Optimal. Leaf size=458 \[ -\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.33, antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1112, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{9/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (7 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^3 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (21 d^3 \left (a b+b^2 x^2\right )\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 )}{32 b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^3 \left (a b+b^2 x^2\right )\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 )}{32 b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt {2} \sqrt [4]{a} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt {2} \sqrt [4]{a} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^5 \left (a b+b^2 x^2\right )\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 )}{64 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^5 \left (a b+b^2 x^2\right )\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 )}{64 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt {2} \sqrt [4]{a} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (21 d^{9/2} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt {2} \sqrt [4]{a} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {7 d^3 (d x)^{3/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} \sqrt [4]{a} b^{11/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 84, normalized size = 0.18 \[ \frac {2 d^3 (d x)^{3/2} \left (7 \left (a+b x^2\right )^2 \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-\frac {b x^2}{a}\right )-a \left (7 a+5 b x^2\right )\right )}{5 a b^2 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 312, normalized size = 0.68 \[ -\frac {84 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {1}{4}} \sqrt {d x} b^{3} d^{13} - \sqrt {d^{27} x - \sqrt {-\frac {d^{18}}{a b^{11}}} a b^{5} d^{18}} \left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {1}{4}} b^{3}}{d^{18}}\right ) - 21 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {1}{4}} \log \left (9261 \, \sqrt {d x} d^{13} + 9261 \, \left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {3}{4}} a b^{8}\right ) + 21 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {1}{4}} \log \left (9261 \, \sqrt {d x} d^{13} - 9261 \, \left (-\frac {d^{18}}{a b^{11}}\right )^{\frac {3}{4}} a b^{8}\right ) + 4 \, {\left (11 \, b d^{4} x^{3} + 7 \, a d^{4} x\right )} \sqrt {d x}}{64 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 380, normalized size = 0.83 \[ -\frac {1}{128} \, d^{4} {\left (\frac {8 \, {\left (11 \, \sqrt {d x} b d^{4} x^{3} + 7 \, \sqrt {d x} a d^{4} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {42 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{5} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {42 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{5} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{5} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {21 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{5} d \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 612, normalized size = 1.34 \[ -\frac {\left (-42 \sqrt {2}\, b^{2} d^{4} x^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-42 \sqrt {2}\, b^{2} d^{4} x^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-21 \sqrt {2}\, b^{2} d^{4} x^{4} \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 )-84 \sqrt {2}\, a b \,d^{4} x^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-84 \sqrt {2}\, a b \,d^{4} x^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-42 \sqrt {2}\, a b \,d^{4} x^{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 )-42 \sqrt {2}\, a^{2} d^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-42 \sqrt {2}\, a^{2} d^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )-21 \sqrt {2}\, a^{2} d^{4} \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 )+56 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {3}{2}} a b \,d^{2}+88 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (d x \right )^{\frac {7}{2}} b^{2}\right ) \left (b \,x^{2}+a \right ) d}{128 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{3}} \]
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 \[ \frac {a d^{\frac {9}{2}} x^{\frac {3}{2}}}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2} + {\left (b^{4} x^{2} + a b^{3}\right )} x^{2}\right )}} + d^{\frac {9}{2}} \int \frac {\sqrt {x}}{b^{3} x^{2} + a b^{2}}\,{d x} - \frac {11 \, d^{\frac {9}{2}} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, b^{2}} - \frac {11 \, b d^{\frac {9}{2}} x^{\frac {7}{2}} + 15 \, a d^{\frac {9}{2}} x^{\frac {3}{2}}}{16 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
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 \[ \int \frac {{\left (d\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d x\right )^{\frac {9}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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