\(\int \frac {f+g x}{\sqrt {d+e x} (a+c x^2)} \, dx\) [94]

Optimal result

Integrand size = 24, antiderivative size = 447 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=-\frac {\left (g+\frac {\sqrt {c} (e f-d g)}{\sqrt {c d^2+a e^2}}\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {-\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {-\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (g+\frac {\sqrt {c} (e f-d g)}{\sqrt {c d^2+a e^2}}\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {-\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} c^{3/4} \sqrt {-\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (g-\frac {\sqrt {c} (e f-d g)}{\sqrt {c d^2+a e^2}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt {c d^2+a e^2}+\sqrt {c} (d+e x)}\right )}{\sqrt {2} c^{3/4} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \] Output:

-1/2*(g+c^(1/2)*(-d*g+e*f)/(a*e^2+c*d^2)^(1/2))*arctan(((c^(1/2)*d+(a*e^2+ 
c*d^2)^(1/2))^(1/2)-2^(1/2)*c^(1/4)*(e*x+d)^(1/2))/(-c^(1/2)*d+(a*e^2+c*d^ 
2)^(1/2))^(1/2))*2^(1/2)/c^(3/4)/(-c^(1/2)*d+(a*e^2+c*d^2)^(1/2))^(1/2)+1/ 
2*(g+c^(1/2)*(-d*g+e*f)/(a*e^2+c*d^2)^(1/2))*arctan(((c^(1/2)*d+(a*e^2+c*d 
^2)^(1/2))^(1/2)+2^(1/2)*c^(1/4)*(e*x+d)^(1/2))/(-c^(1/2)*d+(a*e^2+c*d^2)^ 
(1/2))^(1/2))*2^(1/2)/c^(3/4)/(-c^(1/2)*d+(a*e^2+c*d^2)^(1/2))^(1/2)-1/2*( 
g-c^(1/2)*(-d*g+e*f)/(a*e^2+c*d^2)^(1/2))*arctanh(2^(1/2)*c^(1/4)*(c^(1/2) 
*d+(a*e^2+c*d^2)^(1/2))^(1/2)*(e*x+d)^(1/2)/((a*e^2+c*d^2)^(1/2)+c^(1/2)*( 
e*x+d)))*2^(1/2)/c^(3/4)/(c^(1/2)*d+(a*e^2+c*d^2)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.69 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.47 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\frac {i \left (\frac {\left (\sqrt {c} f+i \sqrt {a} g\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {\left (\sqrt {c} f-i \sqrt {a} g\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {-c d+i \sqrt {a} \sqrt {c} e}}\right )}{\sqrt {a} \sqrt {c}} \] Input:

Integrate[(f + g*x)/(Sqrt[d + e*x]*(a + c*x^2)),x]
 

Output:

(I*(((Sqrt[c]*f + I*Sqrt[a]*g)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]* 
Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c] 
*e] - ((Sqrt[c]*f - I*Sqrt[a]*g)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e 
]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[ 
c]*e]))/(Sqrt[a]*Sqrt[c])
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {654, 1483, 27, 1142, 25, 27, 1083, 219, 1103}

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.

\(\displaystyle \int \frac {f+g x}{\left (a+c x^2\right ) \sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 654

\(\displaystyle 2 \int \frac {e f-d g+g (d+e x)}{c d^2-2 c (d+e x) d+a e^2+c (d+e x)^2}d\sqrt {d+e x}\)

