Optimal. Leaf size=254 \[ \frac {4 \sqrt {c} x^{3/2} \left (b+c x^2\right )}{\left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}}-\frac {4 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b x^2+c x^4}}+\frac {2 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.15, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2045, 2057,
335, 311, 226, 1210} \begin {gather*} \frac {2 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b x^2+c x^4}}-\frac {4 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b x^2+c x^4}}+\frac {4 \sqrt {c} x^{3/2} \left (b+c x^2\right )}{\left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2045
Rule 2057
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^{5/2}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}}+(2 c) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}}+\frac {\left (2 c x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{\sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}}+\frac {\left (4 c x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}}+\frac {\left (4 \sqrt {b} \sqrt {c} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x^2+c x^4}}-\frac {\left (4 \sqrt {b} \sqrt {c} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x^2+c x^4}}\\ &=\frac {4 \sqrt {c} x^{3/2} \left (b+c x^2\right )}{\left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{x^{3/2}}-\frac {4 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b x^2+c x^4}}+\frac {2 \sqrt [4]{b} \sqrt [4]{c} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 55, normalized size = 0.22 \begin {gather*} -\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};-\frac {c x^2}{b}\right )}{x^{3/2} \sqrt {1+\frac {c x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 202, normalized size = 0.80
method | result | size |
default | \(\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (2 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b -\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b -c \,x^{2}-b \right )}{x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(202\) |
risch | \(-\frac {2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x^{\frac {3}{2}}}+\frac {2 \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{\sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 48, normalized size = 0.19 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {c} x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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