Optimal. Leaf size=202 \[ \frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {756, 846, 793,
635, 212} \begin {gather*} -\frac {b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac {\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt {a+b x+c x^2}}{1920 c^5}+\frac {x^2 \left (63 b^2-64 a c\right ) \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 756
Rule 793
Rule 846
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {a+b x+c x^2}} \, dx &=\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^3 \left (-4 a-\frac {9 b x}{2}\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x^2 \left (\frac {27 a b}{2}+\frac {1}{4} \left (63 b^2-64 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\int \frac {x \left (-\frac {1}{2} a \left (63 b^2-64 a c\right )-\frac {7}{8} b \left (45 b^2-92 a c\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right )\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 )}{128 c^5}\\ &=\frac {\left (63 b^2-64 a c\right ) x^2 \sqrt {a+b x+c x^2}}{240 c^3}-\frac {9 b x^3 \sqrt {a+b x+c x^2}}{40 c^2}+\frac {x^4 \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{1920 c^5}-\frac {b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 153, normalized size = 0.76 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^4-630 b^3 c x+8 b c^2 x \left (161 a-54 c x^2\right )+84 b^2 c \left (-35 a+6 c x^2\right )+128 c^2 \left (8 a^2-4 a c x^2+3 c^2 x^4\right )\right )+15 \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{3840 c^{11/2}} \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. \(540\) vs.
\(2(176)=352\).
time = 0.75, size = 541, normalized size = 2.68
method | result | size |
risch | \(\frac {\left (384 c^{4} x^{4}-432 b \,c^{3} x^{3}-512 x^{2} c^{3} a +504 b^{2} c^{2} x^{2}+1288 x a b \,c^{2}-630 b^{3} c x +1024 a^{2} c^{2}-2940 a c \,b^{2}+945 b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{5}}-\frac {15 b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2}}{16 c^{\frac {7}{2}}}+\frac {35 b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a}{32 c^{\frac {9}{2}}}-\frac {63 b^{5} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {11}{2}}}\) | \(193\) |
default | \(\frac {x^{4} \sqrt {c \,x^{2}+b x +a}}{5 c}-\frac {9 b \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{10 c}-\frac {4 a \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{5 c}\) | \(541\) |
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 = 3.46, size = 347, normalized size = 1.72 \begin {gather*} \left [\frac {15 \, {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \sqrt {c} \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 (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \, {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, c^{6}}, \frac {15 \, {\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \sqrt {-c} \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 (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \, {\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \, {\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, c^{6}}\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*} \int \frac {x^{5}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 161, normalized size = 0.80 \begin {gather*} \frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, x {\left (\frac {8 \, x}{c} - \frac {9 \, b}{c^{2}}\right )} + \frac {63 \, b^{2} c^{2} - 64 \, a c^{3}}{c^{5}}\right )} x - \frac {7 \, {\left (45 \, b^{3} c - 92 \, a b c^{2}\right )}}{c^{5}}\right )} x + \frac {945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2}}{c^{5}}\right )} + \frac {{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b c^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {11}{2}}} \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 {x^5}{\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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