Optimal. Leaf size=394 \[ -\frac {77 d^{7/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{15/4} b^{9/4}}-\frac {77 d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.45, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}-\frac {77 d^{7/2} \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^{15/4} b^{9/4}}+\frac {77 d^{7/2} \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^{15/4} b^{9/4}}-\frac {77 d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 288
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{4} \left (b^4 d^2\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {1}{64} \left (b^2 d^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {\left (11 b d^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{768 a}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {\left (77 d^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{6144 a^2}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac {\left (77 d^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^3 b}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac {\left (77 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^3 b}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}+\frac {\left (77 d^2\right ) \operatorname {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^{7/2} b}+\frac {\left (77 d^2\right ) \operatorname {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^{7/2} b}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}-\frac {\left (77 d^{7/2}\right ) \operatorname {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^{15/4} b^{9/4}}-\frac {\left (77 d^{7/2}\right ) \operatorname {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^{15/4} b^{9/4}}+\frac {\left (77 d^4\right ) \operatorname {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^{7/2} b^{5/2}}+\frac {\left (77 d^4\right ) \operatorname {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^{7/2} b^{5/2}}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}-\frac {77 d^{7/2} \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^{15/4} b^{9/4}}+\frac {77 d^{7/2} \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^{15/4} b^{9/4}}+\frac {\left (77 d^{7/2}\right ) \operatorname {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^{15/4} b^{9/4}}-\frac {\left (77 d^{7/2}\right ) \operatorname {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^{15/4} b^{9/4}}\\ &=-\frac {d (d x)^{5/2}}{10 b \left (a+b x^2\right )^5}-\frac {d^3 \sqrt {d x}}{32 b^2 \left (a+b x^2\right )^4}+\frac {d^3 \sqrt {d x}}{384 a b^2 \left (a+b x^2\right )^3}+\frac {11 d^3 \sqrt {d x}}{3072 a^2 b^2 \left (a+b x^2\right )^2}+\frac {77 d^3 \sqrt {d x}}{12288 a^3 b^2 \left (a+b x^2\right )}-\frac {77 d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}-\frac {77 d^{7/2} \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^{15/4} b^{9/4}}+\frac {77 d^{7/2} \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^{15/4} b^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 317, normalized size = 0.80 \begin {gather*} \frac {d^3 \sqrt {d x} \left (-\frac {1155 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{15/4} \sqrt {x}}+\frac {1155 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{15/4} \sqrt {x}}-\frac {2310 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{15/4} \sqrt {x}}+\frac {2310 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{15/4} \sqrt {x}}+\frac {3080 \sqrt [4]{b}}{a^3 \left (a+b x^2\right )}+\frac {1760 \sqrt [4]{b}}{a^2 \left (a+b x^2\right )^2}-\frac {65536 b^{5/4} x^2}{\left (a+b x^2\right )^5}+\frac {1280 \sqrt [4]{b}}{a \left (a+b x^2\right )^3}+\frac {1024 \sqrt [4]{b}}{\left (a+b x^2\right )^4}-\frac {16384 a \sqrt [4]{b}}{\left (a+b x^2\right )^5}\right )}{491520 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.15, size = 244, normalized size = 0.62 \begin {gather*} -\frac {77 d^{7/2} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {d}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} \sqrt {d} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {d x}}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}+\frac {77 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}}{\sqrt {a} d+\sqrt {b} d x}\right )}{8192 \sqrt {2} a^{15/4} b^{9/4}}-\frac {d^5 \sqrt {d x} \left (1155 a^4 d^8+5544 a^3 b d^8 x^2-3130 a^2 b^2 d^8 x^4-1760 a b^3 d^8 x^6-385 b^4 d^8 x^8\right )}{61440 a^3 b^2 \left (a d^2+b d^2 x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 513, normalized size = 1.30 \begin {gather*} \frac {4620 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{11} b^{7} d^{3} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {3}{4}} - \sqrt {a^{8} b^{4} \sqrt {-\frac {d^{14}}{a^{15} b^{9}}} + d^{7} x} a^{11} b^{7} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {3}{4}}}{d^{14}}\right ) + 1155 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \log \left (77 \, a^{4} b^{2} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d^{3}\right ) - 1155 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} \log \left (-77 \, a^{4} b^{2} \left (-\frac {d^{14}}{a^{15} b^{9}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d^{3}\right ) + 4 \, {\left (385 \, b^{4} d^{3} x^{8} + 1760 \, a b^{3} d^{3} x^{6} + 3130 \, a^{2} b^{2} d^{3} x^{4} - 5544 \, a^{3} b d^{3} x^{2} - 1155 \, a^{4} d^{3}\right )} \sqrt {d x}}{245760 \, {\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 342, normalized size = 0.87 \begin {gather*} \frac {1}{491520} \, d^{3} {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{a^{4} b^{3}} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{a^{4} b^{3}} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{a^{4} b^{3}} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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 )}{a^{4} b^{3}} + \frac {8 \, {\left (385 \, \sqrt {d x} b^{4} d^{10} x^{8} + 1760 \, \sqrt {d x} a b^{3} d^{10} x^{6} + 3130 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} - 5544 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 1155 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{3} b^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 339, normalized size = 0.86 \begin {gather*} -\frac {77 \sqrt {d x}\, a \,d^{13}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {231 \left (d x \right )^{\frac {5}{2}} d^{11}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {313 \left (d x \right )^{\frac {9}{2}} d^{9}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {11 \left (d x \right )^{\frac {13}{2}} b \,d^{7}}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {77 \left (d x \right )^{\frac {17}{2}} b^{2} d^{5}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{4} b^{2}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{4} b^{2}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \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^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.15, size = 394, normalized size = 1.00 \begin {gather*} \frac {\frac {8 \, {\left (385 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{6} + 1760 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{8} + 3130 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{10} - 5544 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{12} - 1155 \, \sqrt {d x} a^{4} d^{14}\right )}}{a^{3} b^{7} d^{10} x^{10} + 5 \, a^{4} b^{6} d^{10} x^{8} + 10 \, a^{5} b^{5} d^{10} x^{6} + 10 \, a^{6} b^{4} d^{10} x^{4} + 5 \, a^{7} b^{3} d^{10} x^{2} + a^{8} b^{2} d^{10}} + \frac {1155 \, {\left (\frac {\sqrt {2} d^{6} \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 {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{6} \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 {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{5} \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 {a}} + \frac {2 \, \sqrt {2} d^{5} \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 {a}}\right )}}{a^{3} b^{2}}}{491520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.27, size = 207, normalized size = 0.53 \begin {gather*} \frac {\frac {313\,d^9\,{\left (d\,x\right )}^{9/2}}{6144\,a}-\frac {231\,d^{11}\,{\left (d\,x\right )}^{5/2}}{2560\,b}+\frac {77\,b^2\,d^5\,{\left (d\,x\right )}^{17/2}}{12288\,a^3}-\frac {77\,a\,d^{13}\,\sqrt {d\,x}}{4096\,b^2}+\frac {11\,b\,d^7\,{\left (d\,x\right )}^{13/2}}{384\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {77\,d^{7/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{15/4}\,b^{9/4}}+\frac {77\,d^{7/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{15/4}\,b^{9/4}} \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 {\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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