Integrand size = 28, antiderivative size = 725 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 e}-\frac {2 \sqrt [4]{c} \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} (e f-3 d g) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3 e^2 g^2 \sqrt {a+c x^2}}+\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \left (2 a e^2 g-\sqrt {-a} \sqrt {c} e (e f-3 d g)-3 c d (e f-d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{3 \sqrt [4]{c} e^3 g \sqrt {a+c x^2}}-\frac {2 \left (c d^2+a e^2\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} e^3 \sqrt {a+c x^2}} \] Output:
2/3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/e-2/3*c^(1/4)*(c^(1/2)*f-(-a)^(1/2)*g)*( c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(-3*d*g+e*f)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-( -a)^(1/2)*g))^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*Ell ipticE(c^(1/4)*(g*x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(- a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/e^2/g^2/(c*x^2+a)^(1/2)+2/3*( c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(2*a*e^2*g-(-a)^(1/2)*c^(1/2)*e*(-3*d*g+e*f) -3*c*d*(-d*g+e*f))*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2)*(1-c ^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*EllipticF(c^(1/4)*(g*x+f)^( 1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-( -a)^(1/2)*g))^(1/2))/c^(1/4)/e^3/g/(c*x^2+a)^(1/2)-2*(a*e^2+c*d^2)*(c^(1/2 )*f+(-a)^(1/2)*g)^(1/2)*(1-c^(1/2)*(g*x+f)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2) *(1-c^(1/2)*(g*x+f)/(c^(1/2)*f+(-a)^(1/2)*g))^(1/2)*EllipticPi(c^(1/4)*(g* x+f)^(1/2)/(c^(1/2)*f+(-a)^(1/2)*g)^(1/2),e*(f+(-a)^(1/2)*g/c^(1/2))/(-d*g +e*f),((c^(1/2)*f+(-a)^(1/2)*g)/(c^(1/2)*f-(-a)^(1/2)*g))^(1/2))/c^(1/4)/e ^3/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.35 (sec) , antiderivative size = 1216, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx =\text {Too large to display} \] Input:
Integrate[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x),x]
Output:
(2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3*e) + ((f + g*x)^(3/2)*(2*c*e^2*f*Sqrt [-f - (I*Sqrt[a]*g)/Sqrt[c]] - 6*c*d*e*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + (2*c*e^2*f^3*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (6*c*d*e*f^ 2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 + (2*a*e^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x)^2 - (6*a*d*e*g^3*Sqrt[-f - (I*Sqrt[a] *g)/Sqrt[c]])/(f + g*x)^2 - (4*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]) /(f + g*x) + (12*c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])/(f + g*x) + ( 2*Sqrt[c]*e*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(e*f - 3*d*g)*Sqrt[1 - f/(f + g*x ) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g )/(Sqrt[c]*(f + g*x))]*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c] ]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sq rt[f + g*x] + (2*e*(3*Sqrt[c]*d - I*Sqrt[a]*e)*g*((-I)*Sqrt[c]*f + Sqrt[a] *g)*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/( f + g*x) + (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c ]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*I)*c*d^2*g^2*Sqrt[1 - f/(f + g*x) - (I*Sqrt[a]*g)/(Sqrt[c]*(f + g*x))]*Sqrt[1 - f/(f + g*x) + (I*Sqrt[a]*g) /(Sqrt[c]*(f + g*x))]*EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*S qrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sq rt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)])/Sqrt[f + g*x] + ((6*...
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.
| \(\displaystyle \int \frac {\sqrt {a+c x^2} \sqrt {f+g x}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 722 |
| \(\displaystyle \frac {\int \frac {c (e f-3 d g) x^2-2 (c d f-a e g) x+a (3 e f-d g)}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}+\int \frac {\frac {3 c g d^2}{e^2}-\frac {3 c f d}{e}+2 a g+\left (c f-\frac {3 c d g}{e}\right ) x}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}-\frac {2 \int -\frac {2 a g^2-c \left (f^2-\frac {3 d^2 g^2}{e^2}\right )+c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \int -\frac {c f^2-2 a g^2-\frac {3 c d^2 g^2}{e^2}-c \left (f-\frac {3 d g}{e}\right ) (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}+\frac {3 \left (a e^2+c d^2\right ) (e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e^2}}{3 e}+\frac {2 \sqrt {a+c x^2} \sqrt {f+g x}}{3 e}\) |
Input:
Int[(Sqrt[f + g*x]*Sqrt[a + c*x^2])/(d + e*x),x]
Output:
$Aborted
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*( x_)^2], x_Symbol] :> Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2 ]/(e*(2*m + 5))), x] + Simp[1/(e*(2*m + 5)) Int[((d + e*x)^m/(Sqrt[f + g* x]*Sqrt[a + c*x^2]))*Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 3*c*d*g)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegerQ[2*m ] && !LtQ[m, -1]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] && !IntegerQ[n ] && IntegersQ[2*m, 2*n, 2*p]
Time = 4.24 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{3 e}+\frac {2 \left (\frac {a \,e^{2} g +c \,d^{2} g -c d e f}{e^{3}}-\frac {a g}{3 e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {2 \left (-\frac {\left (d g -e f \right ) c}{e^{2}}-\frac {2 c f}{3 e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}-\frac {2 \left (a d \,e^{2} g -a \,e^{3} f +c \,d^{3} g -c \,d^{2} e f \right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{4} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(922\) |
| risch | \(\text {Expression too large to display}\) | \(1318\) |
| default | \(\text {Expression too large to display}\) | \(2496\) |
Input:
int((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/3/e*(c*g*x^3+c* f*x^2+a*g*x+a*f)^(1/2)+2*((a*e^2*g+c*d^2*g-c*d*e*f)/e^3-1/3/e*a*g)*(f/g-(- a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f /g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2) /(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c) )^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+2*(-1/e^2*(d* g-e*f)*c-2/3/e*c*f)*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1 /2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/( -f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)*((-f/g-(-a*c )^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1 /2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*EllipticF(((x+f/g)/(f/ g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/ 2)))-2*(a*d*e^2*g-a*e^3*f+c*d^3*g-c*d^2*e*f)/e^4*(f/g-(-a*c)^(1/2)/c)*((x+ f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c) )^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+ a*g*x+a*f)^(1/2)/(-f/g+d/e)*EllipticPi(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2 ),(-f/g+(-a*c)^(1/2)/c)/(-f/g+d/e),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/ 2)/c))^(1/2)))
Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {a + c x^{2}} \sqrt {f + g x}}{d + e x}\, dx \] Input:
integrate((g*x+f)**(1/2)*(c*x**2+a)**(1/2)/(e*x+d),x)
Output:
Integral(sqrt(a + c*x**2)*sqrt(f + g*x)/(d + e*x), x)
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{2} + a} \sqrt {g x + f}}{e x + d} \,d x } \] Input:
integrate((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d), x)
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{2} + a} \sqrt {g x + f}}{e x + d} \,d x } \] Input:
integrate((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d),x, algorithm="giac")
Output:
integrate(sqrt(c*x^2 + a)*sqrt(g*x + f)/(e*x + d), x)
Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}}{d+e\,x} \,d x \] Input:
int(((f + g*x)^(1/2)*(a + c*x^2)^(1/2))/(d + e*x),x)
Output:
int(((f + g*x)^(1/2)*(a + c*x^2)^(1/2))/(d + e*x), x)
\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{e x +d}d x \] Input:
int((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d),x)
Output:
int((g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(e*x+d),x)