Optimal. Leaf size=146 \[ -\frac {2 c^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 b^{5/4} \sqrt {b x^2+c x^4}}-\frac {4 c \sqrt {b x^2+c x^4}}{21 b x^{5/2}}-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2020, 2025, 2032, 329, 220} \[ -\frac {2 c^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 b^{5/4} \sqrt {b x^2+c x^4}}-\frac {4 c \sqrt {b x^2+c x^4}}{21 b x^{5/2}}-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2020
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^{11/2}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}}+\frac {1}{7} (2 c) \int \frac {1}{x^{3/2} \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{21 b x^{5/2}}-\frac {\left (2 c^2\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{21 b}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{21 b x^{5/2}}-\frac {\left (2 c^2 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{21 b \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{21 b x^{5/2}}-\frac {\left (4 c^2 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{21 b \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{7 x^{9/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{21 b x^{5/2}}-\frac {2 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^{5/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.39 \[ -\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac {7}{4},-\frac {1}{2};-\frac {3}{4};-\frac {c x^2}{b}\right )}{7 x^{9/2} \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}}}{x^{\frac {11}{2}}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}}}{x^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 142, normalized size = 0.97 \[ -\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (2 c^{2} x^{4}+\sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, c \,x^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+5 b c \,x^{2}+3 b^{2}\right )}{21 \left (c \,x^{2}+b \right ) b \,x^{\frac {9}{2}}} \]
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 \[ \int \frac {\sqrt {c x^{4} + b x^{2}}}{x^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^{11/2}} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{\frac {11}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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