Optimal. Leaf size=455 \[ \frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{80640 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{32768 c^7}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{12288 c^6}+\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{2016 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c} \]
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Rubi [A] time = 0.63, antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{5/2} \left (2048 a^2 B c^2-10 c x \left (504 a A c^2-748 a b B c-594 A b^2 c+429 b^3 B\right )+6696 a A b c^2-7524 a b^2 B c-4158 A b^3 c+3003 b^4 B\right )}{80640 c^5}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{12288 c^6}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right )}{32768 c^7}-\frac {\left (b^2-4 a c\right )^2 \left (-96 a^2 A c^3+240 a^2 b B c^2+432 a A b^2 c^2-440 a b^3 B c-198 A b^4 c+143 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}}+\frac {x^2 \left (a+b x+c x^2\right )^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{2016 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{5/2} (13 b B-18 A c)}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int x^4 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\int x^3 \left (-4 a B-\frac {1}{2} (13 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{9 c}\\ &=-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\int x^2 \left (\frac {3}{2} a (13 b B-18 A c)+\frac {1}{4} \left (143 b^2 B-198 A b c-128 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{72 c^2}\\ &=\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\int x \left (-\frac {1}{2} a \left (143 b^2 B-198 A b c-128 a B c\right )-\frac {3}{8} \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{504 c^3}\\ &=\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^5}\\ &=-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{8192 c^6}\\ &=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{65536 c^7}\\ &=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{32768 c^7}\\ &=\frac {\left (b^2-4 a c\right ) \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^7}-\frac {\left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^6}+\frac {\left (143 b^2 B-198 A b c-128 a B c\right ) x^2 \left (a+b x+c x^2\right )^{5/2}}{2016 c^3}-\frac {(13 b B-18 A c) x^3 \left (a+b x+c x^2\right )^{5/2}}{144 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{5/2}}{9 c}+\frac {\left (3003 b^4 B-4158 A b^3 c-7524 a b^2 B c+6696 a A b c^2+2048 a^2 B c^2-10 c \left (429 b^3 B-594 A b^2 c-748 a b B c+504 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{80640 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (143 b^5 B-198 A b^4 c-440 a b^3 B c+432 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{15/2}}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 338, normalized size = 0.74 \begin {gather*} \frac {\frac {3 \left (96 a^2 A c^3-240 a^2 b B c^2-432 a A b^2 c^2+440 a b^3 B c+198 A b^4 c-143 b^5 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{65536 c^{13/2}}+\frac {x^2 (a+x (b+c x))^{5/2} \left (-128 a B c-198 A b c+143 b^2 B\right )}{224 c^2}+\frac {(a+x (b+c x))^{5/2} \left (396 b^2 c (15 A c x-19 a B)+8 a b c^2 (837 A+935 B x)+16 a c^2 (128 a B-315 A c x)-66 b^3 c (63 A+65 B x)+3003 b^4 B\right )}{8960 c^4}+\frac {x^3 (a+x (b+c x))^{5/2} (18 A c-13 b B)}{16 c}+B x^4 (a+x (b+c x))^{5/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.50, size = 662, normalized size = 1.45 