\(\Big \downarrow \) 1483

\(\displaystyle 2 \left (\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} (e f-d g)+\sqrt [4]{c} \left (e f-d g-\frac {\sqrt {c d^2+a e^2} g}{\sqrt {c}}\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} (e f-d g)-\sqrt [4]{c} \left (e f-\left (d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) g\right ) \sqrt {d+e x}}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} (e f-d g)+\sqrt [4]{c} \left (e f-d g-\frac {\sqrt {c d^2+a e^2} g}{\sqrt {c}}\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\int \frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} (e f-d g)-\sqrt [4]{c} \left (e f-\left (d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) g\right ) \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle 2 \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\sqrt {c d^2+a e^2} g+\sqrt {c} (e f-d g)\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}-\frac {1}{2} \sqrt [4]{c} \left (e f-\left (d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) g\right ) \int -\frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\sqrt {c d^2+a e^2} g+\sqrt {c} (e f-d g)\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (e f-d g-\frac {\sqrt {c d^2+a e^2} g}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\sqrt {c d^2+a e^2} g+\sqrt {c} (e f-d g)\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (e f-\left (d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) g\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\sqrt {c d^2+a e^2} g+\sqrt {c} (e f-d g)\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {1}{2} \sqrt [4]{c} \left (e f-d g-\frac {\sqrt {c d^2+a e^2} g}{\sqrt {c}}\right ) \int \frac {\sqrt {2} \left (\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}\right )}{\sqrt [4]{c} \left (d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}\right )}d\sqrt {d+e x}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \left (g \sqrt {a e^2+c d^2}+\sqrt {c} (e f-d g)\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {\left (e f-g \left (\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}+d\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \left (g \sqrt {a e^2+c d^2}+\sqrt {c} (e f-d g)\right ) \int \frac {1}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2} \sqrt {c}}+\frac {\left (-\frac {g \sqrt {a e^2+c d^2}}{\sqrt {c}}-d g+e f\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle 2 \left (\frac {\frac {\left (e f-g \left (\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}+d\right )\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \left (g \sqrt {a e^2+c d^2}+\sqrt {c} (e f-d g)\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {c}}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\frac {\left (-\frac {g \sqrt {a e^2+c d^2}}{\sqrt {c}}-d g+e f\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \left (g \sqrt {a e^2+c d^2}+\sqrt {c} (e f-d g)\right ) \int \frac {1}{-d+2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-e x}d\left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {c}}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (\frac {\frac {\left (e f-\left (d+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right ) g\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\sqrt {c d^2+a e^2} g+\sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\frac {\left (e f-d g-\frac {\sqrt {c d^2+a e^2} g}{\sqrt {c}}\right ) \int \frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{d+e x+\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}}{\sqrt [4]{c}}}d\sqrt {d+e x}}{\sqrt {2}}-\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\sqrt {c d^2+a e^2} g+\sqrt {c} (e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (\frac {-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (2 \sqrt {d+e x}-\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right ) \left (g \sqrt {a e^2+c d^2}+\sqrt {c} (e f-d g)\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {1}{2} \sqrt [4]{c} \left (e f-g \left (\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}+d\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\frac {1}{2} \sqrt [4]{c} \left (-\frac {g \sqrt {a e^2+c d^2}}{\sqrt {c}}-d g+e f\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )-\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d} \text {arctanh}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\sqrt {2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right ) \left (g \sqrt {a e^2+c d^2}+\sqrt {c} (e f-d g)\right )}{\sqrt [4]{c} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}}{2 \sqrt {2} \sqrt {c} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}\right )\)

Input:

Int[(f + g*x)/(Sqrt[d + e*x]*(a + c*x^2)),x]
 

Output:

2*((-((Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*(Sqrt[c*d^2 + a*e^2]*g + Sqrt 
[c]*(e*f - d*g))*ArcTanh[(c^(1/4)*(-((Sqrt[2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 
+ a*e^2]])/c^(1/4)) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqrt[c]*d - Sqrt[c*d 
^2 + a*e^2]])])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]])) - (c^(1/4 
)*(e*f - (d + Sqrt[c*d^2 + a*e^2]/Sqrt[c])*g)*Log[Sqrt[c*d^2 + a*e^2] - Sq 
rt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c 
]*(d + e*x)])/2)/(2*Sqrt[2]*Sqrt[c]*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + S 
qrt[c*d^2 + a*e^2]]) + (-((Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*(Sqrt[c*d 
^2 + a*e^2]*g + Sqrt[c]*(e*f - d*g))*ArcTanh[(c^(1/4)*((Sqrt[2]*Sqrt[Sqrt[ 
c]*d + Sqrt[c*d^2 + a*e^2]])/c^(1/4) + 2*Sqrt[d + e*x]))/(Sqrt[2]*Sqrt[Sqr 
t[c]*d - Sqrt[c*d^2 + a*e^2]])])/(c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a* 
e^2]])) + (c^(1/4)*(e*f - d*g - (Sqrt[c*d^2 + a*e^2]*g)/Sqrt[c])*Log[Sqrt[ 
c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqr 
t[d + e*x] + Sqrt[c]*(d + e*x)])/2)/(2*Sqrt[2]*Sqrt[c]*Sqrt[c*d^2 + a*e^2] 
*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1483

