Optimal. Leaf size=761 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac {\left (f \left (a f \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f+2 d e f \sqrt {e^2-4 d f}-e^3 \sqrt {e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (f \left (a f \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-4 d e^2 f-2 d e f \sqrt {e^2-4 d f}+e^3 \sqrt {e^2-4 d f}+e^4\right )\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} f^3 \sqrt {e^2-4 d f} \sqrt {f \left (2 a f-b \left (\sqrt {e^2-4 d f}+e\right )\right )+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \]
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Rubi [A] time = 3.14, antiderivative size = 761, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1067, 1076, 621, 206, 1032, 724} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c f (b e-a f)+b^2 f^2-8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}-\frac {\left (f \left (a f \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )-b \left (-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2-e^3 \sqrt {e^2-4 d f}-4 d e^2 f+2 d e f \sqrt {e^2-4 d f}+e^4\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 {f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (f \left (a f \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )-b \left (e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}-3 d e f+e^3\right )\right )+c \left (2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-4 d e^2 f-2 d e f \sqrt {e^2-4 d f}+e^4\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 {f \left (2 a f-b \left (\sqrt {e^2-4 d f}+e\right )\right )+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {\sqrt {a+b x+c x^2} (-b f+4 c e-2 c f x)}{4 c f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 1032
Rule 1067
Rule 1076
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {a+b x+c x^2}}{d+e x+f x^2} \, dx &=-\frac {(4 c e-b f-2 c f x) \sqrt {a+b x+c x^2}}{4 c f^2}-\frac {\int \frac {-\frac {1}{4} d \left (4 b c e-b^2 f-4 a c f\right )-\frac {1}{4} \left (8 c^2 d e-b^2 e f-4 a c e f+4 b c \left (e^2-2 d f\right )\right ) x+\frac {1}{4} \left (b^2 f^2+4 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 c f^2}\\ &=-\frac {(4 c e-b f-2 c f x) \sqrt {a+b x+c x^2}}{4 c f^2}-\frac {\int \frac {-\frac {1}{4} d f \left (4 b c e-b^2 f-4 a c f\right )-\frac {1}{4} d \left (b^2 f^2+4 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+b^2 e f+4 a c e f-4 b c \left (e^2-2 d f\right )\right )-\frac {1}{4} e \left (b^2 f^2+4 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 c f^3}-\frac {\left (b^2 f^2+4 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 c f^3}\\ &=-\frac {(4 c e-b f-2 c f x) \sqrt {a+b x+c x^2}}{4 c f^2}-\frac {\left (b^2 f^2+4 c f (b e-a f)-8 c^2 \left (e^2-d f\right )\right ) \operatorname {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 c f^3}-\frac {\left (c \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 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 (c \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 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 {(4 c e-b f-2 c f x) \sqrt {a+b x+c x^2}}{4 c f^2}-\frac {\left (b^2 f^2+4 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 c^{3/2} f^3}+\frac {\left (2 \left (c \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 d f}\right )\right )\right )\right ) \operatorname {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 \left (c \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 d f}\right )\right )\right )\right ) \operatorname {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-b f-2 c f x) \sqrt {a+b x+c x^2}}{4 c f^2}-\frac {\left (b^2 f^2+4 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 c^{3/2} f^3}-\frac {\left (c \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f-e^2 \sqrt {e^2-4 d f}+d f \sqrt {e^2-4 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 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+f \left (a f \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )-b \left (e^3-3 d e f+e^2 \sqrt {e^2-4 d f}-d f \sqrt {e^2-4 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 [A] time = 2.31, size = 552, normalized size = 0.73 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (4 c f (a f-b e)-b^2 f^2+8 c^2 \left (e^2-d f\right )\right )}{8 c^{3/2} f^3}+\frac {f \sqrt {e^2-4 d f} \sqrt {a+x (b+c x)} (b f-4 c e+2 c f x)+\sqrt {2} c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right ) \sqrt {f \left (2 a f-b \left (\sqrt {e^2-4 d f}+e\right )\right )+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac {4 a f-b \left (\sqrt {e^2-4 d f}+e-2 f x\right )-2 c x \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+x (b+c x)} \sqrt {f \left (2 a f-b \left (\sqrt {e^2-4 d f}+e\right )\right )+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )+\sqrt {2} c \left (e \sqrt {e^2-4 d f}+2 d f-e^2\right ) \sqrt {f \left (2 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right )+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac {4 a f+b \left (\sqrt {e^2-4 d f}-e+2 f x\right )+2 c x \left (\sqrt {e^2-4 d f}-e\right )}{2 \sqrt {2} \sqrt {a+x (b+c x)} \sqrt {f \left (2 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right )+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 c f^3 \sqrt {e^2-4 d f}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.45, size = 869, normalized size = 1.14 \begin {gather*} \frac {\sqrt {c x^2+b x+a} (-4 c e+b f+2 c f x)}{4 c f^2}+\frac {\left (-8 e^2 c^2+8 d f c^2-4 a f^2 c+4 b e f c+b^2 f^2\right ) \log \left (2 x c^2-2 \sqrt {c x^2+b x+a} c^{3/2}+b c\right )}{8 c^{3/2} f^3}-\frac {\text {RootSum}\left [f \text {$\#$1}^4-2 \sqrt {c} e \text {$\#$1}^3+4 c d \text {$\#$1}^2+b e \text {$\#$1}^2-2 a f \text {$\#$1}^2-4 b \sqrt {c} d \text {$\#$1}+2 a \sqrt {c} e \text {$\#$1}+b^2 d-a b e+a^2 f\&,\frac {-c \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 e^3+a c \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e^3+b f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 e^2-b c d \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e^2-a b f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e^2+2 c^{3/2} d \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} e^2-a f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 e+2 c d f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2 e+a^2 f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e+b^2 d f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e-2 a c d f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) e-2 b \sqrt {c} d f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1} e-b d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}^2+b c d^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right )+2 a \sqrt {c} d f^2 \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}-2 c^{3/2} d^2 f \log \left (-\sqrt {c} x-\text {$\#$1}+\sqrt {c x^2+b x+a}\right ) \text {$\#$1}}{-2 f \text {$\#$1}^3+3 \sqrt {c} e \text {$\#$1}^2-4 c d \text {$\#$1}-b e \text {$\#$1}+2 a f \text {$\#$1}+2 b \sqrt {c} d-a \sqrt {c} e}\&\right ]}{f^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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maple [B] time = 0.02, size = 14815, normalized size = 19.47 \begin {gather*} \text {output too large to display} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\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.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \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.
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