Optimal. Leaf size=422 \[ -\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {69615 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}} \]
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Rubi [A]
time = 0.36, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 296, 331,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {69615 b^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 296
Rule 303
Rule 331
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {\left (5 b^5\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^5} \, dx}{4 a}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {\left (105 b^4\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx}{64 a^2}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {\left (595 b^3\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {\left (7735 b^2\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {(69615 b) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (69615 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^6 d^2}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (69615 b^3\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^7 d^4}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (69615 b^3\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^7 d^5}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (69615 b^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{8192 a^7 d^5}+\frac {\left (69615 b^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{8192 a^7 d^5}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (69615 b^{5/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}+\frac {\left (69615 b^{5/4}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}+\frac {(69615 b) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{16384 a^7 d^3}+\frac {(69615 b) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{16384 a^7 d^3}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}+\frac {\left (69615 b^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}-\frac {\left (69615 b^{5/4}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {69615 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 211, normalized size = 0.50 \begin {gather*} \frac {\sqrt {d x} \left (\frac {4 \sqrt [4]{a} \left (-8192 a^6+204800 a^5 b x^2+1317575 a^4 b^2 x^4+2951200 a^3 b^3 x^6+3171350 a^2 b^4 x^8+1670760 a b^5 x^{10}+348075 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}-348075 \sqrt {2} b^{5/4} x^{5/2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-348075 \sqrt {2} b^{5/4} x^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{29/4} d^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 268, normalized size = 0.64
method | result | size |
derivativedivides | \(2 d^{11} \left (\frac {b^{2} \left (\frac {\frac {34139 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{8192}+\frac {3597 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{256}+\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{4096}+\frac {56269 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{5120}+\frac {20463 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {69615 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} d^{14}}-\frac {1}{5 a^{6} d^{12} \left (d x \right )^{\frac {5}{2}}}+\frac {6 b}{a^{7} d^{14} \sqrt {d x}}\right )\) | \(268\) |
default | \(2 d^{11} \left (\frac {b^{2} \left (\frac {\frac {34139 a^{4} d^{8} \left (d x \right )^{\frac {3}{2}}}{8192}+\frac {3597 a^{3} b \,d^{6} \left (d x \right )^{\frac {7}{2}}}{256}+\frac {75471 a^{2} d^{4} b^{2} \left (d x \right )^{\frac {11}{2}}}{4096}+\frac {56269 a \,b^{3} d^{2} \left (d x \right )^{\frac {15}{2}}}{5120}+\frac {20463 b^{4} \left (d x \right )^{\frac {19}{2}}}{8192}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {69615 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} d^{14}}-\frac {1}{5 a^{6} d^{12} \left (d x \right )^{\frac {5}{2}}}+\frac {6 b}{a^{7} d^{14} \sqrt {d x}}\right )\) | \(268\) |
risch | \(-\frac {2 \left (-30 b \,x^{2}+a \right )}{5 a^{7} \sqrt {d x}\, x^{2} d^{3}}+\frac {\frac {34139 b^{2} d^{8} \left (d x \right )^{\frac {3}{2}}}{4096 a^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {3597 b^{3} d^{6} \left (d x \right )^{\frac {7}{2}}}{128 a^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {75471 b^{4} d^{4} \left (d x \right )^{\frac {11}{2}}}{2048 a^{5} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {56269 b^{5} d^{2} \left (d x \right )^{\frac {15}{2}}}{2560 a^{6} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {20463 b^{6} \left (d x \right )^{\frac {19}{2}}}{4096 a^{7} \left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {69615 b \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 a^{7} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {69615 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{7} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+\frac {69615 b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{7} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}}{d^{3}}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 410, normalized size = 0.97 \begin {gather*} \frac {\frac {8 \, {\left (348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}\right )}}{\left (d x\right )^{\frac {25}{2}} a^{7} b^{5} d^{2} + 5 \, \left (d x\right )^{\frac {21}{2}} a^{8} b^{4} d^{4} + 10 \, \left (d x\right )^{\frac {17}{2}} a^{9} b^{3} d^{6} + 10 \, \left (d x\right )^{\frac {13}{2}} a^{10} b^{2} d^{8} + 5 \, \left (d x\right )^{\frac {9}{2}} a^{11} b d^{10} + \left (d x\right )^{\frac {5}{2}} a^{12} d^{12}} + \frac {348075 \, b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{7} d^{2}}}{163840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 591, normalized size = 1.40 \begin {gather*} -\frac {1392300 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {337371570183375 \, \sqrt {d x} a^{7} b^{4} d^{3} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-113819576367995923331126390625 \, a^{15} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{29} d^{14}}} + 113819576367995923331126390625 \, b^{8} d x} a^{7} d^{3} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}}}{337371570183375 \, b^{5}}\right ) - 348075 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (337371570183375 \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) + 348075 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (-337371570183375 \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) - 4 \, {\left (348075 \, b^{6} x^{12} + 1670760 \, a b^{5} x^{10} + 3171350 \, a^{2} b^{4} x^{8} + 2951200 \, a^{3} b^{3} x^{6} + 1317575 \, a^{4} b^{2} x^{4} + 204800 \, a^{5} b x^{2} - 8192 \, a^{6}\right )} \sqrt {d x}}{81920 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\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 {1}{\left (d x\right )^{\frac {7}{2}} \left (a + b x^{2}\right )^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.42, size = 362, normalized size = 0.86 \begin {gather*} \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{8} b d^{5}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{16384 \, a^{8} b d^{5}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{8} b d^{5}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{32768 \, a^{8} b d^{5}} + \frac {348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}}{20480 \, {\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )}^{5} a^{7} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 239, normalized size = 0.57 \begin {gather*} \frac {\frac {10\,b\,d^9\,x^2}{a^2}-\frac {2\,d^9}{5\,a}+\frac {263515\,b^2\,d^9\,x^4}{4096\,a^3}+\frac {18445\,b^3\,d^9\,x^6}{128\,a^4}+\frac {317135\,b^4\,d^9\,x^8}{2048\,a^5}+\frac {41769\,b^5\,d^9\,x^{10}}{512\,a^6}+\frac {69615\,b^6\,d^9\,x^{12}}{4096\,a^7}}{b^5\,{\left (d\,x\right )}^{25/2}+a^5\,d^{10}\,{\left (d\,x\right )}^{5/2}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{13/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{17/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{9/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{21/2}}-\frac {69615\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{29/4}\,d^{7/2}}+\frac {69615\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{29/4}\,d^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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