Optimal. Leaf size=299 \[ -\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \sqrt {a+b x+c x^2}}-\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {700, 706, 705,
704, 313, 227, 1213, 435} \begin {gather*} -\frac {616 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \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 )}{5 \sqrt {a+b x+c x^2}}+\frac {616 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{5 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 313
Rule 435
Rule 700
Rule 704
Rule 705
Rule 706
Rule 1213
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (22 c d^2\right ) \int \frac {(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1}{3} \left (308 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {1}{5} \left (308 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {\left (308 c^2 \left (b^2-4 a c\right ) d^6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {\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}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {\left (616 c \left (b^2-4 a c\right ) d^5 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}-\frac {\left (616 c \left (b^2-4 a c\right )^{3/2} d^6 \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 )}{5 \sqrt {a+b x+c x^2}}+\frac {\left (616 c \left (b^2-4 a c\right )^{3/2} d^6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}-\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \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 )}{5 \sqrt {a+b x+c x^2}}+\frac {\left (616 c \left (b^2-4 a c\right )^{3/2} d^6 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 \sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{11/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {44 c d^3 (b d+2 c d x)^{7/2}}{3 \sqrt {a+b x+c x^2}}+\frac {1232}{15} c^2 d^5 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}+\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{5 \sqrt {a+b x+c x^2}}-\frac {616 c \left (b^2-4 a c\right )^{7/4} d^{13/2} \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 )}{5 \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 = 10.16, size = 202, normalized size = 0.68 \begin {gather*} \frac {4 d^5 (d (b+2 c x))^{3/2} \left (-41 b^4+156 b^3 c x+48 b c^2 x \left (-11 a+2 c x^2\right )+4 b^2 c \left (121 a+51 c x^2\right )+16 c^2 \left (-77 a^2-33 a c x^2+3 c^2 x^4\right )+616 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \left (4 a^2 c-b^2 x (b+c x)+a \left (-b^2+4 b c x+4 c^2 x^2\right )\right ) \, _2F_1\left (\frac {3}{4},\frac {5}{2};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{15 (a+x (b+c x))^{3/2}} \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. \(1327\) vs.
\(2(249)=498\).
time = 0.90, size = 1328, normalized size = 4.44
method | result | size |
default | \(\text {Expression too large to display}\) | \(1328\) |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \sqrt {d \left (2 c x +b \right )}\, \left (\frac {\left (-\frac {4 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) d^{6} x}{3 c}-\frac {2 b \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) d^{6}}{3 c^{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}}{\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}-\frac {2 \left (2 c^{2} d x +b c d \right ) \left (-20 d^{6} c \left (4 a c -b^{2}\right ) x -10 d^{6} b \left (4 a c -b^{2}\right )\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +b c d \right )}}+\frac {128 d^{6} c^{3} x \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}}{5}+\frac {64 b \,c^{2} d^{6} \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}}{5}+\frac {2 \left (-16 b \,c^{2} \left (8 a c -3 b^{2}\right ) d^{7}-20 b \left (4 a c -b^{2}\right ) d^{7} c^{2}-\frac {128 a b \,c^{3} d^{7}}{5}-\frac {64 b \,c^{2} d^{6} \left (a c d +\frac {1}{2} b^{2} d \right )}{5}\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}}+\frac {2 \left (-32 c^{3} \left (8 a c -5 b^{2}\right ) d^{7}-40 c^{3} \left (4 a c -b^{2}\right ) d^{7}-\frac {128 d^{6} c^{3} \left (3 a c d +\frac {3}{2} b^{2} d \right )}{5}-\frac {192 b^{2} c^{3} d^{7}}{5}\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}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right ) \EllipticE \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 )-\frac {b \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 )}{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 )}{\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d}\) | \(1360\) |
risch | \(\text {Expression too large to display}\) | \(3986\) |
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 = 0.57, size = 362, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left (924 \, \sqrt {2} {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{4} + 2 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{6} x^{3} + {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} d^{6} x^{2} + 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d^{6} x + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} d^{6}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (192 \, c^{5} d^{6} x^{5} + 480 \, b c^{4} d^{6} x^{4} + 12 \, {\left (7 \, b^{2} c^{3} + 132 \, a c^{4}\right )} d^{6} x^{3} - 6 \, {\left (59 \, b^{3} c^{2} - 396 \, a b c^{3}\right )} d^{6} x^{2} - 4 \, {\left (40 \, b^{4} c - 143 \, a b^{2} c^{2} - 308 \, a^{2} c^{3}\right )} d^{6} x - {\left (5 \, b^{5} + 110 \, a b^{3} c - 616 \, a^{2} b c^{2}\right )} d^{6}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{15 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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 (b\,d+2\,c\,d\,x\right )}^{13/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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