Optimal. Leaf size=557 \[ \frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.28, 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 = {1126, 294,
296, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \left (a+b x^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} a^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 294
Rule 296
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1126
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {(d x)^{13/2}}{\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)^{13/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (11 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{9/2}}{\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 (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{192 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{2048 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^5 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^5 \left (a b+b^2 x^2\right )\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 )}{2048 a b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^5 \left (a b+b^2 x^2\right )\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 )}{2048 a b^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{13/2} \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} a^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{13/2} \left (a b+b^2 x^2\right )\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 )}{4096 \sqrt {2} a^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^7 \left (a b+b^2 x^2\right )\right ) \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 )}{4096 a b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^7 \left (a b+b^2 x^2\right )\right ) \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 )}{4096 a b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (77 d^{13/2} \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} a^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (77 d^{13/2} \left (a b+b^2 x^2\right )\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 )}{2048 \sqrt {2} a^{5/4} b^{19/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {77 d^5 (d x)^{3/2}}{1024 a b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{11/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {11 d^3 (d x)^{7/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^5 (d x)^{3/2}}{768 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {77 d^{13/2} \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^{5/4} b^{15/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 201, normalized size = 0.36 \begin {gather*} \frac {(d x)^{13/2} \left (a+b x^2\right ) \left (-4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (77 a^3+275 a^2 b x^2+351 a b^2 x^4-231 b^3 x^6\right )-231 \sqrt {2} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-231 \sqrt {2} \left (a+b x^2\right )^4 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{12288 a^{5/4} b^{15/4} x^{13/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1050\) vs.
\(2(361)=722\).
time = 0.05, size = 1051, normalized size = 1.89
method | result | size |
default | \(\text {Expression too large to display}\) | \(1051\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 577, normalized size = 1.04 \begin {gather*} \frac {77 \, d^{\frac {13}{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 )}}{8192 \, a b^{3}} + \frac {77 \, b^{3} d^{\frac {13}{2}} x^{\frac {15}{2}} + 315 \, a b^{2} d^{\frac {13}{2}} x^{\frac {11}{2}} + 495 \, a^{2} b d^{\frac {13}{2}} x^{\frac {7}{2}} + 385 \, a^{3} d^{\frac {13}{2}} x^{\frac {3}{2}}}{1024 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} - \frac {{\left (81 \, b^{4} d^{\frac {13}{2}} x^{5} + 202 \, a b^{3} d^{\frac {13}{2}} x^{3} + 153 \, a^{2} b^{2} d^{\frac {13}{2}} x\right )} x^{\frac {9}{2}} + 2 \, {\left (35 \, a b^{3} d^{\frac {13}{2}} x^{5} + 102 \, a^{2} b^{2} d^{\frac {13}{2}} x^{3} + 99 \, a^{3} b d^{\frac {13}{2}} x\right )} x^{\frac {5}{2}} + {\left (21 \, a^{2} b^{2} d^{\frac {13}{2}} x^{5} + 66 \, a^{3} b d^{\frac {13}{2}} x^{3} + 77 \, a^{4} d^{\frac {13}{2}} x\right )} \sqrt {x}}{192 \, {\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} + {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )} x^{6} + 3 \, {\left (a b^{8} x^{6} + 3 \, a^{2} b^{7} x^{4} + 3 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} x^{4} + 3 \, {\left (a^{2} b^{7} x^{6} + 3 \, a^{3} b^{6} x^{4} + 3 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 448, normalized size = 0.80 \begin {gather*} -\frac {924 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \sqrt {d x} a b^{4} d^{19} - \sqrt {d^{39} x - \sqrt {-\frac {d^{26}}{a^{5} b^{15}}} a^{3} b^{7} d^{26}} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} a b^{4}}{d^{26}}\right ) - 231 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} + 456533 \, \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {3}{4}} a^{4} b^{11}\right ) + 231 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )} \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} - 456533 \, \left (-\frac {d^{26}}{a^{5} b^{15}}\right )^{\frac {3}{4}} a^{4} b^{11}\right ) - 4 \, {\left (231 \, b^{3} d^{6} x^{7} - 351 \, a b^{2} d^{6} x^{5} - 275 \, a^{2} b d^{6} x^{3} - 77 \, a^{3} d^{6} x\right )} \sqrt {d x}}{12288 \, {\left (a b^{7} x^{8} + 4 \, a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.76, size = 386, normalized size = 0.69 \begin {gather*} \frac {1}{24576} \, d^{6} {\left (\frac {462 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {462 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {231 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {231 \, \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^{2} b^{6} d \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {8 \, {\left (231 \, \sqrt {d x} b^{3} d^{8} x^{7} - 351 \, \sqrt {d x} a b^{2} d^{8} x^{5} - 275 \, \sqrt {d x} a^{2} b d^{8} x^{3} - 77 \, \sqrt {d x} a^{3} d^{8} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a b^{3} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \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 )}^{13/2}}{{\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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