Optimal. Leaf size=400 \[ \frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {756, 793, 626,
635, 212} \begin {gather*} -\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{12288 c^5}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e) \left (-4 c e (3 a e+8 b d)+11 b^2 e^2+32 c^2 d^2\right )}{768 c^4}+\frac {e \left (a+b x+c x^2\right )^{7/2} \left (-2 c e (32 a e+243 b d)+99 b^2 e^2+154 c e x (2 c d-b e)+640 c^2 d^2\right )}{2016 c^3}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 756
Rule 793
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\int (d+e x) \left (\frac {1}{2} \left (18 c d^2-e (7 b d+4 a e)\right )+\frac {11}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left ((2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{8192 c^5}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{65536 c^6}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\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 )}{32768 c^6}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{32768 c^6}-\frac {5 \left (b^2-4 a c\right ) (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{12288 c^5}+\frac {(2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{768 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {e \left (640 c^2 d^2+99 b^2 e^2-2 c e (243 b d+32 a e)+154 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {5 \left (b^2-4 a c\right )^3 (2 c d-b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(807\) vs. \(2(400)=800\).
time = 4.71, size = 807, normalized size = 2.02 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^8 e^3+210 b^7 c e^2 (81 d+11 e x)-84 b^6 c e \left (-485 a e^2+c \left (360 d^2+135 d e x+22 e^2 x^2\right )\right )+72 b^5 c^2 \left (-7 a e^2 (375 d+49 e x)+2 c \left (140 d^3+140 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )\right )-16 b^4 c^2 \left (10143 a^2 e^3-9 a c e \left (2240 d^2+791 d e x+124 e^2 x^2\right )+2 c^2 x \left (420 d^3+504 d^2 e x+243 d e^2 x^2+44 e^3 x^3\right )\right )+32 b^3 c^3 \left (9 a^2 e^2 (2359 d+293 e x)+8 c^2 x^2 \left (42 d^3+54 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )-4 a c \left (1680 d^3+1512 d^2 e x+639 d e^2 x^2+107 e^3 x^3\right )\right )+192 b^2 c^3 \left (1221 a^3 e^3-a^2 c e \left (5544 d^2+1791 d e x+266 e^2 x^2\right )+4 a c^2 x \left (168 d^3+180 d^2 e x+81 d e^2 x^2+14 e^3 x^3\right )+8 c^3 x^3 \left (378 d^3+888 d^2 e x+729 d e^2 x^2+206 e^3 x^3\right )\right )+128 b c^4 \left (-13 a^3 e^2 (459 d+53 e x)+6 a^2 c \left (924 d^3+684 d^2 e x+261 d e^2 x^2+41 e^3 x^3\right )+24 a c^2 x^2 \left (546 d^3+1182 d^2 e x+921 d e^2 x^2+251 e^3 x^3\right )+16 c^3 x^4 \left (420 d^3+1044 d^2 e x+891 d e^2 x^2+259 e^3 x^3\right )\right )+256 c^4 \left (-256 a^4 e^3+a^3 c e \left (3456 d^2+945 d e x+128 e^2 x^2\right )+16 c^4 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+8 a c^3 x^3 \left (546 d^3+1296 d^2 e x+1071 d e^2 x^2+304 e^3 x^3\right )+6 a^2 c^2 x \left (924 d^3+1728 d^2 e x+1239 d e^2 x^2+320 e^3 x^3\right )\right )\right )-315 \left (b^2-4 a c\right )^3 (-2 c d+b e) \left (32 c^2 d^2+11 b^2 e^2-4 c e (8 b d+3 a e)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{4128768 c^{13/2}} \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. \(1206\) vs.
\(2(370)=740\).
time = 0.91, size = 1207, normalized size = 3.02
method | result | size |
default | \(\text {Expression too large to display}\) | \(1207\) |
risch | \(\text {Expression too large to display}\) | \(1683\) |
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. 1039 vs.
\(2 (384) = 768\).
time = 4.05, size = 2081, normalized size = 5.20 \begin {gather*} \text {Too large to display} \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 \left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \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. 1160 vs.
\(2 (384) = 768\).
