Optimal. Leaf size=147 \[ \frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {706, 708, 335,
304, 209, 212} \begin {gather*} 2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \text {ArcTan}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+\frac {4}{3} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 706
Rule 708
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx &=\frac {4}{7} d (b d+2 c d x)^{7/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx\\ &=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx\\ &=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+\frac {\left (\left (b^2-4 a c\right )^2 d^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c}\\ &=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+\frac {\left (\left (b^2-4 a c\right )^2 d^3\right ) \text {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{c}\\ &=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}-\left (2 \left (b^2-4 a c\right )^2 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )+\left (2 \left (b^2-4 a c\right )^2 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )\\ &=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.35, size = 217, normalized size = 1.48 \begin {gather*} \frac {\left (\frac {1}{21}+\frac {i}{21}\right ) (d (b+2 c x))^{9/2} \left ((2-2 i) (b+2 c x)^{3/2} \left (7 b^2-28 a c+3 (b+2 c x)^2\right )-21 \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+21 \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-21 \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{9/2}} \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. \(302\) vs.
\(2(119)=238\).
time = 0.72, size = 303, normalized size = 2.06
method | result | size |
derivativedivides | \(4 d \left (-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {d^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(303\) |
default | \(4 d \left (-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {d^{4} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(303\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1320 vs.
\(2 (119) = 238\).
time = 4.16, size = 1320, normalized size = 8.98 \begin {gather*} \frac {8}{21} \, {\left (12 \, c^{3} d^{4} x^{3} + 18 \, b c^{2} d^{4} x^{2} + 4 \, {\left (4 \, b^{2} c - 7 \, a c^{2}\right )} d^{4} x + {\left (5 \, b^{3} - 14 \, a b c\right )} d^{4}\right )} \sqrt {2 \, c d x + b d} - 4 \, \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} {\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} + \sqrt {2 \, {\left (b^{20} c - 40 \, a b^{18} c^{2} + 720 \, a^{2} b^{16} c^{3} - 7680 \, a^{3} b^{14} c^{4} + 53760 \, a^{4} b^{12} c^{5} - 258048 \, a^{5} b^{10} c^{6} + 860160 \, a^{6} b^{8} c^{7} - 1966080 \, a^{7} b^{6} c^{8} + 2949120 \, a^{8} b^{4} c^{9} - 2621440 \, a^{9} b^{2} c^{10} + 1048576 \, a^{10} c^{11}\right )} d^{27} x + {\left (b^{21} - 40 \, a b^{19} c + 720 \, a^{2} b^{17} c^{2} - 7680 \, a^{3} b^{15} c^{3} + 53760 \, a^{4} b^{13} c^{4} - 258048 \, a^{5} b^{11} c^{5} + 860160 \, a^{6} b^{9} c^{6} - 1966080 \, a^{7} b^{7} c^{7} + 2949120 \, a^{8} b^{5} c^{8} - 2621440 \, a^{9} b^{3} c^{9} + 1048576 \, a^{10} b c^{10}\right )} d^{27} + \sqrt {{\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}} {\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}} \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}}}{{\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}}\right ) + \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \log \left (-{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} + \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {3}{4}}\right ) - \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \log \left (-{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} - \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {3}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 531 vs.
\(2 (119) = 238\).
time = 1.88, size = 531, normalized size = 3.61 \begin {gather*} \frac {4}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{3} - \frac {16}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c d^{3} + \frac {4}{7} \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 168, normalized size = 1.14 \begin {gather*} \frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{7}-\frac {4\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (4\,a\,c-b^2\right )}{3}+2\,d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}+d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}\,1{}\mathrm {i}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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