Optimal. Leaf size=117 \[ \frac {3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {1}{2} d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}+\frac {3 \left (b^2-4 a c\right )^2 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}} \]
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Rubi [A]
time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {706, 635, 212}
\begin {gather*} \frac {3}{4} d^4 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}+\frac {3 d^4 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}}+\frac {1}{2} d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 706
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^4}{\sqrt {a+b x+c x^2}} \, dx &=\frac {1}{2} d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}+\frac {1}{4} \left (3 \left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx\\ &=\frac {3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {1}{2} d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}+\frac {1}{8} \left (3 \left (b^2-4 a c\right )^2 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx\\ &=\frac {3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {1}{2} d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}+\frac {1}{4} \left (3 \left (b^2-4 a c\right )^2 d^4\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 )\\ &=\frac {3}{4} \left (b^2-4 a c\right ) d^4 (b+2 c x) \sqrt {a+b x+c x^2}+\frac {1}{2} d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}+\frac {3 \left (b^2-4 a c\right )^2 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 97, normalized size = 0.83 \begin {gather*} \frac {1}{8} d^4 \left (2 (b+2 c x) \sqrt {a+x (b+c x)} \left (5 b^2+8 b c x+4 c \left (-3 a+2 c x^2\right )\right )-\frac {3 \left (b^2-4 a c\right )^2 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{\sqrt {c}}\right ) \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. \(721\) vs.
\(2(99)=198\).
time = 0.69, size = 722, normalized size = 6.17
method | result | size |
risch | \(-\frac {\left (-16 c^{3} x^{3}-24 b \,c^{2} x^{2}+24 a \,c^{2} x -18 b^{2} c x +12 a b c -5 b^{3}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}}{4}+\left (\frac {3 b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}+6 a^{2} c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-3 a \sqrt {c}\, b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )\right ) d^{4}\) | \(166\) |
default | \(d^{4} \left (16 c^{4} \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )+32 b \,c^{3} \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+24 b^{2} c^{2} \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+8 b^{3} c \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {b^{4} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}\right )\) | \(722\) |
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 [A]
time = 2.84, size = 313, normalized size = 2.68 \begin {gather*} \left [\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c} d^{4} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (16 \, c^{4} d^{4} x^{3} + 24 \, b c^{3} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{4} x + {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{16 \, c}, -\frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-c} d^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (16 \, c^{4} d^{4} x^{3} + 24 \, b c^{3} d^{4} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} d^{4} x + {\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{8 \, c}\right ] \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*} d^{4} \left (\int \frac {b^{4}}{\sqrt {a + b x + c x^{2}}}\, dx + \int \frac {16 c^{4} x^{4}}{\sqrt {a + b x + c x^{2}}}\, dx + \int \frac {32 b c^{3} x^{3}}{\sqrt {a + b x + c x^{2}}}\, dx + \int \frac {24 b^{2} c^{2} x^{2}}{\sqrt {a + b x + c x^{2}}}\, dx + \int \frac {8 b^{3} c x}{\sqrt {a + b x + c x^{2}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.07, size = 159, normalized size = 1.36 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, c^{3} d^{4} x + 3 \, b c^{2} d^{4}\right )} x + \frac {3 \, {\left (3 \, b^{2} c^{4} d^{4} - 4 \, a c^{5} d^{4}\right )}}{c^{3}}\right )} x + \frac {5 \, b^{3} c^{3} d^{4} - 12 \, a b c^{4} d^{4}}{c^{3}}\right )} - \frac {3 \, {\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, \sqrt {c}} \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 = -0.01 \begin {gather*} \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^4}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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