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r)   In 
t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r)   Int[(d*r 
 + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N 
eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.51

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}-2 c d}\, \left (\left (-\sqrt {a \,e^{2}+c \,d^{2}}\, g -\sqrt {c}\, \left (d g -e f \right )\right ) \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+\left (\sqrt {a \,e^{2}+c \,d^{2}}\, c g +c^{\frac {3}{2}} \left (d g -e f \right )\right ) d \right ) \sqrt {2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+2 c d}\, \ln \left (\sqrt {c}\, \left (e x +d \right )-\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+2 c d}+\sqrt {a \,e^{2}+c \,d^{2}}\right )}{4}-\frac {\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}-2 c d}\, \left (\left (-\sqrt {a \,e^{2}+c \,d^{2}}\, g -\sqrt {c}\, \left (d g -e f \right )\right ) \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+\left (\sqrt {a \,e^{2}+c \,d^{2}}\, c g +c^{\frac {3}{2}} \left (d g -e f \right )\right ) d \right ) \sqrt {2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+2 c d}\, \ln \left (\sqrt {c}\, \left (e x +d \right )+\sqrt {e x +d}\, \sqrt {2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+2 c d}+\sqrt {a \,e^{2}+c \,d^{2}}\right )}{4}+e^{2} \left (-\sqrt {a \,e^{2}+c \,d^{2}}\, c g +c^{\frac {3}{2}} \left (d g -e f \right )\right ) \left (\arctan \left (\frac {2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}-2 c d}}\right )-\arctan \left (\frac {-2 \sqrt {c}\, \sqrt {e x +d}+\sqrt {2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}+2 c d}}{\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}-2 c d}}\right )\right ) a}{\sqrt {a \,e^{2}+c \,d^{2}}\, c^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {c}-2 \sqrt {\left (a \,e^{2}+c \,d^{2}\right ) c}-2 c d}\, a \,e^{2}}\)	\(673\)
derivativedivides	\(\text {Expression too large to display}\)	\(1577\)
default	\(\text {Expression too large to display}\)	\(1577\)

Input:

int((g*x+f)/(e*x+d)^(1/2)/(c*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/(a*e^2+c*d^2)^(1/2)/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c* 
d^2)*c)^(1/2)-2*c*d)^(1/2)*(1/4*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c 
*d^2)*c)^(1/2)-2*c*d)^(1/2)*((-(a*e^2+c*d^2)^(1/2)*g-c^(1/2)*(d*g-e*f))*(( 
a*e^2+c*d^2)*c)^(1/2)+((a*e^2+c*d^2)^(1/2)*c*g+c^(3/2)*(d*g-e*f))*d)*(2*(( 
a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*ln(c^(1/2)*(e*x+d)-(e*x+d)^(1/2)*(2*((a 
*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))-1/4*(4*(a*e^2+c*d^2 
)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*((-(a*e^2+c*d^2)^(1 
/2)*g-c^(1/2)*(d*g-e*f))*((a*e^2+c*d^2)*c)^(1/2)+((a*e^2+c*d^2)^(1/2)*c*g+ 
c^(3/2)*(d*g-e*f))*d)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*ln(c^(1/2)*( 
e*x+d)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2) 
^(1/2))+e^2*(-(a*e^2+c*d^2)^(1/2)*c*g+c^(3/2)*(d*g-e*f))*(arctan((2*c^(1/2 
)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^ 
(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2))-arctan((-2*c^(1/2)*( 
e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/ 
2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))*a)/a/e^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2385 vs. \(2 (357) = 714\).

