Optimal. Leaf size=179 \[ -\frac {15 b^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{13/4} \sqrt {b x^2+c x^4}}+\frac {30 b^2 \sqrt {b x^2+c x^4}}{77 c^3 \sqrt {x}}-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c} \]
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Rubi [A] time = 0.24, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2024, 2032, 329, 220} \[ \frac {30 b^2 \sqrt {b x^2+c x^4}}{77 c^3 \sqrt {x}}-\frac {15 b^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{13/4} \sqrt {b x^2+c x^4}}-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \frac {x^{13/2}}{\sqrt {b x^2+c x^4}} \, dx &=\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c}-\frac {(9 b) \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx}{11 c}\\ &=-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c}+\frac {\left (45 b^2\right ) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^2}\\ &=\frac {30 b^2 \sqrt {b x^2+c x^4}}{77 c^3 \sqrt {x}}-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c}-\frac {\left (15 b^3\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^3}\\ &=\frac {30 b^2 \sqrt {b x^2+c x^4}}{77 c^3 \sqrt {x}}-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c}-\frac {\left (15 b^3 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{77 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {30 b^2 \sqrt {b x^2+c x^4}}{77 c^3 \sqrt {x}}-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c}-\frac {\left (30 b^3 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{77 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {30 b^2 \sqrt {b x^2+c x^4}}{77 c^3 \sqrt {x}}-\frac {18 b x^{3/2} \sqrt {b x^2+c x^4}}{77 c^2}+\frac {2 x^{7/2} \sqrt {b x^2+c x^4}}{11 c}-\frac {15 b^{11/4} 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 )}{77 c^{13/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 97, normalized size = 0.54 \[ \frac {2 x^{3/2} \left (-15 b^3 \sqrt {\frac {c x^2}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )+15 b^3+6 b^2 c x^2-2 b c^2 x^4+7 c^3 x^6\right )}{77 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} x^{\frac {9}{2}}}{c x^{2} + b}, x\right ) \]
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 \[ \int \frac {x^{\frac {13}{2}}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 148, normalized size = 0.83 \[ -\frac {\left (-14 c^{4} x^{7}+4 b \,c^{3} x^{5}-12 b^{2} c^{2} x^{3}-30 b^{3} c x +15 \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, b^{3} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {x}}{77 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{4}} \]
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 \[ \int \frac {x^{\frac {13}{2}}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
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 \[ \int \frac {x^{13/2}}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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