Optimal. Leaf size=227 \[ -\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{231 c^2 \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 227, 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 = {699, 706, 705,
703, 227} \begin {gather*} -\frac {5 d^{7/2} \left (b^2-4 a c\right )^{13/4} \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 )}{231 c^2 \sqrt {a+b x+c x^2}}-\frac {10 d^3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{231 c}-\frac {2 d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 699
Rule 703
Rule 705
Rule 706
Rubi steps
\begin {align*} \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx &=\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx}{22 c}\\ &=-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^2 d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{154 c}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{462 c}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^4 \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}{462 c \sqrt {a+b x+c x^2}}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^3 \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 )}{231 c^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} d^{7/2} \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 )}{231 c^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 = 10.29, size = 155, normalized size = 0.68 \begin {gather*} \frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (7 (b+2 c x)^2 (a+x (b+c x))-10 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))+\frac {\left (b^2-4 a c\right ) \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )\right )}{77 (b+2 c x)^3} \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. \(797\) vs.
\(2(193)=386\).
time = 0.99, size = 798, normalized size = 3.52
method | result | size |
risch | \(-\frac {\left (-336 c^{4} x^{4}-672 b \,c^{3} x^{3}-96 x^{2} c^{3} a -480 b^{2} c^{2} x^{2}-96 x a b \,c^{2}-144 b^{3} c x +160 a^{2} c^{2}-104 a c \,b^{2}-5 b^{4}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{4}}{231 c \sqrt {d \left (2 c x +b \right )}}+\frac {5 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{231 c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(555\) |
default | \(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{3} \left (1344 c^{7} x^{7}+4704 b \,c^{6} x^{6}+1728 a \,c^{6} x^{5}+6624 b^{2} c^{5} x^{5}+320 \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^{3} c^{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^{2} b^{2} c^{2}+60 \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^{4} c -5 \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}}\, b^{6}+4320 a b \,c^{5} x^{4}+4800 b^{3} c^{4} x^{4}-256 a^{2} c^{5} x^{3}+4448 a \,b^{2} c^{4} x^{3}+1844 b^{4} c^{3} x^{3}-384 a^{2} b \,c^{4} x^{2}+2352 a \,b^{3} c^{3} x^{2}+318 b^{5} c^{2} x^{2}-640 a^{3} c^{4} x +288 a^{2} b^{2} c^{3} x +516 a \,b^{4} c^{2} x +10 b^{6} c x -320 a^{3} b \,c^{3}+208 a^{2} b^{3} c^{2}+10 a \,b^{5} c \right )}{462 c^{2} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right )}\) | \(798\) |
elliptic | \(\text {Expression too large to display}\) | \(2307\) |
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 = 0.80, size = 199, normalized size = 0.88 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (336 \, c^{6} d^{3} x^{4} + 672 \, b c^{5} d^{3} x^{3} + 96 \, {\left (5 \, b^{2} c^{4} + a c^{5}\right )} d^{3} x^{2} + 48 \, {\left (3 \, b^{3} c^{3} + 2 \, a b c^{4}\right )} d^{3} x + {\left (5 \, b^{4} c^{2} + 104 \, a b^{2} c^{3} - 160 \, a^{2} c^{4}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{462 \, c^{3}} \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 \left (b + 2 c x\right )\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{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 {\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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