time = 3.91, size = 1160, normalized size = 2.90 \begin {gather*} \frac {1}{2064384} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, c^{2} x e^{3} + \frac {54 \, c^{10} d e^{2} + 37 \, b c^{9} e^{3}}{c^{8}}\right )} x + \frac {864 \, c^{10} d^{2} e + 1782 \, b c^{9} d e^{2} + 309 \, b^{2} c^{8} e^{3} + 608 \, a c^{9} e^{3}}{c^{8}}\right )} x + \frac {1344 \, c^{10} d^{3} + 8352 \, b c^{9} d^{2} e + 4374 \, b^{2} c^{8} d e^{2} + 8568 \, a c^{9} d e^{2} + 5 \, b^{3} c^{7} e^{3} + 3012 \, a b c^{8} e^{3}}{c^{8}}\right )} x + \frac {6720 \, b c^{9} d^{3} + 10656 \, b^{2} c^{8} d^{2} e + 20736 \, a c^{9} d^{2} e + 54 \, b^{3} c^{7} d e^{2} + 22104 \, a b c^{8} d e^{2} - 11 \, b^{4} c^{6} e^{3} + 84 \, a b^{2} c^{7} e^{3} + 3840 \, a^{2} c^{8} e^{3}}{c^{8}}\right )} x + \frac {36288 \, b^{2} c^{8} d^{3} + 69888 \, a c^{9} d^{3} + 864 \, b^{3} c^{7} d^{2} e + 226944 \, a b c^{8} d^{2} e - 486 \, b^{4} c^{6} d e^{2} + 3888 \, a b^{2} c^{7} d e^{2} + 118944 \, a^{2} c^{8} d e^{2} + 99 \, b^{5} c^{5} e^{3} - 856 \, a b^{3} c^{6} e^{3} + 1968 \, a^{2} b c^{7} e^{3}}{c^{8}}\right )} x + \frac {1344 \, b^{3} c^{7} d^{3} + 209664 \, a b c^{8} d^{3} - 2016 \, b^{4} c^{6} d^{2} e + 17280 \, a b^{2} c^{7} d^{2} e + 331776 \, a^{2} c^{8} d^{2} e + 1134 \, b^{5} c^{5} d e^{2} - 10224 \, a b^{3} c^{6} d e^{2} + 25056 \, a^{2} b c^{7} d e^{2} - 231 \, b^{6} c^{4} e^{3} + 2232 \, a b^{4} c^{5} e^{3} - 6384 \, a^{2} b^{2} c^{6} e^{3} + 4096 \, a^{3} c^{7} e^{3}}{c^{8}}\right )} x - \frac {6720 \, b^{4} c^{6} d^{3} - 64512 \, a b^{2} c^{7} d^{3} - 709632 \, a^{2} c^{8} d^{3} - 10080 \, b^{5} c^{5} d^{2} e + 96768 \, a b^{3} c^{6} d^{2} e - 262656 \, a^{2} b c^{7} d^{2} e + 5670 \, b^{6} c^{4} d e^{2} - 56952 \, a b^{4} c^{5} d e^{2} + 171936 \, a^{2} b^{2} c^{6} d e^{2} - 120960 \, a^{3} c^{7} d e^{2} - 1155 \, b^{7} c^{3} e^{3} + 12348 \, a b^{5} c^{4} e^{3} - 42192 \, a^{2} b^{3} c^{5} e^{3} + 44096 \, a^{3} b c^{6} e^{3}}{c^{8}}\right )} x + \frac {20160 \, b^{5} c^{5} d^{3} - 215040 \, a b^{3} c^{6} d^{3} + 709632 \, a^{2} b c^{7} d^{3} - 30240 \, b^{6} c^{4} d^{2} e + 322560 \, a b^{4} c^{5} d^{2} e - 1064448 \, a^{2} b^{2} c^{6} d^{2} e + 884736 \, a^{3} c^{7} d^{2} e + 17010 \, b^{7} c^{3} d e^{2} - 189000 \, a b^{5} c^{4} d e^{2} + 679392 \, a^{2} b^{3} c^{5} d e^{2} - 763776 \, a^{3} b c^{6} d e^{2} - 3465 \, b^{8} c^{2} e^{3} + 40740 \, a b^{6} c^{3} e^{3} - 162288 \, a^{2} b^{4} c^{4} e^{3} + 234432 \, a^{3} b^{2} c^{5} e^{3} - 65536 \, a^{4} c^{6} e^{3}}{c^{8}}\right )} + \frac {5 \, {\left (64 \, b^{6} c^{3} d^{3} - 768 \, a b^{4} c^{4} d^{3} + 3072 \, a^{2} b^{2} c^{5} d^{3} - 4096 \, a^{3} c^{6} d^{3} - 96 \, b^{7} c^{2} d^{2} e + 1152 \, a b^{5} c^{3} d^{2} e - 4608 \, a^{2} b^{3} c^{4} d^{2} e + 6144 \, a^{3} b c^{5} d^{2} e + 54 \, b^{8} c d e^{2} - 672 \, a b^{6} c^{2} d e^{2} + 2880 \, a^{2} b^{4} c^{3} d e^{2} - 4608 \, a^{3} b^{2} c^{4} d e^{2} + 1536 \, a^{4} c^{5} d e^{2} - 11 \, b^{9} e^{3} + 144 \, a b^{7} c e^{3} - 672 \, a^{2} b^{5} c^{2} e^{3} + 1280 \, a^{3} b^{3} c^{3} e^{3} - 768 \, a^{4} b c^{4} e^{3}\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 {13}{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 (d+e\,x\right )}^3\,{\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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