Integrand size = 30, antiderivative size = 260 \[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {b^2 x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}+\frac {2 a b \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {e+f x^2}}\right )}{\sqrt {d} \sqrt {f}}-\frac {b^2 \sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {a^2 \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \] Output:
b^2*x*(d*x^2+c)^(1/2)/d/(f*x^2+e)^(1/2)+2*a*b*arctanh(f^(1/2)*(d*x^2+c)^(1 /2)/d^(1/2)/(f*x^2+e)^(1/2))/d^(1/2)/f^(1/2)-b^2*e^(1/2)*(d*x^2+c)^(1/2)*E llipticE(f^(1/2)*x/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))/d/f^(1/2)/ (e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+a^2*e^(1/2)*(d*x^2+c)^(1/2 )*InverseJacobiAM(arctan(f^(1/2)*x/e^(1/2)),(1-d*e/c/f)^(1/2))/c/f^(1/2)/( e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
Result contains complex when optimal does not.
Time = 4.60 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {2 a b \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x^2}}{\sqrt {f} \sqrt {c+d x^2}}\right )}{\sqrt {d} \sqrt {f}}-\frac {i b^2 e \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )}{\sqrt {\frac {d}{c}} f \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {i \left (b^2 e-a^2 f\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{\sqrt {\frac {d}{c}} f \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:
Integrate[(a + b*x)^2/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
(2*a*b*ArcTanh[(Sqrt[d]*Sqrt[e + f*x^2])/(Sqrt[f]*Sqrt[c + d*x^2])])/(Sqrt [d]*Sqrt[f]) - (I*b^2*e*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[ I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]) + (I*(b^2*e - a^2*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Ell ipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*f*Sqrt[c + d*x^2]* Sqrt[e + f*x^2])
Time = 0.76 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {2 a b x}{\sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b^2 x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {2 a b \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {e+f x^2}}\right )}{\sqrt {d} \sqrt {f}}-\frac {b^2 \sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {b^2 x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}\) |
Input:
Int[(a + b*x)^2/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
(b^2*x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2]) + (2*a*b*ArcTanh[(Sqrt[f]*Sqrt [c + d*x^2])/(Sqrt[d]*Sqrt[e + f*x^2])])/(Sqrt[d]*Sqrt[f]) - (b^2*Sqrt[e]* Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/( d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (a^2*Sq rt[e]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c* f)])/(c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
Time = 6.25 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.19
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {a^{2} \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {a b \ln \left (\frac {2 d f \,x^{2}+c f +d e}{\sqrt {d f}}+2 \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\right )}{\sqrt {d f}}-\frac {b^{2} e \sqrt {1+\frac {x^{2} d}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(309\) |
| default | \(\frac {\left (\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) \sqrt {d f}\, a^{2} f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) \sqrt {d f}\, b^{2} e +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \sqrt {d f}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b^{2} e +a b \ln \left (\frac {2 d f \,x^{2}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \sqrt {d f}+c f +d e}{\sqrt {d f}}\right ) \sqrt {-\frac {d}{c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, f \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{f \sqrt {d f}\, \sqrt {-\frac {d}{c}}\, \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(310\) |
Input:
int((b*x+a)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x,method=_RETURNVERBOSE)
Output:
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(a^2/(-d/c)^(1 /2)*(1+1/c*x^2*d)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1 /2)*EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+a*b*ln((2*d*f*x^2+c *f+d*e)/(d*f)^(1/2)+2*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2))/(d*f)^(1/2)-b^2 *e/(-d/c)^(1/2)*(1+1/c*x^2*d)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e *x^2+c*e)^(1/2)/f*(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))-Elli pticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))))
Time = 0.46 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {2 \, \sqrt {d f} b^{2} e^{2} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - \sqrt {d f} a b e f x \log \left (-8 \, d^{2} f^{2} x^{4} - d^{2} e^{2} - 6 \, c d e f - c^{2} f^{2} - 8 \, {\left (d^{2} e f + c d f^{2}\right )} x^{2} - 4 \, {\left (2 \, d f x^{2} + d e + c f\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} \sqrt {d f}\right ) - 2 \, \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} b^{2} e f - 2 \, {\left (b^{2} e^{2} + a^{2} f^{2}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e})}{2 \, d e f^{2} x} \] Input:
integrate((b*x+a)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="fricas")
Output:
-1/2*(2*sqrt(d*f)*b^2*e^2*x*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x), c* f/(d*e)) - sqrt(d*f)*a*b*e*f*x*log(-8*d^2*f^2*x^4 - d^2*e^2 - 6*c*d*e*f - c^2*f^2 - 8*(d^2*e*f + c*d*f^2)*x^2 - 4*(2*d*f*x^2 + d*e + c*f)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*sqrt(d*f)) - 2*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*b^2*e *f - 2*(b^2*e^2 + a^2*f^2)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_f(arcsin(sqrt(- e/f)/x), c*f/(d*e)))/(d*e*f^2*x)
\[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {\left (a + b x\right )^{2}}{\sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \] Input:
integrate((b*x+a)**2/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
Output:
Integral((a + b*x)**2/(sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)
\[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{2}}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x+a)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x + a)^2/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
\[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {{\left (b x + a\right )}^{2}}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((b*x+a)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorithm="giac")
Output:
integrate((b*x + a)^2/(sqrt(d*x^2 + c)*sqrt(f*x^2 + e)), x)
Timed out. \[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {{\left (a+b\,x\right )}^2}{\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:
int((a + b*x)^2/((c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)),x)
Output:
int((a + b*x)^2/((c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
\[ \int \frac {(a+b x)^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {2 \sqrt {f}\, \sqrt {d}\, \mathrm {log}\left (-\sqrt {d}\, \sqrt {d \,x^{2}+c}\, f -\sqrt {f}\, \sqrt {f \,x^{2}+e}\, d \right ) a b +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, x^{2}}{d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}d x \right ) b^{2} d f +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}d x \right ) a^{2} d f}{d f} \] Input:
int((b*x+a)^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
Output:
(2*sqrt(f)*sqrt(d)*log( - sqrt(d)*sqrt(c + d*x**2)*f - sqrt(f)*sqrt(e + f* x**2)*d)*a*b + int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*x**2)/(c*e + c*f*x** 2 + d*e*x**2 + d*f*x**4),x)*b**2*d*f + int((sqrt(e + f*x**2)*sqrt(c + d*x* *2))/(c*e + c*f*x**2 + d*e*x**2 + d*f*x**4),x)*a**2*d*f)/(d*f)