Integrand size = 28, antiderivative size = 754 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {\sqrt [4]{c} \left (\sqrt {c} f-\sqrt {-a} g\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} E\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right )|\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\left (c d^2+a e^2\right ) g \sqrt {a+c x^2}}+\frac {\sqrt [4]{c} \left (\sqrt {c} d-\sqrt {-a} e\right ) \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{e \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \left (a e^2 g+c d (2 e f-d g)\right ) \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} e \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}} \] Output:
-e*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)-c^(1/4)*(c^(1/2)*f- (-a)^(1/2)*g)*(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)*E llipticE(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))/(a*e^2+c*d^2)/g/(c*x^2+a)^( 1/2)+c^(1/4)*(c^(1/2)*d-(-a)^(1/2)*e)*(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)*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))/e/( a*e^2+c*d^2)/(c*x^2+a)^(1/2)-(c^(1/2)*f+(-a)^(1/2)*g)^(1/2)*(a*e^2*g+c*d*( -d*g+2*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)*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/(a*e^2+c*d ^2)/(-d*g+e*f)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 28.09 (sec) , antiderivative size = 1330, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[f + g*x]/((d + e*x)^2*Sqrt[a + c*x^2]),x]
Output:
(Sqrt[f + g*x]*(-((e^2*(a + c*x^2))/(d + e*x)) - (-(c*e^2*f^3*Sqrt[-f - (I *Sqrt[a]*g)/Sqrt[c]]) + c*d*e*f^2*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] - a*e ^2*f*g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]] + a*d*e*g^3*Sqrt[-f - (I*Sqrt[a] *g)/Sqrt[c]] + 2*c*e^2*f^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - 2* c*d*e*f*g*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x) - c*e^2*f*Sqrt[-f - ( I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)^2 + c*d*e*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) + d*g)*Sqr t[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*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)] + e*(I*Sqrt[c]*d + Sqrt[a]*e)*g*(Sqrt[c]*f + I*Sqrt[a]*g)*Sqrt[(g* ((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x) /(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/S qrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g )] - (2*I)*c*d*e*f*g*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-( ((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticPi[(Sqrt [c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f - (I*Sqr t[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I* Sqrt[a]*g)] + I*c*d^2*g^2*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sq rt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*Elliptic...
Leaf count is larger than twice the leaf count of optimal. \(1567\) vs. \(2(754)=1508\).
Time = 4.69 (sec) , antiderivative size = 1567, normalized size of antiderivative = 2.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {737, 25, 2349, 599, 27, 729, 25, 1511, 1416, 1509, 1540, 1416, 2222}
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.
| \(\displaystyle \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 737 |
| \(\displaystyle -\frac {\int -\frac {c e g x^2+2 c d g x+2 c d f+a e g}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c e g x^2+2 c d g x+2 c d f+a e g}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx+\int \frac {c x g+\frac {c d g}{e}}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 \int \frac {c g (e f-d g-e (f+g x))}{e \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{g^2}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx-\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e g}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 729 |
| \(\displaystyle \frac {2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e g}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {2 c \int \frac {e f-d g-e (f+g x)}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e g}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {-2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {2 c \left (\frac {e \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\left (d g-e \left (f-\frac {\sqrt {a g^2+c f^2}}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}\right )}{e g}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {-\frac {2 c \left (\frac {e \sqrt {a g^2+c f^2} \int \frac {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{\sqrt {c}}-\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (d g-e \left (f-\frac {\sqrt {a g^2+c f^2}}{\sqrt {c}}\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{e g}-2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {-2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}-\frac {2 c \left (\frac {e \sqrt {a g^2+c f^2} \left (\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}-\frac {\sqrt {f+g x} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{a g^2+c f^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right ) \sqrt {\frac {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}{\left (a+\frac {c f^2}{g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {a g^2+c f^2}}+1\right )^2}} \left (d g-e \left (f-\frac {\sqrt {a g^2+c f^2}}{\sqrt {c}}\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}}}\right )}{e g}}{2 \left (a e^2+c d^2\right )}-\frac {e \sqrt {a+c x^2} \sqrt {f+g x}}{(d+e x) \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1}{(e f-d g-e (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 \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt {c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \int \frac {1}{\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 \sqrt {c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right )}\right )-\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \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}}-\frac {\sqrt {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}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \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}}\right )}{e g}}{2 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{\left (c d^2+a e^2\right ) (d+e x)}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1}{(e f-d g-e (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 \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right ) \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}}\right )-\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \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}}-\frac {\sqrt {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}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \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}}\right )}{e g}}{2 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{\left (c d^2+a e^2\right ) (d+e x)}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {2 \left (a e g+\frac {c d (2 e f-d g)}{e}\right ) \left (\frac {e \sqrt {c f^2+a g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+a g^2}\right ) \left (\frac {\left (e+\frac {\sqrt {c} (e f-d g)}{\sqrt {c f^2+a g^2}}\right ) \text {arctanh}\left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {e} \sqrt {e f-d g} \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}}\right )}{2 \sqrt {e} \sqrt {c d^2+a e^2} \sqrt {e f-d g}}-\frac {\left (\frac {\sqrt {c}}{e}-\frac {\sqrt {c f^2+a g^2}}{e f-d g}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c f^2+a g^2} e+\sqrt {c} (e f-d g)\right )^2}{4 \sqrt {c} e (e f-d g) \sqrt {c f^2+a g^2}},2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{4 \sqrt [4]{c} \sqrt [4]{c f^2+a g^2} \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}}\right )}{g \left (a g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+a e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+a g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+a g^2} \left (a g e^2+c d (2 e f-d g)\right ) \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}}\right )-\frac {2 c \left (\frac {e \sqrt {c f^2+a g^2} \left (\frac {\sqrt [4]{c f^2+a g^2} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{\sqrt [4]{c} \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}}-\frac {\sqrt {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}}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{c f^2+a g^2} \left (d g-e \left (f-\frac {\sqrt {c f^2+a g^2}}{\sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}{\left (\frac {c f^2}{g^2}+a\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+a g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+a g^2}}+1\right )\right )}{2 \sqrt [4]{c} \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}}\right )}{e g}}{2 \left (c d^2+a e^2\right )}-\frac {e \sqrt {f+g x} \sqrt {c x^2+a}}{\left (c d^2+a e^2\right ) (d+e x)}\) |
Input:
Int[Sqrt[f + g*x]/((d + e*x)^2*Sqrt[a + c*x^2]),x]
Output:
-((e*Sqrt[f + g*x]*Sqrt[a + c*x^2])/((c*d^2 + a*e^2)*(d + e*x))) + ((-2*c* ((e*Sqrt[c*f^2 + a*g^2]*(-((Sqrt[f + g*x]*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2]))) + ((c*f^2 + a*g^2)^(1/4)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f ^2 + a*g^2])^2)]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2 )^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(c^(1/4)*Sqrt[a + (c*f ^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])))/Sqrt[c] - ((c*f^ 2 + a*g^2)^(1/4)*(d*g - e*(f - Sqrt[c*f^2 + a*g^2]/Sqrt[c]))*(1 + (Sqrt[c] *(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x)) /g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sq rt[c*f^2 + a*g^2])^2)]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(2*c^(1/4)*Sqrt[ a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])))/(e*g) + 2*(a*e*g + (c*d*(2*e*f - d*g))/e)*(-1/2*(c^(1/4)*(c*e*f^2 + a*e*g^2 - Sqrt [c]*(e*f - d*g)*Sqrt[c*f^2 + a*g^2])*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g ^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*E llipticF[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + ...
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_))/(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[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) ^2]), x_Symbol] :> Simp[2 Subst[Int[1/((d*e - c*f + f*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] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)* (x_)^2], x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^ 2]/((m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/(2*(m + 1)*(c*d^2 + a*e^2)) In t[((d + e*x)^(m + 1)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[2*c*d*f*(m + 1) - e*(a*g) + 2*c*(d*g*(m + 1) - e*f*(m + 2))*x - c*e*g*(2*m + 5)*x^2, x], x] , x] /; FreeQ[{a, c, d, e, f, g}, x] && IntegerQ[2*m] && LeQ[m, -2]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
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 = 4.80 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}{\left (a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )}+\frac {c d g \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 )}{\left (a \,e^{2}+c \,d^{2}\right ) e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {c g \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 )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}+\frac {\left (a \,e^{2} g -c \,d^{2} g +2 c d 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^{2} \left (a \,e^{2}+c \,d^{2}\right ) \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}}\) | \(922\) |
| default | \(\text {Expression too large to display}\) | \(5743\) |
Input:
int((g*x+f)^(1/2)/(e*x+d)^2/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(-e/(a*e^2+c*d^2)* (c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)/(e*x+d)+c*d*g/(a*e^2+c*d^2)/e*(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))+c*g/(a*e^2+c*d^2 )*(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)*Ellipti cE(((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)))+(a*e^2*g-c*d^ 2*g+2*c*d*e*f)/e^2/(a*e^2+c*d^2)*(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. \[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)^(1/2)/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f + g x}}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate((g*x+f)**(1/2)/(e*x+d)**2/(c*x**2+a)**(1/2),x)
Output:
Integral(sqrt(f + g*x)/(sqrt(a + c*x**2)*(d + e*x)**2), x)
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((g*x+f)^(1/2)/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(g*x + f)/(sqrt(c*x^2 + a)*(e*x + d)^2), x)
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((g*x+f)^(1/2)/(e*x+d)^2/(c*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(g*x + f)/(sqrt(c*x^2 + a)*(e*x + d)^2), x)
Timed out. \[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {f+g\,x}}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \] Input:
int((f + g*x)^(1/2)/((a + c*x^2)^(1/2)*(d + e*x)^2),x)
Output:
int((f + g*x)^(1/2)/((a + c*x^2)^(1/2)*(d + e*x)^2), x)
\[ \int \frac {\sqrt {f+g x}}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c \,e^{2} x^{4}+2 c d e \,x^{3}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+2 a d e x +a \,d^{2}}d x \] Input:
int((g*x+f)^(1/2)/(e*x+d)^2/(c*x^2+a)^(1/2),x)
Output:
int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*d**2 + 2*a*d*e*x + a*e**2*x**2 + c *d**2*x**2 + 2*c*d*e*x**3 + c*e**2*x**4),x)