Integrand size = 31, antiderivative size = 346 \[ \int \frac {A+B x}{x^2 \sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\frac {2 b (3 a B c-2 A (b c+a d)) x \sqrt {c+d x}}{3 a^2 c^2 \sqrt {a x+b x^2}}-\frac {2 A \sqrt {c+d x} \sqrt {a x+b x^2}}{3 a c x^2}-\frac {2 (3 a B c-2 A (b c+a d)) \sqrt {c+d x} \sqrt {a x+b x^2}}{3 a^2 c^2 x}-\frac {2 \sqrt {b} (3 a B c-2 A (b c+a d)) \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}-\frac {2 A \sqrt {b} d \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} c^2 \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:
2/3*b*(3*B*a*c-2*A*(a*d+b*c))*x*(d*x+c)^(1/2)/a^2/c^2/(b*x^2+a*x)^(1/2)-2/ 3*A*(d*x+c)^(1/2)*(b*x^2+a*x)^(1/2)/a/c/x^2-2/3*(3*B*a*c-2*A*(a*d+b*c))*(d *x+c)^(1/2)*(b*x^2+a*x)^(1/2)/a^2/c^2/x-2/3*b^(1/2)*(3*B*a*c-2*A*(a*d+b*c) )*x^(1/2)*(d*x+c)^(1/2)*EllipticE(b^(1/2)*x^(1/2)/a^(1/2)/(1+b*x/a)^(1/2), (1-a*d/b/c)^(1/2))/a^(3/2)/c^2/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/ 2)-2/3*A*b^(1/2)*d*x^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x^ (1/2)/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/c^2/(a*(d*x+c)/c/(b*x+a))^(1/2)/ (b*x^2+a*x)^(1/2)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x}{x^2 \sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\frac {-2 a A c (a+b x) (c+d x)-2 i \sqrt {\frac {a}{b}} b d (-3 a B c+2 A (b c+a d)) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )+2 i \sqrt {\frac {a}{b}} b d (A b c-3 a B c+2 a A d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{3 a^2 c^2 x \sqrt {x (a+b x)} \sqrt {c+d x}} \] 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) - (2*I)*Sqrt[a/b]*b*d*(-3*a*B*c + 2*A*(b*c + a*d))*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x^(5/2)*EllipticE[I*ArcSinh[Sqr t[a/b]/Sqrt[x]], (b*c)/(a*d)] + (2*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[Sqrt [a/b]/Sqrt[x]], (b*c)/(a*d)])/(3*a^2*c^2*x*Sqrt[x*(a + b*x)]*Sqrt[c + d*x] )
Time = 1.31 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.41, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, 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 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\int \frac {2 (b c+a d)+b d x}{x^{3/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {b d \sqrt {x} \sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {-\frac {b x}{a}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\int \frac {2 (b c+a d)+b d x}{x^{3/2} \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {-\frac {2 \int -\frac {b d (a c+2 (b c+a d) x)}{2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\frac {b d \int \frac {a c+2 (b c+a d) x}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\frac {b d \left (\frac {2 (a d+b c) \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {c (a d+2 b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\frac {b d \left (\frac {2 \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (a d+2 b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\frac {b d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c (a d+2 b c) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\frac {b d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (a d+2 b c) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {A \sqrt {x} \sqrt {a+b x} \left (-\frac {\frac {b d \left (\frac {4 \sqrt {-a} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} (a d+b c) E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} c \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (a d+2 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a+b x} \sqrt {c+d x}}\right )}{a c}-\frac {4 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c \sqrt {x}}}{3 a c}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{3 a c x^{3/2}}\right )}{\sqrt {a x+b x^2}}+\frac {B \sqrt {x} \sqrt {a+b x} \left (\frac {2 \sqrt {d} \sqrt {x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|1-\frac {b c}{a d}\right )}{a c \sqrt {-\frac {b x}{a}} \sqrt {c+d x}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x}}{a c \sqrt {x}}\right )}{\sqrt {a x+b x^2}}\) |
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)]*Ell ipticE[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]*S qrt[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] )/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - ( 2*Sqrt[-a]*c*(2*b*c + a*d)*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[A rcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*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]
Time = 2.63 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (-\frac {2 A \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}{3 a c \,x^{2}}+\frac {2 \left (b d \,x^{2}+a d x +c b x +a c \right ) \left (2 A a d +2 A b c -3 a B c \right )}{3 a^{2} c^{2} \sqrt {x \left (b d \,x^{2}+a d x +c b x +a c \right )}}-\frac {2 A b \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{3 a \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}-\frac {2 b \left (2 A a d +2 A b c -3 a B c \right ) \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{3 a^{2} c \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\) | \(441\) |
| default | \(\frac {2 \left (2 A \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2} x +A \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{2} d x -2 A \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c \,d^{2} x +2 A \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b^{2} c^{3} x -3 B \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c^{2} d x +3 B \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a^{2} c^{2} d x -3 B \sqrt {\frac {d x +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a b \,c^{3} x +2 A a b \,d^{3} x^{3}+2 A \,b^{2} c \,d^{2} x^{3}-3 B a b c \,d^{2} x^{3}+2 A \,a^{2} d^{3} x^{2}+3 A a b c \,d^{2} x^{2}+2 A \,b^{2} c^{2} d \,x^{2}-3 B \,a^{2} c \,d^{2} x^{2}-3 B a b \,c^{2} d \,x^{2}+A \,a^{2} c \,d^{2} x +A a b \,c^{2} d x -3 B \,a^{2} c^{2} d x -a^{2} A \,c^{2} d \right ) \sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}{3 x^{2} c^{2} a^{2} d \left (b d \,x^{2}+a d x +c b x +a c \right )}\) | \(731\) |
Input:
int((B*x+A)/x^2/(d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(x*(b*x+a)*(d*x+c))^(1/2)/(x*(b*x+a))^(1/2)/(d*x+c)^(1/2)*(-2/3*A/a/c/x^2* (b*d*x^3+a*d*x^2+b*c*x^2+a*c*x)^(1/2)+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)-2/3*A*b/a* ((x+c/d)/c*d)^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)*(-1/c*x*d)^(1/2)/(b*d*x^3+a *d*x^2+b*c*x^2+a*c*x)^(1/2)*EllipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b) )^(1/2))-2/3*b*(2*A*a*d+2*A*b*c-3*B*a*c)/a^2/c*((x+c/d)/c*d)^(1/2)*((x+a/b )/(-c/d+a/b))^(1/2)*(-1/c*x*d)^(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+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))-a/b*El lipticF(((x+c/d)/c*d)^(1/2),(-c/d/(-c/d+a/b))^(1/2))))
Time = 0.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x}{x^2 \sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\frac {2 \, {\left ({\left (2 \, A a^{2} d^{2} - {\left (3 \, B a b - 2 \, A b^{2}\right )} c^{2} - {\left (3 \, B a^{2} - A a b\right )} c d\right )} \sqrt {b d} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (2 \, A a b d^{2} - {\left (3 \, B a b - 2 \, A b^{2}\right )} c d\right )} \sqrt {b d} x^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) - 3 \, {\left (A a b c d - {\left (2 \, A a b d^{2} - {\left (3 \, B a b - 2 \, A b^{2}\right )} c d\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {d x + c}\right )}}{9 \, a^{2} b c^{2} d x^{2}} \] Input:
integrate((B*x+A)/x^2/(d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x, algorithm="fricas ")
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="maxima ")
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)