Optimal. Leaf size=557 \[ \frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.43, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 288
Rule 290
Rule 329
Rule 617
Rule 628
Rule 1112
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (11 b d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{192 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{1536 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{2048 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 \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 )}{2048 a^{7/2} b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 \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 )}{2048 a^{7/2} b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \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 )}{4096 a^{7/2} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^2 \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 )}{4096 a^{7/2} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{9/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d \sqrt {d x}}{3072 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d \sqrt {d x}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {d \sqrt {d x}}{96 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {11 d \sqrt {d x}}{768 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{3/2} \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 )}{4096 \sqrt {2} a^{15/4} b^{5/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 324, normalized size = 0.58 \begin {gather*} \frac {(d x)^{3/2} \left (a+b x^2\right ) \left (616 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^3+352 a^{7/4} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )^2+256 a^{11/4} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right )-3072 a^{15/4} \sqrt [4]{b} \sqrt {x}-231 \sqrt {2} \left (a+b x^2\right )^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+231 \sqrt {2} \left (a+b x^2\right )^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-462 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+462 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )\right )}{24576 a^{15/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 103.79, size = 269, normalized size = 0.48 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (-\frac {77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{5/4}}+\frac {77 d^{3/2} \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 )}{2048 \sqrt {2} a^{15/4} b^{5/4}}-\frac {\sqrt {d x} \left (231 a^3 d^9-351 a^2 b d^9 x^2-275 a b^2 d^9 x^4-77 b^3 d^9 x^6\right )}{3072 a^3 b \left (a d^2+b d^2 x^2\right )^4}\right )}{d^2 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 429, normalized size = 0.77 \begin {gather*} \frac {924 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{11} b^{4} d \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {3}{4}} - \sqrt {a^{8} b^{2} \sqrt {-\frac {d^{6}}{a^{15} b^{5}}} + d^{3} x} a^{11} b^{4} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {3}{4}}}{d^{6}}\right ) + 231 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (77 \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) - 231 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )} \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} \log \left (-77 \, a^{4} b \left (-\frac {d^{6}}{a^{15} b^{5}}\right )^{\frac {1}{4}} + 77 \, \sqrt {d x} d\right ) + 4 \, {\left (77 \, b^{3} d x^{6} + 275 \, a b^{2} d x^{4} + 351 \, a^{2} b d x^{2} - 231 \, a^{3} d\right )} \sqrt {d x}}{12288 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 406, normalized size = 0.73 \begin {gather*} \frac {1}{24576} \, d {\left (\frac {462 \, \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^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {462 \, \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^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {231 \, \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^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {231 \, \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^{2} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {8 \, {\left (77 \, \sqrt {d x} b^{3} d^{8} x^{6} + 275 \, \sqrt {d x} a b^{2} d^{8} x^{4} + 351 \, \sqrt {d x} a^{2} b d^{8} x^{2} - 231 \, \sqrt {d x} a^{3} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{3} b \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1136, normalized size = 2.04
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.64, size = 586, normalized size = 1.05 \begin {gather*} -\frac {385 \, b^{3} d^{\frac {3}{2}} x^{\frac {13}{2}} + 495 \, a b^{2} d^{\frac {3}{2}} x^{\frac {9}{2}} + 315 \, a^{2} b d^{\frac {3}{2}} x^{\frac {5}{2}} + 77 \, a^{3} d^{\frac {3}{2}} \sqrt {x}}{1024 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} + \frac {{\left (77 \, b^{4} d^{\frac {3}{2}} x^{5} + 66 \, a b^{3} d^{\frac {3}{2}} x^{3} + 21 \, a^{2} b^{2} d^{\frac {3}{2}} x\right )} x^{\frac {11}{2}} + 2 \, {\left (99 \, a b^{3} d^{\frac {3}{2}} x^{5} + 102 \, a^{2} b^{2} d^{\frac {3}{2}} x^{3} + 35 \, a^{3} b d^{\frac {3}{2}} x\right )} x^{\frac {7}{2}} + {\left (153 \, a^{2} b^{2} d^{\frac {3}{2}} x^{5} + 202 \, a^{3} b d^{\frac {3}{2}} x^{3} + 81 \, a^{4} d^{\frac {3}{2}} x\right )} x^{\frac {3}{2}}}{192 \, {\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{4} + 3 \, a^{8} b x^{2} + a^{9} + {\left (a^{3} b^{6} x^{6} + 3 \, a^{4} b^{5} x^{4} + 3 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )} x^{6} + 3 \, {\left (a^{4} b^{5} x^{6} + 3 \, a^{5} b^{4} x^{4} + 3 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )} x^{4} + 3 \, {\left (a^{5} b^{4} x^{6} + 3 \, a^{6} b^{3} x^{4} + 3 \, a^{7} b^{2} x^{2} + a^{8} b\right )} x^{2}\right )}} + \frac {77 \, d {\left (\frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {d} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {d} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \sqrt {d} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{8192 \, a^{3} b} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d\,x\right )}^{3/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \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 {3}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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