Optimal. Leaf size=280 \[ -\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^3}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^3 \left (c d^2-b d e+a e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {824, 857, 635,
212, 738} \begin {gather*} -\frac {(2 c d-b e) \left (-4 c e (2 b d-3 a e)-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 e^3 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 738
Rule 824
Rule 857
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx &=-\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )-8 c^2 \left (c d^2-b d e+a e^2\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{4 e^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{e^3}-\frac {\left (8 c^2 d \left (c d^2-b d e+a e^2\right )+\frac {1}{2} e \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{4 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {\left (4 c^2\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 )}{e^3}+\frac {\left (8 c^2 d \left (c d^2-b d e+a e^2\right )+\frac {1}{2} e \left (6 b^2 c d e+8 a c^2 d e+b^3 e^2-4 b c \left (2 c d^2+3 a e^2\right )\right )\right ) \text {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 )}{2 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\left (8 c^2 d^3-b e^2 (b d-2 a e)-2 c d e (3 b d-2 a e)+e \left (12 c^2 d^2+b^2 e^2-4 c e (3 b d-2 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 c^{3/2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e^3}-\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 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 e^3 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 10.85, size = 389, normalized size = 1.39 \begin {gather*} \frac {\frac {4 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{3/2}}{(d+e x)^2}+\frac {2 \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) (a+x (b+c x))^{3/2}}{d+e x}+\frac {-2 c e \sqrt {a+x (b+c x)} \left (b^3 e^3+4 c^3 d^2 (2 d-e x)+b c e^2 (5 b d-10 a e+b e x)-2 c^2 e (b d (7 d-2 e x)+2 a e (-3 d+2 e x))\right )+16 c^{5/2} \left (c d^2+e (-b d+a e)\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+c (2 c d-b e) \sqrt {c d^2+e (-b d+a e)} \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{c e^3}}{8 \left (c d^2+e (-b d+a e)\right )^2} \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. \(1822\) vs.
\(2(258)=516\).
time = 1.09, size = 1823, normalized size = 6.51
method | result | size |
default | \(\text {Expression too large to display}\) | \(1823\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b + 2 c x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{3}}\, dx \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 (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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