Optimal. Leaf size=298 \[ \frac {5 \sqrt [4]{a} d^{7/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} b^{9/4}}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac {5 d^3 \sqrt {d x}}{2 b^2} \]
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Rubi [A] time = 0.29, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {5 \sqrt [4]{a} d^{7/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} d^{7/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{8 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{4 \sqrt {2} b^{9/4}}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac {5 d^3 \sqrt {d x}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 211
Rule 288
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{7/2}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac {1}{4} \left (5 d^2\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx\\ &=\frac {5 d^3 \sqrt {d x}}{2 b^2}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (5 a d^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{4 b}\\ &=\frac {5 d^3 \sqrt {d x}}{2 b^2}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (5 a d^3\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2 b}\\ &=\frac {5 d^3 \sqrt {d x}}{2 b^2}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}-\frac {\left (5 \sqrt {a} 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 )}{4 b}-\frac {\left (5 \sqrt {a} 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 )}{4 b}\\ &=\frac {5 d^3 \sqrt {d x}}{2 b^2}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\left (5 \sqrt [4]{a} 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 )}{8 \sqrt {2} b^{9/4}}+\frac {\left (5 \sqrt [4]{a} 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 )}{8 \sqrt {2} b^{9/4}}-\frac {\left (5 \sqrt {a} 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 )}{8 b^{5/2}}-\frac {\left (5 \sqrt {a} 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 )}{8 b^{5/2}}\\ &=\frac {5 d^3 \sqrt {d x}}{2 b^2}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac {5 \sqrt [4]{a} 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 )}{8 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} 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 )}{8 \sqrt {2} b^{9/4}}-\frac {\left (5 \sqrt [4]{a} 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 )}{4 \sqrt {2} b^{9/4}}+\frac {\left (5 \sqrt [4]{a} 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 )}{4 \sqrt {2} b^{9/4}}\\ &=\frac {5 d^3 \sqrt {d x}}{2 b^2}-\frac {d (d x)^{5/2}}{2 b \left (a+b x^2\right )}+\frac {5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} d^{7/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} b^{9/4}}+\frac {5 \sqrt [4]{a} 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 )}{8 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} 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 )}{8 \sqrt {2} b^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 244, normalized size = 0.82 \[ \frac {d^3 \sqrt {d x} \left (\frac {32 b^{5/4} x^2}{a+b x^2}+\frac {40 a \sqrt [4]{b}}{a+b x^2}+\frac {5 \sqrt {2} \sqrt [4]{a} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {x}}-\frac {5 \sqrt {2} \sqrt [4]{a} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {x}}+\frac {10 \sqrt {2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {x}}-\frac {10 \sqrt {2} \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {x}}\right )}{16 b^{9/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 247, normalized size = 0.83 \[ -\frac {20 \, \left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {1}{4}} {\left (b^{3} x^{2} + a b^{2}\right )} \arctan \left (-\frac {\left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {3}{4}} \sqrt {d x} b^{7} d^{3} - \sqrt {d^{7} x + \sqrt {-\frac {a d^{14}}{b^{9}}} b^{4}} \left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {3}{4}} b^{7}}{a d^{14}}\right ) + 5 \, \left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {1}{4}} {\left (b^{3} x^{2} + a b^{2}\right )} \log \left (5 \, \sqrt {d x} d^{3} + 5 \, \left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2}\right ) - 5 \, \left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {1}{4}} {\left (b^{3} x^{2} + a b^{2}\right )} \log \left (5 \, \sqrt {d x} d^{3} - 5 \, \left (-\frac {a d^{14}}{b^{9}}\right )^{\frac {1}{4}} b^{2}\right ) - 4 \, {\left (4 \, b d^{3} x^{2} + 5 \, a d^{3}\right )} \sqrt {d x}}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 263, normalized size = 0.88 \[ \frac {1}{16} \, d^{3} {\left (\frac {8 \, \sqrt {d x} a d^{2}}{{\left (b d^{2} x^{2} + a d^{2}\right )} b^{2}} - \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}} - \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{3}} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{3}} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{3}} + \frac {32 \, \sqrt {d x}}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 223, normalized size = 0.75 \[ \frac {\sqrt {d x}\, a \,d^{5}}{2 \left (b \,d^{2} x^{2}+d^{2} a \right ) b^{2}}-\frac {5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{2}}-\frac {5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{2}}-\frac {5 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{3} \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{16 b^{2}}+\frac {2 \sqrt {d x}\, d^{3}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 282, normalized size = 0.95 \[ \frac {\frac {8 \, \sqrt {d x} a d^{6}}{b^{3} d^{2} x^{2} + a b^{2} d^{2}} + \frac {32 \, \sqrt {d x} d^{4}}{b^{2}} - \frac {5 \, {\left (\frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{6} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a}{b^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 112, normalized size = 0.38 \[ \frac {2\,d^3\,\sqrt {d\,x}}{b^2}-\frac {5\,{\left (-a\right )}^{1/4}\,d^{7/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{4\,b^{9/4}}+\frac {a\,d^5\,\sqrt {d\,x}}{2\,\left (b^3\,d^2\,x^2+a\,b^2\,d^2\right )}+\frac {{\left (-a\right )}^{1/4}\,d^{7/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,5{}\mathrm {i}}{4\,b^{9/4}} \]
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 \[ \int \frac {\left (d x\right )^{\frac {7}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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