Optimal. Leaf size=239 \[ -\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}-\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}} \]
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Rubi [A]
time = 0.12, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {294, 335, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {5 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}-\frac {5 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 294
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}+\frac {5 \int \frac {x^{3/2}}{\left (a+b x^2\right )^2} \, dx}{8 b}\\ &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}+\frac {5 \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{32 b^2}\\ &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}+\frac {5 \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 b^2}\\ &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}+\frac {5 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {a} b^2}+\frac {5 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 \sqrt {a} b^2}\\ &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}+\frac {5 \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 )}{64 \sqrt {a} b^{5/2}}+\frac {5 \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 )}{64 \sqrt {a} b^{5/2}}-\frac {5 \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 )}{64 \sqrt {2} a^{3/4} b^{9/4}}-\frac {5 \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 )}{64 \sqrt {2} a^{3/4} b^{9/4}}\\ &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}-\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}\\ &=-\frac {x^{5/2}}{4 b \left (a+b x^2\right )^2}-\frac {5 \sqrt {x}}{16 b^2 \left (a+b x^2\right )}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{3/4} b^{9/4}}-\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}+\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{3/4} b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 138, normalized size = 0.58 \begin {gather*} \frac {-\frac {4 \sqrt [4]{b} \sqrt {x} \left (5 a+9 b x^2\right )}{\left (a+b x^2\right )^2}-\frac {5 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {5 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{64 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 139, normalized size = 0.58
method | result | size |
derivativedivides | \(\frac {-\frac {9 x^{\frac {5}{2}}}{16 b}-\frac {5 a \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a}\) | \(139\) |
default | \(\frac {-\frac {9 x^{\frac {5}{2}}}{16 b}-\frac {5 a \sqrt {x}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{2} a}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 218, normalized size = 0.91 \begin {gather*} -\frac {9 \, b x^{\frac {5}{2}} + 5 \, a \sqrt {x}}{16 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {5 \, {\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 {a} \sqrt {\sqrt {a} \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 {a} \sqrt {\sqrt {a} \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 {3}{4}} b^{\frac {1}{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 {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.66, size = 254, normalized size = 1.06 \begin {gather*} \frac {20 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{2} b^{4} \sqrt {-\frac {1}{a^{3} b^{9}}} + x} a^{2} b^{7} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {3}{4}} - a^{2} b^{7} \sqrt {x} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {3}{4}}\right ) + 5 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (a b^{2} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 5 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {1}{4}} \log \left (-a b^{2} \left (-\frac {1}{a^{3} b^{9}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 4 \, {\left (9 \, b x^{2} + 5 \, a\right )} \sqrt {x}}{64 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.50, size = 209, normalized size = 0.87 \begin {gather*} \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \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 )}{64 \, a b^{3}} + \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \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 )}{64 \, a b^{3}} + \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a b^{3}} - \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a b^{3}} - \frac {9 \, b x^{\frac {5}{2}} + 5 \, a \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.69, size = 87, normalized size = 0.36 \begin {gather*} -\frac {\frac {9\,x^{5/2}}{16\,b}+\frac {5\,a\,\sqrt {x}}{16\,b^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {5\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{3/4}\,b^{9/4}}-\frac {5\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{3/4}\,b^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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