Time = 0.31 (sec) , antiderivative size = 2385, normalized size of antiderivative = 5.34 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)/(e*x+d)^(1/2)/(c*x^2+a),x, algorithm="fricas")
 

Output:

-1/2*sqrt(-(c*d*f^2 + 2*a*e*f*g - a*d*g^2 + (a*c^2*d^2 + a^2*c*e^2)*sqrt(- 
(c^2*e^2*f^4 - 4*c^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e^2*g^4 + 2*(2*c^2* 
d^2 - a*c*e^2)*f^2*g^2)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a 
*c^2*d^2 + a^2*c*e^2))*log(-(c^2*e*f^4 - 2*c^2*d*f^3*g - 2*a*c*d*f*g^3 - a 
^2*e*g^4)*sqrt(e*x + d) + (a*c^2*e^2*f^3 - 3*a*c^2*d*e*f^2*g + a^2*c*d*e*g 
^3 + (2*a*c^2*d^2 - a^2*c*e^2)*f*g^2 + ((a*c^4*d^3 + a^2*c^3*d*e^2)*f + (a 
^2*c^3*d^2*e + a^3*c^2*e^3)*g)*sqrt(-(c^2*e^2*f^4 - 4*c^2*d*e*f^3*g + 4*a* 
c*d*e*f*g^3 + a^2*e^2*g^4 + 2*(2*c^2*d^2 - a*c*e^2)*f^2*g^2)/(a*c^5*d^4 + 
2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))*sqrt(-(c*d*f^2 + 2*a*e*f*g - a*d*g^2 + 
(a*c^2*d^2 + a^2*c*e^2)*sqrt(-(c^2*e^2*f^4 - 4*c^2*d*e*f^3*g + 4*a*c*d*e*f 
*g^3 + a^2*e^2*g^4 + 2*(2*c^2*d^2 - a*c*e^2)*f^2*g^2)/(a*c^5*d^4 + 2*a^2*c 
^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 + a^2*c*e^2))) + 1/2*sqrt(-(c*d*f^2 
 + 2*a*e*f*g - a*d*g^2 + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(c^2*e^2*f^4 - 4*c^ 
2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e^2*g^4 + 2*(2*c^2*d^2 - a*c*e^2)*f^2* 
g^2)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 + a^2*c*e^ 
2))*log(-(c^2*e*f^4 - 2*c^2*d*f^3*g - 2*a*c*d*f*g^3 - a^2*e*g^4)*sqrt(e*x 
+ d) - (a*c^2*e^2*f^3 - 3*a*c^2*d*e*f^2*g + a^2*c*d*e*g^3 + (2*a*c^2*d^2 - 
 a^2*c*e^2)*f*g^2 + ((a*c^4*d^3 + a^2*c^3*d*e^2)*f + (a^2*c^3*d^2*e + a^3* 
c^2*e^3)*g)*sqrt(-(c^2*e^2*f^4 - 4*c^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e 
^2*g^4 + 2*(2*c^2*d^2 - a*c*e^2)*f^2*g^2)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^...
 

