Optimal. Leaf size=209 \[ \frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac {3 e \sqrt {a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {e \sqrt {a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.22, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {744, 806, 724, 206} \[ \frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac {3 e \sqrt {a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {e \sqrt {a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 744
Rule 806
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx &=-\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} (-4 c d+3 b e)+c e x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{4 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {e \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 205, normalized size = 0.98 \[ -\frac {\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{5/2}}-\frac {3 e \sqrt {a+x (b+c x)} (2 c d-b e)}{4 (d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac {e \sqrt {a+x (b+c x)}}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.92, size = 1362, normalized size = 6.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 785, normalized size = 3.76 \[ \frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2} - 4 \, a c e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} - \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c^{2} d^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{2} d^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c d e^{2} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c d^{2} e - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{2} d^{2} e + 6 \, b^{2} c^{\frac {3}{2}} d^{3} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} - 3 \, b^{3} \sqrt {c} d^{2} e - 20 \, a b c^{\frac {3}{2}} d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} e^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c e^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} d e^{2} + 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c d e^{2} + 11 \, a b^{2} \sqrt {c} d e^{2} + 12 \, a^{2} c^{\frac {3}{2}} d e^{2} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} e^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c e^{3} - 8 \, a^{2} b \sqrt {c} e^{3}}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 959, normalized size = 4.59 \[ -\frac {3 b^{2} e \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{8 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {3 b c d \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {3 c^{2} d^{2} \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e}+\frac {3 \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, b e}{4 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}-\frac {3 \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, c d}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}+\frac {c \ln \left (\frac {\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e}-\frac {\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} e} \]
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 \[ \int \frac {1}{{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
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 \[ \int \frac {1}{\left (d + e x\right )^{3} \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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