3.1.19 \(\int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx\) [19]

3.1.19.1 Optimal result

Integrand size = 32, antiderivative size = 617 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\frac {B \sqrt {a+b x+c x^2}}{f}-\frac {(2 B c e-b B f-2 A c f) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f^2}+\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f)+B \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^2 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e+\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f)+B \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^2 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]

output

-1/2*(-2*A*c*f-B*b*f+2*B*c*e)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^ 
(1/2))/f^2/c^(1/2)+B*(c*x^2+b*x+a)^(1/2)/f+1/2*arctanh(1/4*(4*a*f+2*x*(b*f 
-c*(e-(-4*d*f+e^2)^(1/2)))-b*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+b*x+a) 
^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2))* 
(2*f*(A*f*(-a*f+c*d)-B*d*(-b*f+c*e))-(A*f*(-b*f+c*e)+B*(f*(-a*f+b*e)-c*(-d 
*f+e^2)))*(e-(-4*d*f+e^2)^(1/2)))/f^2*2^(1/2)/(-4*d*f+e^2)^(1/2)/(c*e^2-2* 
c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)-1/2*arctanh(1/4*( 
4*a*f-b*(e+(-4*d*f+e^2)^(1/2))+2*x*(b*f-c*(e+(-4*d*f+e^2)^(1/2))))*2^(1/2) 
/(c*x^2+b*x+a)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^ 
(1/2))^(1/2))*(2*f*(A*f*(-a*f+c*d)-B*d*(-b*f+c*e))-(A*f*(-b*f+c*e)+B*(f*(- 
a*f+b*e)-c*(-d*f+e^2)))*(e+(-4*d*f+e^2)^(1/2)))/f^2*2^(1/2)/(-4*d*f+e^2)^( 
1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)
 

3.1.19.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.28 (sec) , antiderivative size = 1115, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\frac {B f \sqrt {a+x (b+c x)}+\frac {(-2 B c e+b B f+2 A c f) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}-\text {RootSum}\left [c^2 d-b c e+b^2 f+2 \sqrt {a} c e \text {$\#$1}-4 \sqrt {a} b f \text {$\#$1}-2 c d \text {$\#$1}^2+b e \text {$\#$1}^2+4 a f \text {$\#$1}^2-2 \sqrt {a} e \text {$\#$1}^3+d \text {$\#$1}^4\&,\frac {-B c^2 d e \log (x)+b B c e^2 \log (x)+A c^2 d f \log (x)-b^2 B e f \log (x)-A b c e f \log (x)+A b^2 f^2 \log (x)+a b B f^2 \log (x)-a A c f^2 \log (x)+B c^2 d e \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-b B c e^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-A c^2 d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+b^2 B e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+A b c e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-A b^2 f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-a b B f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )+a A c f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right )-2 \sqrt {a} B c e^2 \log (x) \text {$\#$1}+2 \sqrt {a} B c d f \log (x) \text {$\#$1}+2 \sqrt {a} b B e f \log (x) \text {$\#$1}+2 \sqrt {a} A c e f \log (x) \text {$\#$1}-2 \sqrt {a} A b f^2 \log (x) \text {$\#$1}-2 a^{3/2} B f^2 \log (x) \text {$\#$1}+2 \sqrt {a} B c e^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \sqrt {a} B c d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \sqrt {a} b B e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}-2 \sqrt {a} A c e f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 \sqrt {a} A b f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+2 a^{3/2} B f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}+B c d e \log (x) \text {$\#$1}^2-b B d f \log (x) \text {$\#$1}^2-A c d f \log (x) \text {$\#$1}^2+a A f^2 \log (x) \text {$\#$1}^2-B c d e \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+b B d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2+A c d f \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-a A f^2 \log \left (-\sqrt {a}+\sqrt {a+b x+c x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-\sqrt {a} c e+2 \sqrt {a} b f+2 c d \text {$\#$1}-b e \text {$\#$1}-4 a f \text {$\#$1}+3 \sqrt {a} e \text {$\#$1}^2-2 d \text {$\#$1}^3}\&\right ]}{f^2} \]