Sympy [F]

\[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\int \frac {f + g x}{\left (a + c x^{2}\right ) \sqrt {d + e x}}\, dx \] Input:

integrate((g*x+f)/(e*x+d)**(1/2)/(c*x**2+a),x)
 

Output:

Integral((f + g*x)/((a + c*x**2)*sqrt(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\int { \frac {g x + f}{{\left (c x^{2} + a\right )} \sqrt {e x + d}} \,d x } \] Input:

integrate((g*x+f)/(e*x+d)^(1/2)/(c*x^2+a),x, algorithm="maxima")
 

Output:

integrate((g*x + f)/((c*x^2 + a)*sqrt(e*x + d)), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.62 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=-\frac {{\left (a c e f {\left | c \right |} {\left | e \right |} - a c d g {\left | c \right |} {\left | e \right |} + \sqrt {-a c} c d e f {\left | c \right |} + \sqrt {-a c} a e^{2} g {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} + a e^{2}\right )} c}}{c}}}\right )}{{\left (a c^{2} d - \sqrt {-a c} a c e\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} - \frac {{\left (a c e f {\left | c \right |} {\left | e \right |} - a c d g {\left | c \right |} {\left | e \right |} - \sqrt {-a c} c d e f {\left | c \right |} - \sqrt {-a c} a e^{2} g {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} + a e^{2}\right )} c}}{c}}}\right )}{{\left (a c^{2} d + \sqrt {-a c} a c e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} \] Input:

integrate((g*x+f)/(e*x+d)^(1/2)/(c*x^2+a),x, algorithm="giac")
 

Output:

-(a*c*e*f*abs(c)*abs(e) - a*c*d*g*abs(c)*abs(e) + sqrt(-a*c)*c*d*e*f*abs(c 
) + sqrt(-a*c)*a*e^2*g*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c*d + sqrt(c^2* 
d^2 - (c*d^2 + a*e^2)*c))/c))/((a*c^2*d - sqrt(-a*c)*a*c*e)*sqrt(-c^2*d - 
sqrt(-a*c)*c*e)*abs(e)) - (a*c*e*f*abs(c)*abs(e) - a*c*d*g*abs(c)*abs(e) - 
 sqrt(-a*c)*c*d*e*f*abs(c) - sqrt(-a*c)*a*e^2*g*abs(c))*arctan(sqrt(e*x + 
d)/sqrt(-(c*d - sqrt(c^2*d^2 - (c*d^2 + a*e^2)*c))/c))/((a*c^2*d + sqrt(-a 
*c)*a*c*e)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(e))
 

Mupad [B] (verification not implemented)

Time = 6.72 (sec) , antiderivative size = 2133, normalized size of antiderivative = 4.77 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\text {Too large to display} \] Input:

int((f + g*x)/((a + c*x^2)*(d + e*x)^(1/2)),x)
 

Output:

- atan((a^2*c^5*d^3*((a^2*c^2*d*g^2 + a*e*g^2*(-a^3*c^3)^(1/2) - c*e*f^2*( 
-a^3*c^3)^(1/2) - a*c^3*d*f^2 - 2*a^2*c^2*e*f*g + 2*c*d*f*g*(-a^3*c^3)^(1/ 
2))/(4*a^2*c^4*d^2 + 4*a^3*c^3*e^2))^(3/2)*(d + e*x)^(1/2)*8i - a^2*c^3*d^ 
2*g^2*((a^2*c^2*d*g^2 + a*e*g^2*(-a^3*c^3)^(1/2) - c*e*f^2*(-a^3*c^3)^(1/2 
) - a*c^3*d*f^2 - 2*a^2*c^2*e*f*g + 2*c*d*f*g*(-a^3*c^3)^(1/2))/(4*a^2*c^4 
*d^2 + 4*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i + a^2*c^3*e^2*f^2*((a^2*c^ 
2*d*g^2 + a*e*g^2*(-a^3*c^3)^(1/2) - c*e*f^2*(-a^3*c^3)^(1/2) - a*c^3*d*f^ 
2 - 2*a^2*c^2*e*f*g + 2*c*d*f*g*(-a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 + 4*a^3*c 
^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i - a^3*c^2*e^2*g^2*((a^2*c^2*d*g^2 + a*e* 
g^2*(-a^3*c^3)^(1/2) - c*e*f^2*(-a^3*c^3)^(1/2) - a*c^3*d*f^2 - 2*a^2*c^2* 
e*f*g + 2*c*d*f*g*(-a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 + 4*a^3*c^3*e^2))^(1/2) 
*(d + e*x)^(1/2)*2i + a^3*c^4*d*e^2*((a^2*c^2*d*g^2 + a*e*g^2*(-a^3*c^3)^( 
1/2) - c*e*f^2*(-a^3*c^3)^(1/2) - a*c^3*d*f^2 - 2*a^2*c^2*e*f*g + 2*c*d*f* 
g*(-a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 + 4*a^3*c^3*e^2))^(3/2)*(d + e*x)^(1/2) 
*8i + a*c^4*d^2*f^2*((a^2*c^2*d*g^2 + a*e*g^2*(-a^3*c^3)^(1/2) - c*e*f^2*( 
-a^3*c^3)^(1/2) - a*c^3*d*f^2 - 2*a^2*c^2*e*f*g + 2*c*d*f*g*(-a^3*c^3)^(1/ 
2))/(4*a^2*c^4*d^2 + 4*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i)/(a^3*c*e^2* 
g^3 - c*e^2*f^3*(-a^3*c^3)^(1/2) - a*c^3*d*e*f^3 + 2*a*c^3*d^2*f^2*g - a*d 
*e*g^3*(-a^3*c^3)^(1/2) - a^2*c^2*e^2*f^2*g + a*e^2*f*g^2*(-a^3*c^3)^(1/2) 
 - 2*c*d^2*f*g^2*(-a^3*c^3)^(1/2) + 3*a^2*c^2*d*e*f*g^2 + 3*c*d*e*f^2*g...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1483, normalized size of antiderivative = 3.32 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx =\text {Too large to display} \] Input:

