Optimal. Leaf size=330 \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}+\frac {\sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.18, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1929, 12, 1914, 1139, 1103, 1195} \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1103
Rule 1139
Rule 1195
Rule 1914
Rule 1929
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx &=-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}+\frac {\int \frac {c x^{5/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{a}\\ &=-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}+\frac {c \int \frac {x^{5/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{a}\\ &=-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}+\frac {\left (c \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a \sqrt {a x+b x^3+c x^5}}\\ &=-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}+\frac {\left (\sqrt {c} \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \sqrt {a x+b x^3+c x^5}}-\frac {\left (\sqrt {c} \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}-\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 303, normalized size = 0.92 \[ \frac {-4 \left (a+b x^2+c x^4\right )+\frac {i \sqrt {2} x \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}}}}{4 a \sqrt {x} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{c x^{7} + b x^{5} + a x^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c x^{5} + b x^{3} + a x} x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 508, normalized size = 1.54 \[ \frac {\left (-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b c \,x^{4}-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, c \,x^{4}+\sqrt {-\frac {2 \left (-b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}-2 a \right )}{a}}\, \sqrt {\frac {b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}+2 a}{a}}\, a c x \EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {2}\, \sqrt {\frac {-2 a c +b^{2}+\sqrt {-4 a c +b^{2}}\, b}{a c}}}{2}\right )-\sqrt {-\frac {2 \left (-b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}-2 a \right )}{a}}\, \sqrt {\frac {b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}+2 a}{a}}\, a c x \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {2}\, \sqrt {\frac {-2 a c +b^{2}+\sqrt {-4 a c +b^{2}}\, b}{a c}}}{2}\right )-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} x^{2}-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, b \,x^{2}-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b -\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, a \right ) \sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}}{\left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right ) a \,x^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{\sqrt {c x^{5} + b x^{3} + a x} x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^{3/2}\,\sqrt {c\,x^5+b\,x^3+a\,x}} \,d x \]
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 \[ \int \frac {1}{x^{\frac {3}{2}} \sqrt {x \left (a + b x^{2} + c x^{4}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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