input

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

output

(B*f*Sqrt[a + x*(b + c*x)] + ((-2*B*c*e + b*B*f + 2*A*c*f)*ArcTanh[(Sqrt[c 
]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/Sqrt[c] - RootSum[c^2*d - b*c*e 
+ b^2*f + 2*Sqrt[a]*c*e*#1 - 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4* 
a*f*#1^2 - 2*Sqrt[a]*e*#1^3 + d*#1^4 & , (-(B*c^2*d*e*Log[x]) + b*B*c*e^2* 
Log[x] + A*c^2*d*f*Log[x] - b^2*B*e*f*Log[x] - A*b*c*e*f*Log[x] + A*b^2*f^ 
2*Log[x] + a*b*B*f^2*Log[x] - a*A*c*f^2*Log[x] + B*c^2*d*e*Log[-Sqrt[a] + 
Sqrt[a + b*x + c*x^2] - x*#1] - b*B*c*e^2*Log[-Sqrt[a] + Sqrt[a + b*x + c* 
x^2] - x*#1] - A*c^2*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + b^ 
2*B*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + A*b*c*e*f*Log[-Sqrt 
[a] + Sqrt[a + b*x + c*x^2] - x*#1] - A*b^2*f^2*Log[-Sqrt[a] + Sqrt[a + b* 
x + c*x^2] - x*#1] - a*b*B*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1 
] + a*A*c*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 2*Sqrt[a]*B*c 
*e^2*Log[x]*#1 + 2*Sqrt[a]*B*c*d*f*Log[x]*#1 + 2*Sqrt[a]*b*B*e*f*Log[x]*#1 
 + 2*Sqrt[a]*A*c*e*f*Log[x]*#1 - 2*Sqrt[a]*A*b*f^2*Log[x]*#1 - 2*a^(3/2)*B 
*f^2*Log[x]*#1 + 2*Sqrt[a]*B*c*e^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - 
x*#1]*#1 - 2*Sqrt[a]*B*c*d*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]* 
#1 - 2*Sqrt[a]*b*B*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - 2 
*Sqrt[a]*A*c*e*f*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + 2*Sqrt[ 
a]*A*b*f^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + 2*a^(3/2)*B*f 
^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 + B*c*d*e*Log[x]*#1^...
 

3.1.19.3 Rubi [A] (warning: unable to verify)

Time = 1.48 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {1352, 27, 2143, 27, 1092, 219, 1365, 1154, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx\)

\(\Big \downarrow \) 1352

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\int \frac {(2 B c e-b B f-2 A c f) x^2-(2 A b f-B (2 c d+b e-2 a f)) x+b B d-2 a A f}{2 \sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\int \frac {(2 B c e-b B f-2 A c f) x^2-(2 A b f-B (2 c d+b e-2 a f)) x+b B d-2 a A f}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{2 f}\)

\(\Big \downarrow \) 2143

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {\int \frac {2 \left (A f (c d-a f)-\frac {1}{2} B (2 c d e-2 b d f)+\left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right ) x\right )}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}+\frac {(-2 A c f-b B f+2 B c e) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {2 \int \frac {A f (c d-a f)-B (c d e-b d f)+\left (A f (c e-b f)-B \left (c e^2-b f e+a f^2-c d f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}+\frac {(-2 A c f-b B f+2 B c e) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{f}}{2 f}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {2 \int \frac {A f (c d-a f)-B (c d e-b d f)+\left (A f (c e-b f)-B \left (c e^2-b f e+a f^2-c d f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}+\frac {2 (-2 A c f-b B f+2 B c e) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{f}}{2 f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {2 \int \frac {A f (c d-a f)-B (c d e-b d f)+\left (A f (c e-b f)-B \left (c e^2-b f e+a f^2-c d f\right )\right ) x}{\sqrt {c x^2+b x+a} \left (f x^2+e x+d\right )}dx}{f}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (-2 A c f-b B f+2 B c e)}{\sqrt {c} f}}{2 f}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {2 \left (\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt {e^2-4 d f}\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (\sqrt {e^2-4 d f}+e\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}\right )}{f}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (-2 A c f-b B f+2 B c e)}{\sqrt {c} f}}{2 f}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {2 \left (\frac {2 \left (2 f (A f (c d-a f)-B d (c e-b f))-\left (\sqrt {e^2-4 d f}+e\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {2 \left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt {e^2-4 d f}\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}\right )}{f}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (-2 A c f-b B f+2 B c e)}{\sqrt {c} f}}{2 f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\frac {2 \left (\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (\sqrt {e^2-4 d f}+e\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt {e^2-4 d f}\right ) \left (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{f}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (-2 A c f-b B f+2 B c e)}{\sqrt {c} f}}{2 f}\)

