Optimal. Leaf size=317 \[ \frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}} \]
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Rubi [A]
time = 0.21, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {398, 393, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^2 (3 a d+b c)}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (3 a d+b c)}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{4 a b^3 \left (a+b x^4\right )}+\frac {d^3 x^5}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 393
Rule 398
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx &=\int \left (\frac {d^2 (3 b c-2 a d)}{b^3}+\frac {d^3 x^4}{b^2}+\frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{b^3 \left (a+b x^4\right )^2}\right ) \, dx\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {\int \frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{\left (a+b x^4\right )^2} \, dx}{b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {1}{a+b x^4} \, dx}{4 a b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^3}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{7/2}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{7/2}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} b^{13/4}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{16 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 301, normalized size = 0.95 \begin {gather*} \frac {160 \sqrt [4]{b} d^2 (3 b c-2 a d) x+32 b^{5/4} d^3 x^5+\frac {40 \sqrt [4]{b} (b c-a d)^3 x}{a \left (a+b x^4\right )}-\frac {30 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {30 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}}{160 b^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 220, normalized size = 0.69
method | result | size |
risch | \(\frac {d^{3} x^{5}}{5 b^{2}}-\frac {2 d^{3} a x}{b^{3}}+\frac {3 d^{2} c x}{b^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{4 a \,b^{3} \left (b \,x^{4}+a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16 b^{4} a}\) | \(151\) |
default | \(-\frac {d^{2} \left (-\frac {1}{5} b d \,x^{5}+2 a d x -3 b c x \right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{4 a \left (b \,x^{4}+a \right )}+\frac {3 \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}}{b^{3}}\) | \(220\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 405, normalized size = 1.28 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{4 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )}} + \frac {b d^{3} x^{5} + 5 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{5 \, b^{3}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{32 \, a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1937 vs.
\(2 (248) = 496\).
time = 2.66, size = 1937, normalized size = 6.11 \begin {gather*} \frac {16 \, a b^{2} d^{3} x^{9} + 48 \, {\left (5 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x^{5} + 60 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} b^{6} \sqrt {-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}} + {\left (b^{6} c^{6} + 2 \, a b^{5} c^{5} d - 9 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + 31 \, a^{4} b^{2} c^{2} d^{4} - 30 \, a^{5} b c d^{5} + 9 \, a^{6} d^{6}\right )} x^{2}} a^{5} b^{10} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {3}{4}} - {\left (a^{5} b^{13} c^{3} + a^{6} b^{12} c^{2} d - 5 \, a^{7} b^{11} c d^{2} + 3 \, a^{8} b^{10} d^{3}\right )} x \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {3}{4}}}{b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}\right ) + 15 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (3 \, a^{2} b^{3} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} + 3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x\right ) - 15 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{2} b^{3} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} + 3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x\right ) + 20 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3}\right )} x}{80 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 82.86, size = 337, normalized size = 1.06 \begin {gather*} x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}\right )}{4 a^{2} b^{3} + 4 a b^{4} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{13} + 6561 a^{12} d^{12} - 43740 a^{11} b c d^{11} + 118098 a^{10} b^{2} c^{2} d^{10} - 156492 a^{9} b^{3} c^{3} d^{9} + 84159 a^{8} b^{4} c^{4} d^{8} + 26568 a^{7} b^{5} c^{5} d^{7} - 52164 a^{6} b^{6} c^{6} d^{6} + 11016 a^{5} b^{7} c^{7} d^{5} + 10287 a^{4} b^{8} c^{8} d^{4} - 3564 a^{3} b^{9} c^{9} d^{3} - 1134 a^{2} b^{10} c^{10} d^{2} + 324 a b^{11} c^{11} d + 81 b^{12} c^{12}, \left ( t \mapsto t \log {\left (\frac {16 t a^{2} b^{3}}{9 a^{3} d^{3} - 15 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 3 b^{3} c^{3}} + x \right )} \right )\right )} + \frac {d^{3} x^{5}}{5 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 496, normalized size = 1.56 \begin {gather*} \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{4}} + \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{4 \, {\left (b x^{4} + a\right )} a b^{3}} + \frac {b^{8} d^{3} x^{5} + 15 \, b^{8} c d^{2} x - 10 \, a b^{7} d^{3} x}{5 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.53, size = 1616, normalized size = 5.10 \begin {gather*} \frac {d^3\,x^5}{5\,b^2}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{4\,a\,\left (b^4\,x^4+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}{\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}{\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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