Optimal. Leaf size=779 \[ \frac {2 \left (c x \left (\left (e^2-d f\right ) (a b f-2 a c e+b c d)-d e \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )\right )-\left (a d f-a e^2+b d e\right ) \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )+c d e (a b f-2 a c e+b c d)\right )}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {2 e (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) \left (a \left (e^2-d f\right )-b d e+c d^2\right )+2 d f (b d-a e)\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) \left (a \left (e^2-d f\right )-b d e+c d^2\right )+2 d f (b d-a e)\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}} \]
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Rubi [A] time = 14.17, antiderivative size = 779, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6728, 613, 636, 1016, 1032, 724, 206} \[ \frac {2 \left (c x \left (\left (e^2-d f\right ) (a b f-2 a c e+b c d)-d e \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )\right )-\left (a d f-a e^2+b d e\right ) \left (-c (2 a f+b e)+b^2 f+2 c^2 d\right )+c d e (a b f-2 a c e+b c d)\right )}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {2 e (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (\left (e-\sqrt {e^2-4 d f}\right ) \left (a \left (e^2-d f\right )-b d e+c d^2\right )+2 d f (b d-a e)\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right )-b \left (e-\sqrt {e^2-4 d f}\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2-\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac {\left (\left (\sqrt {e^2-4 d f}+e\right ) \left (a \left (e^2-d f\right )-b d e+c d^2\right )+2 d f (b d-a e)\right ) \tanh ^{-1}\left (\frac {4 a f+2 x \left (b f-c \left (\sqrt {e^2-4 d f}+e\right )\right )-b \left (\sqrt {e^2-4 d f}+e\right )}{2 \sqrt {2} \sqrt {a+b x+c x^2} \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {2 a f^2+\sqrt {e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 613
Rule 636
Rule 724
Rule 1016
Rule 1032
Rule 6728
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx &=\int \left (-\frac {e}{f^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {x}{f \left (a+b x+c x^2\right )^{3/2}}+\frac {d e+\left (e^2-d f\right ) x}{f^2 \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {d e+\left (e^2-d f\right ) x}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx}{f^2}-\frac {e \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{f^2}+\frac {\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{f}\\ &=\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 e (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (c d e (b c d-2 a c e+a b f)-\left (b d e-a e^2+a d f\right ) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left ((b c d-2 a c e+a b f) \left (e^2-d f\right )-d e \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} \left (b^2-4 a c\right ) d (b d-a e) f^2-\frac {1}{2} \left (b^2-4 a c\right ) f^2 \left (c d^2-b d e+a e^2-a d f\right ) x}{\sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{\left (b^2-4 a c\right ) f^2 \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 e (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (c d e (b c d-2 a c e+a b f)-\left (b d e-a e^2+a d f\right ) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left ((b c d-2 a c e+a b f) \left (e^2-d f\right )-d e \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}-\frac {\left (2 d (b d-a e) f+\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2-b d e+a \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}{\sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac {\left (2 d (b d-a e) f+\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2-b d e+a \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}{\sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 e (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (c d e (b c d-2 a c e+a b f)-\left (b d e-a e^2+a d f\right ) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left ((b c d-2 a c e+a b f) \left (e^2-d f\right )-d e \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}+\frac {\left (2 \left (2 d (b d-a e) f+\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2-b d e+a \left (e^2-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 )}{\sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac {\left (2 \left (2 d (b d-a e) f+\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2-b d e+a \left (e^2-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 )}{\sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=\frac {2 (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {a+b x+c x^2}}+\frac {2 e (b+2 c x)}{\left (b^2-4 a c\right ) f^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (c d e (b c d-2 a c e+a b f)-\left (b d e-a e^2+a d f\right ) \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )+c \left ((b c d-2 a c e+a b f) \left (e^2-d f\right )-d e \left (2 c^2 d+b^2 f-c (b e+2 a f)\right )\right ) x\right )}{\left (b^2-4 a c\right ) f^2 \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}+\frac {\left (2 d (b d-a e) f+\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2-b d e+a \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} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \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 d (b d-a e) f+\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2-b d e+a \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} \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \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 = 2.62, size = 1066, normalized size = 1.37 \[ \frac {\frac {4 \left (2 f a^3+(-2 c d-b e+2 c e x+b f x) a^2+b (b (d-e x)-3 c d x) a+b^3 d x\right )}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {\sqrt {2} \left (c \left (\sqrt {e^2-4 d f}-e\right ) d^2+b \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right ) d+a \left (-e^3+\sqrt {e^2-4 d f} e^2+3 d f e-d f \sqrt {e^2-4 d f}\right )\right ) \log \left (-e-2 f x+\sqrt {e^2-4 d f}\right )}{\sqrt {e^2-4 d f} \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right )}}+\frac {\sqrt {2} \left (c \left (e+\sqrt {e^2-4 d f}\right ) d^2-b \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) d+a \left (e^3+\sqrt {e^2-4 d f} e^2-3 d f e-d f \sqrt {e^2-4 d f}\right )\right ) \log \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\sqrt {e^2-4 d f} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}}-\frac {\sqrt {2} \left (c \left (e+\sqrt {e^2-4 d f}\right ) d^2-b \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) d+a \left (e^3+\sqrt {e^2-4 d f} e^2-3 d f e-d f \sqrt {e^2-4 d f}\right )\right ) \log \left (-4 a f+2 c e x+2 c \sqrt {e^2-4 d f} x+b \left (e-2 f x+\sqrt {e^2-4 d f}\right )-2 \sqrt {2} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )} \sqrt {a+x (b+c x)}\right )}{\sqrt {e^2-4 d f} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}}-\frac {\sqrt {2} \left (c \left (\sqrt {e^2-4 d f}-e\right ) d^2+b \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right ) d+a \left (-e^3+\sqrt {e^2-4 d f} e^2+3 d f e-d f \sqrt {e^2-4 d f}\right )\right ) \log \left (b \left (-e+2 f x+\sqrt {e^2-4 d f}\right )+2 \left (2 a f-c e x+c \sqrt {e^2-4 d f} x+\sqrt {2} \sqrt {f \left (-e b+\sqrt {e^2-4 d f} b+2 a f\right )+c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )} \sqrt {a+x (b+c x)}\right )\right )}{\sqrt {e^2-4 d f} \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f+b \left (\sqrt {e^2-4 d f}-e\right )\right )}}}{2 \left (c^2 d^2-b c e d+f \left (f a^2-b e a+b^2 d\right )+a c \left (e^2-2 d f\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 14651, normalized size = 18.81 \[ \text {output too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x^3}{{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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