*x))*sqrt(c - d*x)), x)
\[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=\int { \frac {B x + A}{\sqrt {b x^{2} + a x} \sqrt {-d x + c} x^{2}} \,d x } \] Input:
integrate((B*x+A)/x^2/(-d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x, algorithm="maxim a")
Output:
integrate((B*x + A)/(sqrt(b*x^2 + a*x)*sqrt(-d*x + c)*x^2), x)
\[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=\int { \frac {B x + A}{\sqrt {b x^{2} + a x} \sqrt {-d x + c} x^{2}} \,d x } \] Input:
integrate((B*x+A)/x^2/(-d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x, algorithm="giac" )
Output:
integrate((B*x + A)/(sqrt(b*x^2 + a*x)*sqrt(-d*x + c)*x^2), x)
Timed out. \[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=\int \frac {A+B\,x}{x^2\,\sqrt {b\,x^2+a\,x}\,\sqrt {c-d\,x}} \,d x \] Input:
int((A + B*x)/(x^2*(a*x + b*x^2)^(1/2)*(c - d*x)^(1/2)),x)
Output:
int((A + B*x)/(x^2*(a*x + b*x^2)^(1/2)*(c - d*x)^(1/2)), x)
\[ \int \frac {A+B x}{x^2 \sqrt {c-d x} \sqrt {a x+b x^2}} \, dx=\frac {-2 \sqrt {-d x +c}\, \sqrt {b x +a}\, b +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d x +c}\, \sqrt {b x +a}}{-b d \,x^{5}-a d \,x^{4}+b c \,x^{4}+a c \,x^{3}}d x \right ) a^{2} c -\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d x +c}\, \sqrt {b x +a}}{-b d \,x^{2}-a d x +b c x +a c}d x \right ) b^{2} d}{\sqrt {x}\, a c} \] Input:
int((B*x+A)/x^2/(-d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x)
Output:
( - 2*sqrt(c - d*x)*sqrt(a + b*x)*b + sqrt(x)*int((sqrt(x)*sqrt(c - d*x)*s qrt(a + b*x))/(a*c*x**3 - a*d*x**4 + b*c*x**4 - b*d*x**5),x)*a**2*c - sqrt (x)*int((sqrt(x)*sqrt(c - d*x)*sqrt(a + b*x))/(a*c - a*d*x + b*c*x - b*d*x **2),x)*b**2*d)/(sqrt(x)*a*c)