Optimal. Leaf size=617 \[ \frac {B \sqrt {a+b x+c x^2}}{f}-\frac {(2 B c e-b B f-2 A c f) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} f^2}+\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e-\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f)+B \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^2 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (2 f (A f (c d-a f)-B d (c e-b f))-\left (e+\sqrt {e^2-4 d f}\right ) \left (A f (c e-b f)+B \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^2 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 5.93, antiderivative size = 615, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1033, 1090,
635, 212, 1046, 738} \begin {gather*} \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 ) \tanh ^{-1}\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} f^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 (\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 ) \tanh ^{-1}\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} f^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 {\tanh ^{-1}\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)}{2 \sqrt {c} f^2}+\frac {B \sqrt {a+b x+c x^2}}{f} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 212
Rule 635
Rule 738
Rule 1033
Rule 1046
Rule 1090
Rubi steps
\begin {align*} \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 {\int \frac {\frac {1}{2} (b B d-2 a A f)-\frac {1}{2} (2 A b f-B (2 c d+b e-2 a f)) x+\frac {1}{2} (2 B c e-b B f-2 A c f) x^2}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{f}\\ &=\frac {B \sqrt {a+b x+c x^2}}{f}-\frac {\int \frac {\frac {1}{2} f (b B d-2 a A f)-\frac {1}{2} d (2 B c e-b B f-2 A c f)+\left (-\frac {1}{2} e (2 B c e-b B f-2 A c f)+\frac {1}{2} f (-2 A b f+B (2 c d+b e-2 a f))\right ) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{f^2}-\frac {(2 B c e-b B f-2 A c f) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 f^2}\\ &=\frac {B \sqrt {a+b x+c x^2}}{f}-\frac {(2 B c e-b B f-2 A c f) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{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 (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+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{f^2 \sqrt {e^2-4 d f}}-\frac {\left (2 f \left (\frac {1}{2} f (b B d-2 a A f)-\frac {1}{2} d (2 B c e-b B f-2 A c f)\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (-\frac {1}{2} e (2 B c e-b B f-2 A c f)+\frac {1}{2} f (-2 A b f+B (2 c d+b e-2 a f))\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+b x+c x^2}} \, dx}{f^2 \sqrt {e^2-4 d f}}\\ &=\frac {B \sqrt {a+b x+c x^2}}{f}-\frac {(2 B c e-b B f-2 A c f) \tanh ^{-1}\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 \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 )\right ) \text {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e+\sqrt {e^2-4 d f}\right )+4 c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f^2 \sqrt {e^2-4 d f}}+\frac {\left (2 \left (2 f \left (\frac {1}{2} f (b B d-2 a A f)-\frac {1}{2} d (2 B c e-b B f-2 A c f)\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (-\frac {1}{2} e (2 B c e-b B f-2 A c f)+\frac {1}{2} f (-2 A b f+B (2 c d+b e-2 a f))\right )\right )\right ) \text {Subst}\left (\int \frac {1}{16 a f^2-8 b f \left (e-\sqrt {e^2-4 d f}\right )+4 c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )}{f^2 \sqrt {e^2-4 d f}}\\ &=\frac {B \sqrt {a+b x+c x^2}}{f}-\frac {(2 B c e-b B f-2 A c f) \tanh ^{-1}\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 (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\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 (B f (b e-a f)+A f (c e-b f)-B c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\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}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.93, size = 889, normalized size = 1.44 \begin {gather*} \frac {2 B f \sqrt {a+x (b+c x)}+\frac {(2 B c e-b B f-2 A c f) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{\sqrt {c}}+2 \text {RootSum}\left [b^2 d-a b e+a^2 f-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-b B c d e \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a B c e^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^2 B d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+A b c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a B c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a b B e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a A c e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 B f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 B c^{3/2} d e \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b B \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 A c^{3/2} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a A \sqrt {c} f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-B c e^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+B c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b B e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+A c e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-A b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a B f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 b \sqrt {c} d-a \sqrt {c} e-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+3 \sqrt {c} e \text {$\#$1}^2-2 f \text {$\#$1}^3}\&\right ]}{2 f^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1594\) vs.
\(2(558)=1116\).
time = 0.17, size = 1595, normalized size = 2.59
method | result | size |
default | \(\text {Expression too large to display}\) | \(1595\) |
risch | \(\text {Expression too large to display}\) | \(11585\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{d + e x + f x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{f\,x^2+e\,x+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________