Optimal. Leaf size=249 \[ -\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {3 \left (b^2-4 a c\right )^5 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {699, 706, 635,
212} \begin {gather*} -\frac {3 d^4 \left (b^2-4 a c\right )^5 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}}-\frac {3 d^4 \left (b^2-4 a c\right )^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 699
Rule 706
Rubi steps
\begin {align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}+\frac {\left (3 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^4 \sqrt {a+b x+c x^2} \, dx}{256 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (b^2-4 a c\right )^3 \int \frac {(b d+2 c d x)^4}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (3 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{8192 c^3}\\ &=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (3 \left (b^2-4 a c\right )^5 d^4\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16384 c^3}\\ &=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (3 \left (b^2-4 a c\right )^5 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 )}{8192 c^3}\\ &=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {3 \left (b^2-4 a c\right )^5 d^4 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.26, size = 312, normalized size = 1.25 \begin {gather*} \frac {d^4 \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (15 b^8-40 b^7 c x+8 b^6 c \left (-35 a+11 c x^2\right )+32 b^5 c^2 x \left (23 a+360 c x^2\right )+128 b^3 c^3 x \left (233 a^2+1184 a c x^2+1288 c^2 x^4\right )+32 b^4 c^2 \left (64 a^2+1047 a c x^2+2084 c^2 x^4\right )+512 b c^4 x \left (5 a^3+248 a^2 c x^2+504 a c^2 x^4+256 c^3 x^6\right )+128 b^2 c^3 \left (35 a^3+729 a^2 c x^2+2272 a c^2 x^4+1624 c^3 x^6\right )+256 c^4 \left (-15 a^4+10 a^3 c x^2+248 a^2 c^2 x^4+336 a c^3 x^6+128 c^4 x^8\right )\right )+15 \left (b^2-4 a c\right )^5 \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{81920 c^{7/2}} \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. \(2129\) vs.
\(2(219)=438\).
time = 0.79, size = 2130, normalized size = 8.55
method | result | size |
risch | \(-\frac {\left (-65536 c^{9} x^{9}-294912 b \,c^{8} x^{8}-172032 a \,c^{8} x^{7}-546816 b^{2} c^{7} x^{7}-602112 a b \,c^{7} x^{6}-537600 b^{3} c^{6} x^{6}-126976 a^{2} c^{7} x^{5}-839680 a \,b^{2} c^{6} x^{5}-298240 b^{4} c^{5} x^{5}-317440 a^{2} b \,c^{6} x^{4}-593920 a \,b^{3} c^{5} x^{4}-89728 b^{5} c^{4} x^{4}-5120 a^{3} c^{6} x^{3}-313600 a^{2} b^{2} c^{5} x^{3}-218560 b^{4} c^{4} a \,x^{3}-11696 b^{6} c^{3} x^{3}-7680 a^{3} b \,c^{5} x^{2}-152960 a^{2} b^{3} c^{4} x^{2}-34976 a \,b^{5} c^{3} x^{2}-8 b^{7} c^{2} x^{2}+7680 a^{4} c^{5} x -11520 a^{3} b^{2} c^{4} x -33920 a^{2} b^{4} c^{3} x -176 c^{2} b^{6} a x +10 b^{8} c x +3840 a^{4} b \,c^{4}-4480 a^{3} b^{3} c^{3}-2048 a^{2} b^{5} c^{2}+280 a \,b^{7} c -15 b^{9}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}}{40960 c^{3}}+\left (\frac {3 c^{\frac {3}{2}} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{5}}{16}-\frac {15 \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{4} b^{2}}{64}+\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} b^{4}}{128 \sqrt {c}}-\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{6}}{512 c^{\frac {3}{2}}}+\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{8}}{4096 c^{\frac {5}{2}}}-\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{10}}{16384 c^{\frac {7}{2}}}\right ) d^{4}\) | \(553\) |
default | \(\text {Expression too large to display}\) | \(2130\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 460 vs.
\(2 (219) = 438\).
time = 3.30, size = 923, normalized size = 3.71 \begin {gather*} \left [-\frac {15 \, {\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\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 (65536 \, c^{10} d^{4} x^{9} + 294912 \, b c^{9} d^{4} x^{8} + 6144 \, {\left (89 \, b^{2} c^{8} + 28 \, a c^{9}\right )} d^{4} x^{7} + 21504 \, {\left (25 \, b^{3} c^{7} + 28 \, a b c^{8}\right )} d^{4} x^{6} + 256 \, {\left (1165 \, b^{4} c^{6} + 3280 \, a b^{2} c^{7} + 496 \, a^{2} c^{8}\right )} d^{4} x^{5} + 128 \, {\left (701 \, b^{5} c^{5} + 4640 \, a b^{3} c^{6} + 2480 \, a^{2} b c^{7}\right )} d^{4} x^{4} + 16 \, {\left (731 \, b^{6} c^{4} + 13660 \, a b^{4} c^{5} + 19600 \, a^{2} b^{2} c^{6} + 320 \, a^{3} c^{7}\right )} d^{4} x^{3} + 8 \, {\left (b^{7} c^{3} + 4372 \, a b^{5} c^{4} + 19120 \, a^{2} b^{3} c^{5} + 960 \, a^{3} b