Optimal. Leaf size=309 \[ \frac {2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )-a b^3 f^2+2 a b^2 c e f+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac {f \sqrt {a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2} \]
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Rubi [A] time = 0.45, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 621, 206} \begin {gather*} \frac {2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )+2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^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 (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac {f \sqrt {a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 1660
Rule 1661
Rubi steps
\begin {align*} \int \frac {\left (d+e x+f x^2\right )^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {-\frac {\left (b^2-4 a c\right ) \left (b^2 f^2-c f (2 b e+a f)+c^2 \left (e^2+2 d f\right )\right )}{2 c^3}-\frac {\left (b^2-4 a c\right ) f (2 c e-b f) x}{2 c^2}-\frac {\left (b^2-4 a c\right ) f^2 x^2}{2 c}}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}-\frac {\int \frac {-\frac {\left (b^2-4 a c\right ) \left (2 b^2 f^2-c f (4 b e+3 a f)+2 c^2 \left (e^2+2 d f\right )\right )}{2 c^2}-\frac {\left (b^2-4 a c\right ) f (8 c e-7 b f) x}{4 c}}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+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^3}\\ &=\frac {2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {f (8 c e-7 b f) \sqrt {a+b x+c x^2}}{4 c^3}+\frac {f^2 x \sqrt {a+b x+c x^2}}{2 c^2}+\frac {\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+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^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 288, normalized size = 0.93 \begin {gather*} \frac {4 b c \left (-13 a^2 f^2+a c \left (4 d f+2 e^2+20 e f x-5 f^2 x^2\right )+2 c^2 d (d-2 e x)\right )+8 c^2 \left (a^2 f (8 e+3 f x)+a c \left (x \left (-2 e^2+4 e f x+f^2 x^2\right )-4 d (e+f x)\right )+2 c^2 d^2 x\right )+b^3 f (15 a f+c x (5 f x-24 e))-2 b^2 c \left (a f (12 e+31 f x)+c x \left (-8 d f-4 e^2+4 e f x+f^2 x^2\right )\right )+15 b^4 f^2 x}{4 c^3 \left (4 a c-b^2\right ) \sqrt {a+x (b+c x)}}+\frac {\log \left (2 \sqrt {c} \sqrt {a+x (b+c x)}+b+2 c x\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.16, size = 375, normalized size = 1.21 \begin {gather*} \frac {\log \left (-2 c^{7/2} \sqrt {a+b x+c x^2}+b c^3+2 c^4 x\right ) \left (12 a c f^2-15 b^2 f^2+24 b c e f-16 c^2 d f-8 c^2 e^2\right )}{8 c^{7/2}}-\frac {52 a^2 b c f^2-64 a^2 c^2 e f-24 a^2 c^2 f^2 x-15 a b^3 f^2+24 a b^2 c e f+62 a b^2 c f^2 x-16 a b c^2 d f-8 a b c^2 e^2-80 a b c^2 e f x+20 a b c^2 f^2 x^2+32 a c^3 d e+32 a c^3 d f x+16 a c^3 e^2 x-32 a c^3 e f x^2-8 a c^3 f^2 x^3-15 b^4 f^2 x+24 b^3 c e f x-5 b^3 c f^2 x^2-16 b^2 c^2 d f x-8 b^2 c^2 e^2 x+8 b^2 c^2 e f x^2+2 b^2 c^2 f^2 x^3-8 b c^3 d^2+16 b c^3 d e x-16 c^4 d^2 x}{4 c^3 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 1305, normalized size = 4.22
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 407, normalized size = 1.32 \begin {gather*} \frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} f^{2} - 4 \, a c^{3} f^{2}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} - \frac {5 \, b^{3} c f^{2} - 20 \, a b c^{2} f^{2} - 8 \, b^{2} c^{2} f e + 32 \, a c^{3} f e}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {16 \, c^{4} d^{2} + 16 \, b^{2} c^{2} d f - 32 \, a c^{3} d f + 15 \, b^{4} f^{2} - 62 \, a b^{2} c f^{2} + 24 \, a^{2} c^{2} f^{2} - 16 \, b c^{3} d e - 24 \, b^{3} c f e + 80 \, a b c^{2} f e + 8 \, b^{2} c^{2} e^{2} - 16 \, a c^{3} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {8 \, b c^{3} d^{2} + 16 \, a b c^{2} d f + 15 \, a b^{3} f^{2} - 52 \, a^{2} b c f^{2} - 32 \, a c^{3} d e - 24 \, a b^{2} c f e + 64 \, a^{2} c^{2} f e + 8 \, a b c^{2} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {{\left (16 \, c^{2} d f + 15 \, b^{2} f^{2} - 12 \, a c f^{2} - 24 \, b c f e + 8 \, c^{2} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1011, normalized size = 3.27 \begin {gather*} -\frac {13 a \,b^{2} f^{2} x}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {8 a b e f x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}+\frac {15 b^{4} f^{2} x}{8 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}-\frac {3 b^{3} e f x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {2 b^{2} d f x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}+\frac {b^{2} e^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {4 b d e x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {f^{2} x^{3}}{2 \sqrt {c \,x^{2}+b x +a}\, c}-\frac {13 a \,b^{3} f^{2}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {4 a \,b^{2} e f}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {15 b^{5} f^{2}}{16 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {3 b^{4} e f}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {b^{3} d f}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {b^{3} e^{2}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {2 b^{2} d e}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {5 b \,f^{2} x^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {2 e f \,x^{2}}{\sqrt {c \,x^{2}+b x +a}\, c}+\frac {3 a \,f^{2} x}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {15 b^{2} f^{2} x}{8 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {3 b e f x}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {2 d f x}{\sqrt {c \,x^{2}+b x +a}\, c}-\frac {e^{2} x}{\sqrt {c \,x^{2}+b x +a}\, c}+\frac {2 \left (2 c x +b \right ) d^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {3 a \,f^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}+\frac {15 b^{2} f^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {7}{2}}}-\frac {3 b e f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {5}{2}}}+\frac {2 d f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}+\frac {e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}-\frac {13 a b \,f^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {4 a e f}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {15 b^{3} f^{2}}{16 \sqrt {c \,x^{2}+b x +a}\, c^{4}}-\frac {3 b^{2} e f}{2 \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {b d f}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {b \,e^{2}}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {2 d e}{\sqrt {c \,x^{2}+b x +a}\, c} \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 {{\left (f\,x^2+e\,x+d\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,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 {\left (d + e x + f x^{2}\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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