Optimal. Leaf size=340 \[ \frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}} \]
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Rubi [A]
time = 0.26, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 398, 393,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (3 a d+b c)}{4 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (3 a d+b c)}{4 \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} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{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} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {2 d^2 \sqrt {x} (3 b c-2 a d)}{b^3}+\frac {\sqrt {x} (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {2 d^3 x^{5/2}}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 393
Rule 398
Rule 477
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\sqrt {x} \left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\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,x,\sqrt {x}\right )\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {2 \text {Subst}\left (\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,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a b^3}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} b^3}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} b^3}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{3/2} b^{7/2}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{3/2} b^{7/2}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\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,x,\sqrt {x}\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 {\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,x,\sqrt {x}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\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} \sqrt {x}+\sqrt {b} x\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} \sqrt {x}+\sqrt {b} x\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} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \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} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \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} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \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} \sqrt {x}+\sqrt {b} x\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} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 227, normalized size = 0.67 \begin {gather*} \frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (5 b^3 c^3-45 a^3 d^3+3 a^2 b d^2 \left (25 c-12 d x^2\right )+a b^2 d \left (-15 c^2+60 c d x^2+4 d^2 x^4\right )\right )}{a+b x^2}-15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{40 a^{7/4} b^{13/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 231, normalized size = 0.68
method | result | size |
derivativedivides | \(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 a \left (b \,x^{2}+a \right )}+\frac {3 \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2}}}{b^{3}}\) | \(231\) |
default | \(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 a \left (b \,x^{2}+a \right )}+\frac {3 \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2}}}{b^{3}}\) | \(231\) |
risch | \(-\frac {2 d^{2} \left (-b d \,x^{2}+10 a d -15 b c \right ) \sqrt {x}}{5 b^{3}}-\frac {\sqrt {x}\, d^{3} a^{2}}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {x}\, d^{2} a c}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {3 \sqrt {x}\, d \,c^{2}}{2 b \left (b \,x^{2}+a \right )}+\frac {\sqrt {x}\, c^{3}}{2 a \left (b \,x^{2}+a \right )}+\frac {9 a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{8 b^{3}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{2} c}{8 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d \,c^{2}}{8 b a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{3}}{8 a^{2}}+\frac {9 a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{3}}{16 b^{3}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{2} c}{16 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d \,c^{2}}{16 b a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) c^{3}}{16 a^{2}}+\frac {9 a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{8 b^{3}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{2} c}{8 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d \,c^{2}}{8 b a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{3}}{8 a^{2}}\) | \(689\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 412, normalized size = 1.21 \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 )} \sqrt {x}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, {\left (b d^{3} x^{\frac {5}{2}} + 5 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \sqrt {x}\right )}}{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 (\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} {\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 (\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} {\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 {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} {\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 {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 )}}{16 \, 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. 1944 vs.
\(2 (258) = 516\).
time = 0.55, size = 1944, normalized size = 5.72 \begin {gather*} \frac {60 \, {\left (a b^{4} x^{2} + 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} 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 )} \sqrt {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^{2} + 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 )} \sqrt {x}\right ) - 15 \, {\left (a b^{4} x^{2} + 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 )} \sqrt {x}\right ) + 4 \, {\left (4 \, a b^{2} d^{3} x^{4} + 5 \, b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 75 \, a^{2} b c d^{2} - 45 \, a^{3} d^{3} + 12 \, {\left (5 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1775 vs.
\(2 (323) = 646\).
time = 56.92, size = 1775, normalized size = 5.22 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}}{b^{2}} & \text {for}\: a = 0 \\\frac {2 c^{3} \sqrt {x} + \frac {6 c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 d^{3} x^{\frac {13}{2}}}{13}}{a^{2}} & \text {for}\: b = 0 \\- \frac {180 a^{4} d^{3} \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {45 a^{4} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {45 a^{4} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {90 a^{4} d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {300 a^{3} b c d^{2} \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {75 a^{3} b c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {75 a^{3} b c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {150 a^{3} b c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {144 a^{3} b d^{3} x^{\frac {5}{2}}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {45 a^{3} b d^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {45 a^{3} b d^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {90 a^{3} b d^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {60 a^{2} b^{2} c^{2} d \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 a^{2} b^{2} c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 a^{2} b^{2} c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 a^{2} b^{2} c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {240 a^{2} b^{2} c d^{2} x^{\frac {5}{2}}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {75 a^{2} b^{2} c d^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {75 a^{2} b^{2} c d^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {150 a^{2} b^{2} c d^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {16 a^{2} b^{2} d^{3} x^{\frac {9}{2}}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {20 a b^{3} c^{3} \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 a b^{3} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 a b^{3} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 a b^{3} c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 a b^{3} c^{2} d x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 a b^{3} c^{2} d x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 a b^{3} c^{2} d x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 b^{4} c^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 b^{4} c^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 b^{4} c^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.70, size = 511, normalized size = 1.50 \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 (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, 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 (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, 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 (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\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 (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {2 \, {\left (b^{8} d^{3} x^{\frac {5}{2}} + 15 \, b^{8} c d^{2} \sqrt {x} - 10 \, a b^{7} d^{3} \sqrt {x}\right )}}{5 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 1636, normalized size = 4.81 \begin {gather*} \frac {2\,d^3\,x^{5/2}}{5\,b^2}-\sqrt {x}\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )-\frac {\sqrt {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 )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\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 {\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\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\,\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\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}}-\frac {3\,\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\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}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{4\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\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}}+\frac {3\,\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\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}}}{\frac {\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\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 {\left (\frac {9\,\sqrt {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 )}{a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\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}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{4\,{\left (-a\right )}^{7/4}\,b^{13/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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[Out]
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