Optimal. Leaf size=211 \[ \frac {5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}-\frac {5 \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{84 c^4 d^{9/2} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {698, 699, 705,
703, 227} \begin {gather*} -\frac {5 \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{84 c^4 d^{9/2} \sqrt {a+b x+c x^2}}+\frac {5 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{84 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 698
Rule 699
Rule 703
Rule 705
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{9/2}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {5 \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx}{14 c d^2}\\ &=-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {5 \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx}{28 c^2 d^4}\\ &=\frac {5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{168 c^3 d^4}\\ &=\frac {5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}-\frac {\left (5 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{168 c^3 d^4 \sqrt {a+b x+c x^2}}\\ &=\frac {5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}-\frac {\left (5 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{84 c^4 d^5 \sqrt {a+b x+c x^2}}\\ &=\frac {5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{84 c^3 d^5}-\frac {5 \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{5/2}}{7 c d (b d+2 c d x)^{7/2}}-\frac {5 \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{84 c^4 d^{9/2} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.06, size = 109, normalized size = 0.52 \begin {gather*} -\frac {\left (b^2-4 a c\right )^2 \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {5}{2},-\frac {7}{4};-\frac {3}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{224 c^3 d^5 (b+2 c x)^4 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \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. \(1309\) vs.
\(2(177)=354\).
time = 0.86, size = 1310, normalized size = 6.21
method | result | size |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{1792 c^{7} d^{5} \left (x +\frac {b}{2 c}\right )^{4}}-\frac {\left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{84 c^{5} d^{5} \left (x +\frac {b}{2 c}\right )^{2}}+\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{48 d^{5} c^{3}}+\frac {2 \left (\frac {6 a c -b^{2}}{32 d^{4} c^{3}}-\frac {4 a c -b^{2}}{84 c^{3} d^{4}}-\frac {a c d +\frac {1}{2} b^{2} d}{48 d^{5} c^{3}}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(646\) |
default | \(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (160 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a \,c^{4} x^{3}-40 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2} c^{3} x^{3}+240 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a b \,c^{3} x^{2}-60 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{3} c^{2} x^{2}+56 c^{6} x^{6}+120 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{2} c^{2} x -30 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{4} c x +168 b \,c^{5} x^{5}+20 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{3} c -5 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{5}-72 a \,c^{5} x^{4}+228 b^{2} c^{4} x^{4}-144 a b \,c^{4} x^{3}+176 b^{3} x^{3} c^{3}-152 a^{2} c^{4} x^{2}-32 a \,b^{2} c^{3} x^{2}+70 b^{4} c^{2} x^{2}-152 a^{2} b \,c^{3} x +40 a \,b^{3} c^{2} x +10 b^{5} c x -24 a^{3} c^{3}-20 a^{2} b^{2} c^{2}+10 a \,b^{4} c \right )}{168 d^{5} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right ) \left (2 c x +b \right )^{3} c^{4}}\) | \(1310\) |
risch | \(\text {Expression too large to display}\) | \(1829\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.69, size = 299, normalized size = 1.42 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (b^{6} - 4 \, a b^{4} c + 16 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 32 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 24 \, {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} x^{2} + 8 \, {\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (28 \, c^{6} x^{4} + 56 \, b c^{5} x^{3} + 5 \, b^{4} c^{2} - 10 \, a b^{2} c^{3} - 12 \, a^{2} c^{4} + 2 \, {\left (29 \, b^{2} c^{4} - 32 \, a c^{5}\right )} x^{2} + 2 \, {\left (15 \, b^{3} c^{3} - 32 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{168 \, {\left (16 \, c^{9} d^{5} x^{4} + 32 \, b c^{8} d^{5} x^{3} + 24 \, b^{2} c^{7} d^{5} x^{2} + 8 \, b^{3} c^{6} d^{5} x + b^{4} c^{5} d^{5}\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*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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