Optimal. Leaf size=367 \[ -\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}} \]
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Rubi [A]
time = 0.29, antiderivative size = 367, 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, 479, 584,
217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (5 a d+7 b c)}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (5 a d+7 b c)}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (5 a d+7 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 \sqrt {x} (b c-5 a d)}{2 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{3/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 477
Rule 479
Rule 584
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{x^{5/2} \left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^4 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (7 b c-3 a d)+d (b c-5 a d) x^4\right )}{x^4 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {d^2 (b c-5 a d)}{b}+\frac {c^2 (-7 b c+3 a d)}{a x^4}+\frac {(-b c+a d)^2 (7 b c+5 a d)}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 b^2}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 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^{5/2} b^2}-\frac {\left ((b c-a d)^2 (7 b c+5 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^{5/2} b^2}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (7 b c+5 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^{5/2} b^{5/2}}-\frac {\left ((b c-a d)^2 (7 b c+5 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^{5/2} b^{5/2}}+\frac {\left ((b c-a d)^2 (7 b c+5 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^{11/4} b^{9/4}}+\frac {\left ((b c-a d)^2 (7 b c+5 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^{11/4} b^{9/4}}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {\left ((b c-a d)^2 (7 b c+5 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^{11/4} b^{9/4}}+\frac {\left ((b c-a d)^2 (7 b c+5 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^{11/4} b^{9/4}}\\ &=-\frac {c^2 (7 b c-3 a d)}{6 a^2 b x^{3/2}}-\frac {d^2 (b c-5 a d) \sqrt {x}}{2 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{3/2} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} b^{9/4}}+\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}-\frac {(b c-a d)^2 (7 b c+5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 229, normalized size = 0.62 \begin {gather*} \frac {\frac {4 a^{3/4} \sqrt [4]{b} \left (-7 b^3 c^3 x^2+15 a^3 d^3 x^2+3 a^2 b d^2 x^2 \left (-3 c+4 d x^2\right )+a b^2 c^2 \left (-4 c+9 d x^2\right )\right )}{x^{3/2} \left (a+b x^2\right )}+3 \sqrt {2} (b c-a d)^2 (7 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} (b c-a d)^2 (7 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{24 a^{11/4} b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 225, normalized size = 0.61
method | result | size |
derivativedivides | \(\frac {2 d^{3} \sqrt {x}}{b^{2}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}}\) | \(225\) |
default | \(\frac {2 d^{3} \sqrt {x}}{b^{2}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (5 a^{3} d^{3}-3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} b^{2}}-\frac {2 c^{3}}{3 a^{2} x^{\frac {3}{2}}}\) | \(225\) |
risch | \(\frac {2 a^{2} d^{3} x^{2}-\frac {2 b^{2} c^{3}}{3}}{b^{2} x^{\frac {3}{2}} a^{2}}+\frac {a \sqrt {x}\, d^{3}}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {3 \sqrt {x}\, d^{2} c}{2 b \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {x}\, d \,c^{2}}{2 a \left (b \,x^{2}+a \right )}-\frac {b \sqrt {x}\, c^{3}}{2 a^{2} \left (b \,x^{2}+a \right )}-\frac {5 \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^{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^{2} c}{8 a b}+\frac {9 \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 a^{2}}-\frac {7 b \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^{3}}-\frac {5 \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^{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^{2} c}{8 a b}+\frac {9 \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 a^{2}}-\frac {7 b \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^{3}}-\frac {5 \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^{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^{2} c}{16 a b}+\frac {9 \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 a^{2}}-\frac {7 b \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^{3}}\) | \(691\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 415, normalized size = 1.13 \begin {gather*} \frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3}\right )} x^{2}}{6 \, {\left (a^{2} b^{3} x^{\frac {7}{2}} + a^{3} b^{2} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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 (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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 (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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 (7 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 5 \, 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}}}}{16 \, a^{2} b^{2}} \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. 1967 vs.
\(2 (283) = 566\).
time = 0.51, size = 1967, normalized size = 5.36 \begin {gather*} \text {Too large to display} \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. 2008 vs.
\(2 (340) = 680\).
time = 116.77, size = 2008, normalized size = 5.47 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.95, size = 501, normalized size = 1.37 \begin {gather*} \frac {2 \, d^{3} \sqrt {x}}{b^{2}} - \frac {2 \, c^{3}}{3 \, a^{2} x^{\frac {3}{2}}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} + \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 9 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 5 \, \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^{3} b^{3}} - \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^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 1759, normalized size = 4.79 \begin {gather*} \frac {\frac {x^2\,\left (3\,a^3\,d^3-9\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d-7\,b^3\,c^3\right )}{6\,a^2}-\frac {2\,b^2\,c^3}{3\,a}}{b^3\,x^{7/2}+a\,b^2\,x^{3/2}}+\frac {2\,d^3\,\sqrt {x}}{b^2}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}+\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}}{\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}-\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{11/4}\,b^{9/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}+\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}}{\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )-\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}-\frac {\left (\sqrt {x}\,\left (800\,a^{12}\,b^9\,d^6-960\,a^{11}\,b^{10}\,c\,d^5-2592\,a^{10}\,b^{11}\,c^2\,d^4+3968\,a^9\,b^{12}\,c^3\,d^3+1248\,a^8\,b^{13}\,c^4\,d^2-4032\,a^7\,b^{14}\,c^5\,d+1568\,a^6\,b^{15}\,c^6\right )+\frac {{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,\left (1280\,a^{12}\,b^{11}\,d^3-768\,a^{11}\,b^{12}\,c\,d^2-2304\,a^{10}\,b^{13}\,c^2\,d+1792\,a^9\,b^{14}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{11/4}\,b^{9/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+7\,b\,c\right )}{4\,{\left (-a\right )}^{11/4}\,b^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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