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (262144 a^4 B c^4+1058688 a^3 A b c^4-241920 a^3 A c^5 x-1467072 a^3 b^2 B c^3+473728 a^3 b B c^4 x-131072 a^3 B c^5 x^2-1469664 a^2 A b^3 c^3+680256 a^2 A b^2 c^4 x-347904 a^2 A b c^5 x^2+161280 a^2 A c^6 x^3+1383984 a^2 b^4 B c^2-677664 a^2 b^3 B c^3 x+378240 a^2 b^2 B c^4 x^2-206592 a^2 b B c^5 x^3+98304 a^2 B c^6 x^4+551880 a A b^5 c^2-323568 a A b^4 c^3 x+224640 a A b^3 c^4 x^2-163584 a A b^2 c^5 x^3+119808 a A b c^6 x^4+1935360 a A c^7 x^5-438900 a b^6 B c+260568 a b^5 B c^2 x-183744 a b^4 B c^3 x^2+136576 a b^3 B c^4 x^3-102912 a b^2 B c^5 x^4+76800 a b B c^6 x^5+1638400 a B c^7 x^6-62370 A b^7 c+41580 A b^6 c^2 x-33264 A b^5 c^3 x^2+28512 A b^4 c^4 x^3-25344 A b^3 c^5 x^4+23040 A b^2 c^6 x^5+1566720 A b c^7 x^6+1290240 A c^8 x^7+45045 b^8 B-30030 b^7 B c x+24024 b^6 B c^2 x^2-20592 b^5 B c^3 x^3+18304 b^4 B c^4 x^4-16640 b^3 B c^5 x^5+15360 b^2 B c^6 x^6+1361920 b B c^7 x^7+1146880 B c^8 x^8\right )}{10321920 c^7}+\frac {\left (-1536 a^4 A c^5+3840 a^4 b B c^4+7680 a^3 A b^2 c^4-8960 a^3 b^3 B c^3-6720 a^2 A b^4 c^3+6048 a^2 b^5 B c^2+2016 a A b^6 c^2-1584 a b^7 B c-198 A b^8 c+143 b^9 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{65536 c^{15/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 1263, normalized size = 2.78
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 639, normalized size = 1.40 \begin {gather*} \frac {1}{10321920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, {\left (16 \, B c x + \frac {19 \, B b c^{8} + 18 \, A c^{9}}{c^{8}}\right )} x + \frac {3 \, B b^{2} c^{7} + 320 \, B a c^{8} + 306 \, A b c^{8}}{c^{8}}\right )} x - \frac {13 \, B b^{3} c^{6} - 60 \, B a b c^{7} - 18 \, A b^{2} c^{7} - 1512 \, A a c^{8}}{c^{8}}\right )} x + \frac {143 \, B b^{4} c^{5} - 804 \, B a b^{2} c^{6} - 198 \, A b^{3} c^{6} + 768 \, B a^{2} c^{7} + 936 \, A a b c^{7}}{c^{8}}\right )} x - \frac {1287 \, B b^{5} c^{4} - 8536 \, B a b^{3} c^{5} - 1782 \, A b^{4} c^{5} + 12912 \, B a^{2} b c^{6} + 10224 \, A a b^{2} c^{6} - 10080 \, A a^{2} c^{7}}{c^{8}}\right )} x + \frac {3003 \, B b^{6} c^{3} - 22968 \, B a b^{4} c^{4} - 4158 \, A b^{5} c^{4} + 47280 \, B a^{2} b^{2} c^{5} + 28080 \, A a b^{3} c^{5} - 16384 \, B a^{3} c^{6} - 43488 \, A a^{2} b c^{6}}{c^{8}}\right )} x - \frac {15015 \, B b^{7} c^{2} - 130284 \, B a b^{5} c^{3} - 20790 \, A b^{6} c^{3} + 338832 \, B a^{2} b^{3} c^{4} + 161784 \, A a b^{4} c^{4} - 236864 \, B a^{3} b c^{5} - 340128 \, A a^{2} b^{2} c^{5} + 120960 \, A a^{3} c^{6}}{c^{8}}\right )} x + \frac {45045 \, B b^{8} c - 438900 \, B a b^{6} c^{2} - 62370 \, A b^{7} c^{2} + 1383984 \, B a^{2} b^{4} c^{3} + 551880 \, A a b^{5} c^{3} - 1467072 \, B a^{3} b^{2} c^{4} - 1469664 \, A a^{2} b^{3} c^{4} + 262144 \, B a^{4} c^{5} + 1058688 \, A a^{3} b c^{5}}{c^{8}}\right )} + \frac {{\left (143 \, B b^{9} - 1584 \, B a b^{7} c - 198 \, A b^{8} c + 6048 \, B a^{2} b^{5} c^{2} + 2016 \, A a b^{6} c^{2} - 8960 \, B a^{3} b^{3} c^{3} - 6720 \, A a^{2} b^{4} c^{3} + 3840 \, B a^{4} b c^{4} + 7680 \, A a^{3} b^{2} c^{4} - 1536 \, A a^{4} c^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{65536 \, c^{\frac {15}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1311, normalized size = 2.88
result too large to display
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \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 x^{4} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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