Optimal. Leaf size=679 \[ -\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} f^3}+\frac {\left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left ((c e-b f) \left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \]
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Rubi [A]
time = 7.73, antiderivative size = 678, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {991, 1090, 635,
212, 1046, 738} \begin {gather*} \frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (-f^2 \left (b^2 d-a^2 f\right )+2 c d f (b e-a f)+c^2 (-d) \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^3 \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^3 \left (b^2 d-a^2 f\right )-\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )+4 c d f^2 (b e-a f)-2 c^2 d f \left (e^2-d f\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^3 \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 ) \left (-12 c f (b e-a f)+3 b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 \sqrt {c} f^3}-\frac {\sqrt {a+b x+c x^2} (-5 b f+4 c e-2 c f x)}{4 f^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 212
Rule 635
Rule 738
Rule 991
Rule 1046
Rule 1090
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}-\frac {\int \frac {\frac {1}{4} \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} \left (8 c^2 d e-4 a c e f-b f (5 b e-16 a f)+4 b c \left (e^2-4 d f\right )\right ) x-\frac {1}{4} \left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) x^2}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 f^2}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}-\frac {\int \frac {\frac {1}{4} f \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} d \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )+\left (\frac {1}{4} f \left (-8 c^2 d e+4 a c e f+b f (5 b e-16 a f)-4 b c \left (e^2-4 d f\right )\right )-\frac {1}{4} e \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right ) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{2 f^3}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 f^3}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \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 )}{4 f^3}-\frac {\left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \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^3 \sqrt {e^2-4 d f}}+\frac {\left (2 f \left (\frac {1}{4} f \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} d \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (\frac {1}{4} f \left (-8 c^2 d e+4 a c e f+b f (5 b e-16 a f)-4 b c \left (e^2-4 d f\right )\right )-\frac {1}{4} e \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\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}{2 f^3 \sqrt {e^2-4 d f}}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} f^3}+\frac {\left (2 \left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \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^3 \sqrt {e^2-4 d f}}-\frac {\left (2 f \left (\frac {1}{4} f \left (-4 b c d e+5 b^2 d f+4 a f (c d-2 a f)\right )-\frac {1}{4} d \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (\frac {1}{4} f \left (-8 c^2 d e+4 a c e f+b f (5 b e-16 a f)-4 b c \left (e^2-4 d f\right )\right )-\frac {1}{4} e \left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \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^3 \sqrt {e^2-4 d f}}\\ &=-\frac {(4 c e-5 b f-2 c f x) \sqrt {a+b x+c x^2}}{4 f^2}+\frac {\left (3 b^2 f^2-12 c f (b e-a f)+8 c^2 \left (e^2-d f\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} f^3}+\frac {\left ((c e-b f) \left (e-\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \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^3 \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 ((c e-b f) \left (e+\sqrt {e^2-4 d f}\right ) \left (f (b e-2 a f)-c \left (e^2-2 d f\right )\right )-2 f \left (2 c d f (b e-a f)-f^2 \left (b^2 d-a^2 f\right )-c^2 d \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^3 \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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 1.26, size = 1150, normalized size = 1.69 \begin {gather*} \frac {2 f (-4 c e+5 b f+2 c f x) \sqrt {a+x (b+c x)}+\frac {\left (-3 b^2 f^2+12 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{\sqrt {c}}+8 \text {RootSum}\left [b^2 d-a b e+a^2 f-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {b c^2 d e^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 e^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-b c^2 d^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b^2 c d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 a c^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 a b c e^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^3 d f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a b^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c e f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 b f^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d e^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 c^{5/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 b c^{3/2} d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b^2 \sqrt {c} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c^{3/2} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^2 \sqrt {c} f^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 e^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 c^2 d e f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 b c e^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 b c d f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c e f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 a b f^3 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 b \sqrt {c} d-a \sqrt {c} e-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+3 \sqrt {c} e \text {$\#$1}^2-2 f \text {$\#$1}^3}\&\right ]}{8 f^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2859\) vs.
\(2(618)=1236\).
time = 0.21, size = 2860, normalized size = 4.21
method | result | size |
default | \(\text {Expression too large to display}\) | \(2860\) |
risch | \(\text {Expression too large to display}\) | \(13999\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{f\,x^2+e\,x+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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