Optimal. Leaf size=356 \[ \frac {3 \left (b^2-4 a c\right )^2 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{16384 c^6}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{2048 c^5}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{4480 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c} \]
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Rubi [A] time = 0.40, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{2048 c^5}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{16384 c^6}+\frac {3 \left (b^2-4 a c\right )^2 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{4480 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 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^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\int x^2 \left (-3 a B-\frac {1}{2} (11 b B-16 A c) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{8 c}\\ &=-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}+\frac {\int x \left (a (11 b B-16 A c)+\frac {3}{4} \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{56 c^2}\\ &=-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (231 b^3 B-336 A b^2 c-372 a b B c+256 a A c^2-10 c \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{4480 c^4}+\frac {\left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{256 c^4}\\ &=\frac {\left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (231 b^3 B-336 A b^2 c-372 a b B c+256 a A c^2-10 c \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{4480 c^4}-\frac {\left (3 \left (b^2-4 a c\right ) \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{4096 c^5}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (231 b^3 B-336 A b^2 c-372 a b B c+256 a A c^2-10 c \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{4480 c^4}+\frac {\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^6}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (231 b^3 B-336 A b^2 c-372 a b B c+256 a A c^2-10 c \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{4480 c^4}+\frac {\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\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 )}{16384 c^6}\\ &=-\frac {3 \left (b^2-4 a c\right ) \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^6}+\frac {\left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{2048 c^5}-\frac {(11 b B-16 A c) x^2 \left (a+b x+c x^2\right )^{5/2}}{112 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c}-\frac {\left (231 b^3 B-336 A b^2 c-372 a b B c+256 a A c^2-10 c \left (33 b^2 B-48 A b c-28 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2}}{4480 c^4}+\frac {3 \left (b^2-4 a c\right )^2 \left (33 b^4 B-48 A b^3 c-72 a b^2 B c+64 a A b c^2+16 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 267, normalized size = 0.75 \begin {gather*} \frac {\frac {\left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 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 )}{4096 c^{11/2}}+\frac {(a+x (b+c x))^{5/2} \left (12 b c (31 a B-40 A c x)-8 a c^2 (32 A+35 B x)+6 b^2 c (56 A+55 B x)-231 b^3 B\right )}{560 c^3}+\frac {x^2 (a+x (b+c x))^{5/2} (16 A c-11 b B)}{14 c}+B x^3 (a+x (b+c x))^{5/2}}{8 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.50, size = 535, normalized size = 1.50 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-32768 a^3 A c^4+58816 a^3 b B c^3-13440 a^3 B c^4 x+87808 a^2 A b^2 c^3-37376 a^2 A b c^4 x+16384 a^2 A c^5 x^2-81648 a^2 b^3 B c^2+37792 a^2 b^2 B c^3 x-19328 a^2 b B c^4 x^2+8960 a^2 B c^5 x^3-40320 a A b^4 c^2+23296 a A b^3 c^3 x-15872 a A b^2 c^4 x^2+11264 a A b c^5 x^3+131072 a A c^6 x^4+30660 a b^5 B c-17976 a b^4 B c^2 x+12480 a b^3 B c^3 x^2-9088 a b^2 B c^4 x^3+6656 a b B c^5 x^4+107520 a B c^6 x^5+5040 A b^6 c-3360 A b^5 c^2 x+2688 A b^4 c^3 x^2-2304 A b^3 c^4 x^3+2048 A b^2 c^5 x^4+102400 A b c^6 x^5+81920 A c^7 x^6-3465 b^7 B+2310 b^6 B c x-1848 b^5 B c^2 x^2+1584 b^4 B c^3 x^3-1408 b^3 B c^4 x^4+1280 b^2 B c^5 x^5+87040 b B c^6 x^6+71680 B c^7 x^7\right )}{573440 c^6}-\frac {3 \left (256 a^4 B c^4+1024 a^3 A b c^4-1280 a^3 b^2 B c^3-1280 a^2 A b^3 c^3+1120 a^2 b^4 B c^2+448 a A b^5 c^2-336 a b^6 B c-48 A b^7 c+33 b^8 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{32768 c^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 1037, normalized size = 2.91
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 524, normalized size = 1.47 \begin {gather*} \frac {1}{573440} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, B c x + \frac {17 \, B b c^{7} + 16 \, A c^{8}}{c^{7}}\right )} x + \frac {B b^{2} c^{6} + 84 \, B a c^{7} + 80 \, A b c^{7}}{c^{7}}\right )} x - \frac {11 \, B b^{3} c^{5} - 52 \, B a b c^{6} - 16 \, A b^{2} c^{6} - 1024 \, A a c^{7}}{c^{7}}\right )} x + \frac {99 \, B b^{4} c^{4} - 568 \, B a b^{2} c^{5} - 144 \, A b^{3} c^{5} + 560 \, B a^{2} c^{6} + 704 \, A a b c^{6}}{c^{7}}\right )} x - \frac {231 \, B b^{5} c^{3} - 1560 \, B a b^{3} c^{4} - 336 \, A b^{4} c^{4} + 2416 \, B a^{2} b c^{5} + 1984 \, A a b^{2} c^{5} - 2048 \, A a^{2} c^{6}}{c^{7}}\right )} x + \frac {1155 \, B b^{6} c^{2} - 8988 \, B a b^{4} c^{3} - 1680 \, A b^{5} c^{3} + 18896 \, B a^{2} b^{2} c^{4} + 11648 \, A a b^{3} c^{4} - 6720 \, B a^{3} c^{5} - 18688 \, A a^{2} b c^{5}}{c^{7}}\right )} x - \frac {3465 \, B b^{7} c - 30660 \, B a b^{5} c^{2} - 5040 \, A b^{6} c^{2} + 81648 \, B a^{2} b^{3} c^{3} + 40320 \, A a b^{4} c^{3} - 58816 \, B a^{3} b c^{4} - 87808 \, A a^{2} b^{2} c^{4} + 32768 \, A a^{3} c^{5}}{c^{7}}\right )} - \frac {3 \, {\left (33 \, B b^{8} - 336 \, B a b^{6} c - 48 \, A b^{7} c + 1120 \, B a^{2} b^{4} c^{2} + 448 \, A a b^{5} c^{2} - 1280 \, B a^{3} b^{2} c^{3} - 1280 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{32768 \, c^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1061, normalized size = 2.98
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^3\,\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^{3} \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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