c^{6}\right )} d^{4} x^{2} - 2 \, {\left (5 \, b^{8} c^{2} - 88 \, a b^{6} c^{3} - 16960 \, a^{2} b^{4} c^{4} - 5760 \, a^{3} b^{2} c^{5} + 3840 \, a^{4} c^{6}\right )} d^{4} x + {\left (15 \, b^{9} c - 280 \, a b^{7} c^{2} + 2048 \, a^{2} b^{5} c^{3} + 4480 \, a^{3} b^{3} c^{4} - 3840 \, a^{4} b c^{5}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{163840 \, c^{4}}, \frac {15 \, {\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\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 (65536 \, c^{10} d^{4} x^{9} + 294912 \, b c^{9} d^{4} x^{8} + 6144 \, {\left (89 \, b^{2} c^{8} + 28 \, a c^{9}\right )} d^{4} x^{7} + 21504 \, {\left (25 \, b^{3} c^{7} + 28 \, a b c^{8}\right )} d^{4} x^{6} + 256 \, {\left (1165 \, b^{4} c^{6} + 3280 \, a b^{2} c^{7} + 496 \, a^{2} c^{8}\right )} d^{4} x^{5} + 128 \, {\left (701 \, b^{5} c^{5} + 4640 \, a b^{3} c^{6} + 2480 \, a^{2} b c^{7}\right )} d^{4} x^{4} + 16 \, {\left (731 \, b^{6} c^{4} + 13660 \, a b^{4} c^{5} + 19600 \, a^{2} b^{2} c^{6} + 320 \, a^{3} c^{7}\right )} d^{4} x^{3} + 8 \, {\left (b^{7} c^{3} + 4372 \, a b^{5} c^{4} + 19120 \, a^{2} b^{3} c^{5} + 960 \, a^{3} b c^{6}\right )} d^{4} x^{2} - 2 \, {\left (5 \, b^{8} c^{2} - 88 \, a b^{6} c^{3} - 16960 \, a^{2} b^{4} c^{4} - 5760 \, a^{3} b^{2} c^{5} + 3840 \, a^{4} c^{6}\right )} d^{4} x + {\left (15 \, b^{9} c - 280 \, a b^{7} c^{2} + 2048 \, a^{2} b^{5} c^{3} + 4480 \, a^{3} b^{3} c^{4} - 3840 \, a^{4} b c^{5}\right )} d^{4}\right )} \sqrt {c x^{2} + b x + a}}{81920 \, c^{4}}\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 a^{2} b^{4} \sqrt {a + b x + c x^{2}}\, dx + \int b^{6} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 16 c^{6} x^{8} \sqrt {a + b x + c x^{2}}\, dx + \int 2 a b^{5} x \sqrt {a + b x + c x^{2}}\, dx + \int 32 a c^{5} x^{6} \sqrt {a + b x + c x^{2}}\, dx + \int 16 a^{2} c^{4} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 64 b c^{5} x^{7} \sqrt {a + b x + c x^{2}}\, dx + \int 104 b^{2} c^{4} x^{6} \sqrt {a + b x + c x^{2}}\, dx + \int 88 b^{3} c^{3} x^{5} \sqrt {a + b x + c x^{2}}\, dx + \int 41 b^{4} c^{2} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 10 b^{5} c x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 96 a b c^{4} x^{5} \sqrt {a + b x + c x^{2}}\, dx + \int 112 a b^{2} c^{3} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 64 a b^{3} c^{2} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 18 a b^{4} c x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 32 a^{2} b c^{3} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 24 a^{2} b^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 8 a^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 547 vs.
\(2 (219) = 438\).
time = 1.67, size = 547, normalized size = 2.20 \begin {gather*} \frac {1}{40960} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (16 \, {\left (2 \, c^{6} d^{4} x + 9 \, b c^{5} d^{4}\right )} x + \frac {3 \, {\left (89 \, b^{2} c^{13} d^{4} + 28 \, a c^{14} d^{4}\right )}}{c^{9}}\right )} x + \frac {21 \, {\left (25 \, b^{3} c^{12} d^{4} + 28 \, a b c^{13} d^{4}\right )}}{c^{9}}\right )} x + \frac {1165 \, b^{4} c^{11} d^{4} + 3280 \, a b^{2} c^{12} d^{4} + 496 \, a^{2} c^{13} d^{4}}{c^{9}}\right )} x + \frac {701 \, b^{5} c^{10} d^{4} + 4640 \, a b^{3} c^{11} d^{4} + 2480 \, a^{2} b c^{12} d^{4}}{c^{9}}\right )} x + \frac {731 \, b^{6} c^{9} d^{4} + 13660 \, a b^{4} c^{10} d^{4} + 19600 \, a^{2} b^{2} c^{11} d^{4} + 320 \, a^{3} c^{12} d^{4}}{c^{9}}\right )} x + \frac {b^{7} c^{8} d^{4} + 4372 \, a b^{5} c^{9} d^{4} + 19120 \, a^{2} b^{3} c^{10} d^{4} + 960 \, a^{3} b c^{11} d^{4}}{c^{9}}\right )} x - \frac {5 \, b^{8} c^{7} d^{4} - 88 \, a b^{6} c^{8} d^{4} - 16960 \, a^{2} b^{4} c^{9} d^{4} - 5760 \, a^{3} b^{2} c^{10} d^{4} + 3840 \, a^{4} c^{11} d^{4}}{c^{9}}\right )} x + \frac {15 \, b^{9} c^{6} d^{4} - 280 \, a b^{7} c^{7} d^{4} + 2048 \, a^{2} b^{5} c^{8} d^{4} + 4480 \, a^{3} b^{3} c^{9} d^{4} - 3840 \, a^{4} b c^{10} d^{4}}{c^{9}}\right )} + \frac {3 \, {\left (b^{10} d^{4} - 20 \, a b^{8} c d^{4} + 160 \, a^{2} b^{6} c^{2} d^{4} - 640 \, a^{3} b^{4} c^{3} d^{4} + 1280 \, a^{4} b^{2} c^{4} d^{4} - 1024 \, a^{5} c^{5} d^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16384 \, c^{\frac {7}{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 {\left (b\,d+2\,c\,d\,x\right )}^4\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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