Optimal. Leaf size=149 \[ -\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}}+\frac {10 c \sqrt {b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac {5 c^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 b^{9/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.12, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2050, 2057,
335, 226} \begin {gather*} \frac {5 c^{7/4} 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 )}{21 b^{9/4} \sqrt {b x^2+c x^4}}+\frac {10 c \sqrt {b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2050
Rule 2057
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} \sqrt {b x^2+c x^4}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}}-\frac {(5 c) \int \frac {1}{x^{3/2} \sqrt {b x^2+c x^4}} \, dx}{7 b}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}}+\frac {10 c \sqrt {b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac {\left (5 c^2\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{21 b^2}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}}+\frac {10 c \sqrt {b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac {\left (5 c^2 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{21 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}}+\frac {10 c \sqrt {b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac {\left (10 c^2 x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{21 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 b x^{9/2}}+\frac {10 c \sqrt {b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac {5 c^{7/4} 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 )}{21 b^{9/4} \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 = 10.02, size = 57, normalized size = 0.38 \begin {gather*} -\frac {2 \sqrt {1+\frac {c x^2}{b}} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};-\frac {c x^2}{b}\right )}{7 x^{5/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 134, normalized size = 0.90
method | result | size |
default | \(\frac {5 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, c \,x^{3}+10 c^{2} x^{4}+4 b c \,x^{2}-6 b^{2}}{21 \sqrt {c \,x^{4}+b \,x^{2}}\, x^{\frac {5}{2}} b^{2}}\) | \(134\) |
risch | \(-\frac {2 \left (c \,x^{2}+b \right ) \left (-5 c \,x^{2}+3 b \right )}{21 b^{2} x^{\frac {5}{2}} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {5 c \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{21 b^{2} \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(176\) |
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 = 0.08, size = 53, normalized size = 0.36 \begin {gather*} \frac {2 \, {\left (5 \, c^{\frac {3}{2}} x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + \sqrt {c x^{4} + b x^{2}} {\left (5 \, c x^{2} - 3 \, b\right )} \sqrt {x}\right )}}{21 \, b^{2} x^{5}} \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 {1}{x^{\frac {7}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{x^{7/2}\,\sqrt {c\,x^4+b\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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