Integrand size = 28, antiderivative size = 218 \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\frac {2 B x \sqrt {c+d x}}{d \sqrt {a x+b x^2}}-\frac {2 \sqrt {a} B \sqrt {x} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}}+\frac {2 \sqrt {a} A \sqrt {x} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c \sqrt {\frac {a (c+d x)}{c (a+b x)}} \sqrt {a x+b x^2}} \] Output:
2*B*x*(d*x+c)^(1/2)/d/(b*x^2+a*x)^(1/2)-2*a^(1/2)*B*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))/b^(1/ 2)/d/(a*(d*x+c)/c/(b*x+a))^(1/2)/(b*x^2+a*x)^(1/2)+2*a^(1/2)*A*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))/b^(1/2)/c/(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 = 13.49 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\frac {\frac {2 a B (a+b x) (c+d x)}{b}+2 i a \sqrt {\frac {a}{b}} B d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} (-A b+a B) d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{a d \sqrt {x (a+b x)} \sqrt {c+d x}} \] Input:
Integrate[(A + B*x)/(Sqrt[c + d*x]*Sqrt[a*x + b*x^2]),x]
Output:
((2*a*B*(a + b*x)*(c + d*x))/b + (2*I)*a*Sqrt[a/b]*B*d*Sqrt[1 + a/(b*x)]*S qrt[1 + c/(d*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a* d)] - (2*I)*Sqrt[a/b]*(-(A*b) + a*B)*d*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)] *x^(3/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)])/(a*d*Sqrt[x *(a + b*x)]*Sqrt[c + d*x])
Time = 0.55 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1269, 1169, 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.
\(\displaystyle \int \frac {A+B x}{\sqrt {a x+b x^2} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {B \int \frac {\sqrt {c+d x}}{\sqrt {b x^2+a x}}dx}{d}-\frac {(B c-A d) \int \frac {1}{\sqrt {c+d x} \sqrt {b x^2+a x}}dx}{d}\) |
\(\Big \downarrow \) 1169 |
\(\displaystyle \frac {B \sqrt {x} \sqrt {a+b x} \int \frac {\sqrt {c+d x}}{\sqrt {x} \sqrt {a+b x}}dx}{d \sqrt {a x+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x} (B c-A d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {B \sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} \int \frac {\sqrt {\frac {d x}{c}+1}}{\sqrt {x} \sqrt {\frac {b x}{a}+1}}dx}{d \sqrt {a x+b x^2} \sqrt {\frac {d x}{c}+1}}-\frac {\sqrt {x} \sqrt {a+b x} (B c-A d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {2 \sqrt {-a} B \sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a x+b x^2} \sqrt {\frac {d x}{c}+1}}-\frac {\sqrt {x} \sqrt {a+b x} (B c-A d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} \sqrt {c+d x}}dx}{d \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {2 \sqrt {-a} B \sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a x+b x^2} \sqrt {\frac {d x}{c}+1}}-\frac {\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (B c-A d) \int \frac {1}{\sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1}}dx}{d \sqrt {a x+b x^2} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {2 \sqrt {-a} B \sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a x+b x^2} \sqrt {\frac {d x}{c}+1}}-\frac {2 \sqrt {-a} \sqrt {x} \sqrt {\frac {b x}{a}+1} \sqrt {\frac {d x}{c}+1} (B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a x+b x^2} \sqrt {c+d x}}\) |
Input:
Int[(A + B*x)/(Sqrt[c + d*x]*Sqrt[a*x + b*x^2]),x]
Output:
(2*Sqrt[-a]*B*Sqrt[x]*Sqrt[1 + (b*x)/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sq rt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqrt[b]*d*Sqrt[1 + (d*x)/c]*Sqrt[ a*x + b*x^2]) - (2*Sqrt[-a]*(B*c - A*d)*Sqrt[x]*Sqrt[1 + (b*x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c)])/(Sqr t[b]*d*Sqrt[c + d*x]*Sqrt[a*x + b*x^2])
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[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[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 1.38 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b d -B \operatorname {EllipticF}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d +B \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a d -B \operatorname {EllipticE}\left (\sqrt {\frac {d x +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b c \right ) c \sqrt {-\frac {x d}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {\frac {d x +c}{c}}\, \sqrt {d x +c}\, \sqrt {x \left (b x +a \right )}}{d^{2} b x \left (b d \,x^{2}+a d x +c b x +a c \right )}\) | \(219\) |
elliptic | \(\frac {\sqrt {x \left (b x +a \right ) \left (d x +c \right )}\, \left (\frac {2 A c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}+\frac {2 B c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{d \sqrt {b d \,x^{3}+a d \,x^{2}+b c \,x^{2}+a c x}}\right )}{\sqrt {x \left (b x +a \right )}\, \sqrt {d x +c}}\) | \(322\) |
Input:
int((B*x+A)/(d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(A*EllipticF(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*b*d-B*EllipticF(( (d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*a*d+B*EllipticE(((d*x+c)/c)^(1/2) ,(-b*c/(a*d-b*c))^(1/2))*a*d-B*EllipticE(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c) )^(1/2))*b*c)*c*(-1/c*x*d)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*((d*x+c)/c)^( 1/2)*(d*x+c)^(1/2)*(x*(b*x+a))^(1/2)/d^2/b/x/(b*d*x^2+a*d*x+b*c*x+a*c)
Time = 0.11 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.40 \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b d} B b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right ) + {\left (B b c + {\left (B a - 3 \, A b\right )} d\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )}}{3 \, b^{2} d^{2}} \] Input:
integrate((B*x+A)/(d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x, algorithm="fricas")
Output:
-2/3*(3*sqrt(b*d)*B*b*d*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))) + (B*b*c + (B*a - 3*A*b)*d)*sqrt(b*d)*we ierstrassPInverse(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)))/(b^2*d^2)
\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\int \frac {A + B x}{\sqrt {x \left (a + b x\right )} \sqrt {c + d x}}\, dx \] Input:
integrate((B*x+A)/(d*x+c)**(1/2)/(b*x**2+a*x)**(1/2),x)
Output:
Integral((A + B*x)/(sqrt(x*(a + b*x))*sqrt(c + d*x)), x)
\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\int { \frac {B x + A}{\sqrt {b x^{2} + a x} \sqrt {d x + c}} \,d x } \] Input:
integrate((B*x+A)/(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)
\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\int { \frac {B x + A}{\sqrt {b x^{2} + a x} \sqrt {d x + c}} \,d x } \] Input:
integrate((B*x+A)/(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)
Timed out. \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {b\,x^2+a\,x}\,\sqrt {c+d\,x}} \,d x \] Input:
int((A + B*x)/((a*x + b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int((A + B*x)/((a*x + b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {a x+b x^2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, c +\sqrt {x}\, d x}d x \] Input:
int((B*x+A)/(d*x+c)^(1/2)/(b*x^2+a*x)^(1/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a + b*x))/(sqrt(x)*c + sqrt(x)*d*x),x)