Optimal. Leaf size=147 \[ -\frac {15 \sqrt {b^2-4 a c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{512 c^{7/2} d^5}+\frac {15 \sqrt {a+b x+c x^2}}{256 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4} \]
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Rubi [A] time = 0.10, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {684, 685, 688, 205} \begin {gather*} -\frac {15 \sqrt {b^2-4 a c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{512 c^{7/2} d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}+\frac {15 \sqrt {a+b x+c x^2}}{256 c^3 d^5}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 684
Rule 685
Rule 688
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^5} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx}{16 c d^2}\\ &=-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}+\frac {15 \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx}{128 c^2 d^4}\\ &=\frac {15 \sqrt {a+b x+c x^2}}{256 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}-\frac {\left (15 \left (b^2-4 a c\right )\right ) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{512 c^3 d^4}\\ &=\frac {15 \sqrt {a+b x+c x^2}}{256 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}-\frac {\left (15 \left (b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{128 c^2 d^4}\\ &=\frac {15 \sqrt {a+b x+c x^2}}{256 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{64 c^2 d^5 (b+2 c x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{8 c d^5 (b+2 c x)^4}-\frac {15 \sqrt {b^2-4 a c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{512 c^{7/2} d^5}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 62, normalized size = 0.42 \begin {gather*} \frac {2 (a+x (b+c x))^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {4 c (a+x (b+c x))}{4 a c-b^2}\right )}{7 d^5 \left (b^2-4 a c\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.48, size = 195, normalized size = 1.33 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-32 a^2 c^2-20 a b^2 c-144 a b c^2 x-144 a c^3 x^2+15 b^4+100 b^3 c x+228 b^2 c^2 x^2+256 b c^3 x^3+128 c^4 x^4\right )}{256 c^3 d^5 (b+2 c x)^4}+\frac {15 \sqrt {b^2-4 a c} \tan ^{-1}\left (-\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+\frac {2 c x}{\sqrt {b^2-4 a c}}+\frac {b}{\sqrt {b^2-4 a c}}\right )}{256 c^{7/2} d^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.53, size = 524, normalized size = 3.56 \begin {gather*} \left [\frac {15 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {-\frac {b^{2} - 4 \, a c}{c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 15 \, b^{4} - 20 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (19 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2} + 4 \, {\left (25 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{1024 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}}, \frac {15 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (\frac {\sqrt {\frac {b^{2} - 4 \, a c}{c}}}{2 \, \sqrt {c x^{2} + b x + a}}\right ) + 2 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} + 15 \, b^{4} - 20 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \, {\left (19 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x^{2} + 4 \, {\left (25 \, b^{3} c - 36 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{512 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 900, normalized size = 6.12 \begin {gather*} -\frac {15 a^{3} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{8 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c \,d^{5}}+\frac {45 a^{2} b^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{32 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d^{5}}-\frac {45 a \,b^{4} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{128 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d^{5}}+\frac {15 b^{6} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{512 \left (4 a c -b^{2}\right )^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{4} d^{5}}+\frac {15 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a^{2}}{32 \left (4 a c -b^{2}\right )^{2} c \,d^{5}}-\frac {15 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a \,b^{2}}{64 \left (4 a c -b^{2}\right )^{2} c^{2} d^{5}}+\frac {15 \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{4}}{512 \left (4 a c -b^{2}\right )^{2} c^{3} d^{5}}+\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} a}{16 \left (4 a c -b^{2}\right )^{2} c \,d^{5}}-\frac {5 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b^{2}}{64 \left (4 a c -b^{2}\right )^{2} c^{2} d^{5}}+\frac {3 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{16 \left (4 a c -b^{2}\right )^{2} c \,d^{5}}-\frac {3 \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{16 \left (4 a c -b^{2}\right )^{2} \left (x +\frac {b}{2 c}\right )^{2} c^{2} d^{5}}-\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{32 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{4} c^{4} d^{5}} \end {gather*}
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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^5} \,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*} \frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{5} + 10 b^{4} c x + 40 b^{3} c^{2} x^{2} + 80 b^{2} c^{3} x^{3} + 80 b c^{4} x^{4} + 32 c^{5} x^{5}}\, dx}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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