Optimal. Leaf size=204 \[ \frac {2 x^{3/2}}{3 c}+\frac {b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}-\frac {b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}} \]
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Rubi [A]
time = 0.11, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1598, 327,
335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {b^{3/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}-\frac {b^{3/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} c^{7/4}}-\frac {b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {b^{3/4} \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {2 x^{3/2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{9/2}}{b x^2+c x^4} \, dx &=\int \frac {x^{5/2}}{b+c x^2} \, dx\\ &=\frac {2 x^{3/2}}{3 c}-\frac {b \int \frac {\sqrt {x}}{b+c x^2} \, dx}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {b \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^{3/2}}-\frac {b \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{c^{3/2}}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {b \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 c^2}-\frac {b \text {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{2 c^2}-\frac {b^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{7/4}}-\frac {b^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} c^{7/4}}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}-\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}+\frac {b^{3/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}-\frac {b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{7/4}}-\frac {b^{3/4} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}+\frac {b^{3/4} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} c^{7/4}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 119, normalized size = 0.58 \begin {gather*} \frac {4 c^{3/4} x^{3/2}+3 \sqrt {2} b^{3/4} \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+3 \sqrt {2} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{6 c^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 116, normalized size = 0.57
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {3}{2}}}{3 c}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(116\) |
default | \(\frac {2 x^{\frac {3}{2}}}{3 c}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(116\) |
risch | \(\frac {2 x^{\frac {3}{2}}}{3 c}-\frac {b \sqrt {2}\, \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )}{4 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}-\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{2 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}-\frac {b \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{2 c^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 186, normalized size = 0.91 \begin {gather*} -\frac {b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{4 \, c} + \frac {2 \, x^{\frac {3}{2}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 165, normalized size = 0.81 \begin {gather*} \frac {12 \, c \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {b^{2} c^{2} \sqrt {x} \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {1}{4}} - \sqrt {-b^{3} c^{3} \sqrt {-\frac {b^{3}}{c^{7}}} + b^{4} x} c^{2} \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {1}{4}}}{b^{3}}\right ) - 3 \, c \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {1}{4}} \log \left (c^{5} \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {3}{4}} + b^{2} \sqrt {x}\right ) + 3 \, c \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {1}{4}} \log \left (-c^{5} \left (-\frac {b^{3}}{c^{7}}\right )^{\frac {3}{4}} + b^{2} \sqrt {x}\right ) + 4 \, x^{\frac {3}{2}}}{6 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 53.36, size = 124, normalized size = 0.61 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: b = 0 \wedge c = 0 \\\frac {2 x^{\frac {7}{2}}}{7 b} & \text {for}\: c = 0 \\\frac {2 x^{\frac {3}{2}}}{3 c} & \text {for}\: b = 0 \\- \frac {b \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 c^{2} \sqrt [4]{- \frac {b}{c}}} + \frac {b \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 c^{2} \sqrt [4]{- \frac {b}{c}}} - \frac {b \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{c^{2} \sqrt [4]{- \frac {b}{c}}} + \frac {2 x^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.71, size = 178, normalized size = 0.87 \begin {gather*} \frac {2 \, x^{\frac {3}{2}}}{3 \, c} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{4}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.36, size = 54, normalized size = 0.26 \begin {gather*} \frac {2\,x^{3/2}}{3\,c}+\frac {{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{c^{7/4}}-\frac {{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{c^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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