input

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

output

(B*Sqrt[a + b*x + c*x^2])/f - (((2*B*c*e - b*B*f - 2*A*c*f)*ArcTanh[(b + 2 
*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*f) + (2*(-(((2*f*(A*f*( 
c*d - a*f) - B*d*(c*e - b*f)) - (e - Sqrt[e^2 - 4*d*f])*(B*f*(b*e - a*f) + 
 A*f*(c*e - b*f) - B*c*(e^2 - d*f)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4* 
d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c 
*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x 
^2])])/(Sqrt[2]*Sqrt[e^2 - 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - 
 (c*e - b*f)*Sqrt[e^2 - 4*d*f]])) + ((2*f*(A*f*(c*d - a*f) - B*d*(c*e - b* 
f)) - (e + Sqrt[e^2 - 4*d*f])*(B*f*(b*e - a*f) + A*f*(c*e - b*f) - B*c*(e^ 
2 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sq 
rt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + ( 
c*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*Sqrt[e^2 - 
 4*d*f]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*Sqrt[e^2 - 4* 
d*f]])))/f)/(2*f)
 

3.1.19.3.1 Defintions of rubi rules used

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 1092

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

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1352

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e 
_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[h*(a + b*x + c*x^2)^p*((d 
+ e*x + f*x^2)^(q + 1)/(2*f*(p + q + 1))), x] - Simp[1/(2*f*(p + q + 1)) 
Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[h*p*(b*d - a*e) + a* 
(h*e - 2*g*f)*(p + q + 1) + (2*h*p*(c*d - a*f) + b*(h*e - 2*g*f)*(p + q + 1 
))*x + (h*p*(c*e - b*f) + c*(h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; F 
reeQ[{a, b, c, d, e, f, g, h, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4* 
d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

rule 2143

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ 
.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C 
 = Coeff[Px, x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 
1/c   Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x 
^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
 

3.1.19.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1175\) vs. \(2(558)=1116\).

Time = 0.99 (sec) , antiderivative size = 1176, normalized size of antiderivative = 1.91

method	result	size
risch	\(\text {Expression too large to display}\)	\(1176\)
default	\(\text {Expression too large to display}\)	\(1595\)

input

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

output

B*(c*x^2+b*x+a)^(1/2)/f+1/2/f*(1/f*(2*A*c*f+B*b*f-2*B*c*e)*ln((1/2*b+c*x)/ 
c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-1/2*(2*A*b*f^2*(-4*d*f+e^2)^(1/2)-2*A 
*c*e*f*(-4*d*f+e^2)^(1/2)+4*A*a*f^3-2*A*b*e*f^2-4*A*c*d*f^2+2*A*c*e^2*f+2* 
B*a*f^2*(-4*d*f+e^2)^(1/2)-2*B*b*e*f*(-4*d*f+e^2)^(1/2)-2*B*c*d*f*(-4*d*f+ 
e^2)^(1/2)+2*B*c*e^2*(-4*d*f+e^2)^(1/2)-2*B*a*e*f^2-4*B*b*d*f^2+2*B*b*e^2* 
f+6*B*c*d*e*f-2*B*c*e^3)/f^2/(-4*d*f+e^2)^(1/2)*2^(1/2)/((b*f*(-4*d*f+e^2) 
^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(( 
(b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2 
)/f^2+(c*(-4*d*f+e^2)^(1/2)+b*f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1 
/2*2^(1/2)*((b*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2 
*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c+4*(c*(-4 
*d*f+e^2)^(1/2)+b*f-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(b*f*(-4*d* 
f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2 
))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))-1/2*(2*A*b*f^2*(-4*d*f+e^2)^(1/2)-2* 
A*c*e*f*(-4*d*f+e^2)^(1/2)-4*A*a*f^3+2*A*b*e*f^2+4*A*c*d*f^2-2*A*c*e^2*f+2 
*B*a*f^2*(-4*d*f+e^2)^(1/2)-2*B*b*e*f*(-4*d*f+e^2)^(1/2)-2*B*c*d*f*(-4*d*f 
+e^2)^(1/2)+2*B*c*e^2*(-4*d*f+e^2)^(1/2)+2*B*a*e*f^2+4*B*b*d*f^2-2*B*b*e^2 
*f-6*B*c*d*e*f+2*B*c*e^3)/f^2/(-4*d*f+e^2)^(1/2)*2^(1/2)/((-b*f*(-4*d*f+e^ 
2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f+c*e^2)/f^2)^(1/2)*ln 
(((-b*f*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*c*d*f...
 

3.1.19.5 Fricas [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.19.6 Sympy [F]

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

input

integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/(f*x**2+e*x+d),x)
 

output

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/(d + e*x + f*x**2), x)
 

3.1.19.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` for 
 more deta
 

3.1.19.8 Giac [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

3.1.19.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{f\,x^2+e\,x+d} \,d x \]

input

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

output

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