int((g*x+f)/(e*x+d)^(1/2)/(c*x^2+a),x)
 

Output:

(sqrt(2)*( - 2*sqrt(a*e**2 + c*d**2)*sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - 
c*d)*atan((sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) + c*d)*sqrt(2) - 2*sqrt(c)*s 
qrt(d + e*x))/(sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*sqrt(2)))*a*e*g - 
 2*sqrt(a*e**2 + c*d**2)*sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*atan((s 
qrt(sqrt(c)*sqrt(a*e**2 + c*d**2) + c*d)*sqrt(2) - 2*sqrt(c)*sqrt(d + e*x) 
)/(sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*sqrt(2)))*c*d*f - 2*sqrt(c)*s 
qrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*atan((sqrt(sqrt(c)*sqrt(a*e**2 + 
c*d**2) + c*d)*sqrt(2) - 2*sqrt(c)*sqrt(d + e*x))/(sqrt(sqrt(c)*sqrt(a*e** 
2 + c*d**2) - c*d)*sqrt(2)))*a*e**2*f - 2*sqrt(c)*sqrt(sqrt(c)*sqrt(a*e**2 
 + c*d**2) - c*d)*atan((sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) + c*d)*sqrt(2) 
- 2*sqrt(c)*sqrt(d + e*x))/(sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*sqrt 
(2)))*c*d**2*f + 2*sqrt(a*e**2 + c*d**2)*sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2 
) - c*d)*atan((sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) + c*d)*sqrt(2) + 2*sqrt( 
c)*sqrt(d + e*x))/(sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*sqrt(2)))*a*e 
*g + 2*sqrt(a*e**2 + c*d**2)*sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*ata 
n((sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) + c*d)*sqrt(2) + 2*sqrt(c)*sqrt(d + 
e*x))/(sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*sqrt(2)))*c*d*f + 2*sqrt( 
c)*sqrt(sqrt(c)*sqrt(a*e**2 + c*d**2) - c*d)*atan((sqrt(sqrt(c)*sqrt(a*e** 
2 + c*d**2) + c*d)*sqrt(2) + 2*sqrt(c)*sqrt(d + e*x))/(sqrt(sqrt(c)*sqrt(a 
*e**2 + c*d**2) - c*d)*sqrt(2)))*a*e**2*f + 2*sqrt(c)*sqrt(sqrt(c)*sqrt...