Optimal. Leaf size=268 \[ -\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{462 c^3 \left (b^2-4 a c\right )^{7/4} d^{17/2} \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {698, 707, 705,
703, 227} \begin {gather*} \frac {\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 )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 698
Rule 703
Rule 705
Rule 707
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{17/2}} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx}{10 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}} \, dx}{220 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {1}{(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{308 c^2 \left (b^2-4 a c\right ) d^6}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{924 c^2 \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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}{924 c^2 \left (b^2-4 a c\right )^2 d^8 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{462 c^3 \left (b^2-4 a c\right )^2 d^9 \sqrt {a+b x+c x^2}}\\ &=-\frac {\sqrt {a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac {\sqrt {a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac {\sqrt {a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac {\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 )}{462 c^3 \left (b^2-4 a c\right )^{7/4} d^{17/2} \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 = 4.99, size = 107, normalized size = 0.40 \begin {gather*} \frac {\left (b^2-4 a c\right ) \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {15}{4},-\frac {3}{2};-\frac {11}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{120 c^2 d^9 (b+2 c x)^8 \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. \(1430\) vs.
\(2(228)=456\).
time = 0.91, size = 1431, normalized size = 5.34
method | result | size |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{15360 c^{10} d^{9} \left (x +\frac {b}{2 c}\right )^{8}}-\frac {17 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{42240 c^{8} d^{9} \left (x +\frac {b}{2 c}\right )^{6}}-\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{6160 c^{6} \left (4 a c -b^{2}\right ) d^{9} \left (x +\frac {b}{2 c}\right )^{4}}+\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}{924 c^{4} \left (4 a c -b^{2}\right )^{2} d^{9} \left (x +\frac {b}{2 c}\right )^{2}}+\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{462 c^{2} \left (4 a c -b^{2}\right )^{2} d^{8} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(677\) |
default | \(\text {Expression too large to display}\) | \(1431\) |
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 = 0.76, size = 620, normalized size = 2.31 \begin {gather*} \frac {5 \, \sqrt {2} {\left (256 \, c^{8} x^{8} + 1024 \, b c^{7} x^{7} + 1792 \, b^{2} c^{6} x^{6} + 1792 \, b^{3} c^{5} x^{5} + 1120 \, b^{4} c^{4} x^{4} + 448 \, b^{5} c^{3} x^{3} + 112 \, b^{6} c^{2} x^{2} + 16 \, b^{7} c x + b^{8}\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 (640 \, c^{8} x^{6} + 1920 \, b c^{7} x^{5} - 5 \, b^{6} c^{2} - 10 \, a b^{4} c^{3} + 896 \, a^{2} b^{2} c^{4} - 2464 \, a^{3} c^{5} + 192 \, {\left (13 \, b^{2} c^{6} - 2 \, a c^{7}\right )} x^{4} + 256 \, {\left (7 \, b^{3} c^{5} - 3 \, a b c^{6}\right )} x^{3} + 2 \, {\left (253 \, b^{4} c^{4} + 664 \, a b^{2} c^{5} - 1904 \, a^{2} c^{6}\right )} x^{2} - 2 \, {\left (35 \, b^{5} c^{3} - 856 \, a b^{3} c^{4} + 1904 \, a^{2} b c^{5}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{4620 \, {\left (256 \, {\left (b^{4} c^{12} - 8 \, a b^{2} c^{13} + 16 \, a^{2} c^{14}\right )} d^{9} x^{8} + 1024 \, {\left (b^{5} c^{11} - 8 \, a b^{3} c^{12} + 16 \, a^{2} b c^{13}\right )} d^{9} x^{7} + 1792 \, {\left (b^{6} c^{10} - 8 \, a b^{4} c^{11} + 16 \, a^{2} b^{2} c^{12}\right )} d^{9} x^{6} + 1792 \, {\left (b^{7} c^{9} - 8 \, a b^{5} c^{10} + 16 \, a^{2} b^{3} c^{11}\right )} d^{9} x^{5} + 1120 \, {\left (b^{8} c^{8} - 8 \, a b^{6} c^{9} + 16 \, a^{2} b^{4} c^{10}\right )} d^{9} x^{4} + 448 \, {\left (b^{9} c^{7} - 8 \, a b^{7} c^{8} + 16 \, a^{2} b^{5} c^{9}\right )} d^{9} x^{3} + 112 \, {\left (b^{10} c^{6} - 8 \, a b^{8} c^{7} + 16 \, a^{2} b^{6} c^{8}\right )} d^{9} x^{2} + 16 \, {\left (b^{11} c^{5} - 8 \, a b^{9} c^{6} + 16 \, a^{2} b^{7} c^{7}\right )} d^{9} x + {\left (b^{12} c^{4} - 8 \, a b^{10} c^{5} + 16 \, a^{2} b^{8} c^{6}\right )} d^{9}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{17/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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