Optimal. Leaf size=347 \[ \frac {\sqrt [4]{a} \sqrt {x} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 c^{3/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} b \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 c^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5}+\frac {b x^{3/2} \left (a+b x^2+c x^4\right )}{3 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.22, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1921, 1953, 1197, 1103, 1195} \[ \frac {\sqrt [4]{a} \sqrt {x} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 c^{3/4} \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{a} b \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 c^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {b x^{3/2} \left (a+b x^2+c x^4\right )}{3 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1197
Rule 1921
Rule 1953
Rubi steps
\begin {align*} \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx &=\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5}+\frac {1}{3} \int \frac {\sqrt {x} \left (2 a+b x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx\\ &=\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{3 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5}+\frac {\left (\sqrt {a} \left (2 \sqrt {a}+\frac {b}{\sqrt {c}}\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{3 \sqrt {a x+b x^3+c x^5}}-\frac {\left (\sqrt {a} b \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}{3 \sqrt {c} \sqrt {a x+b x^3+c x^5}}\\ &=\frac {b x^{3/2} \left (a+b x^2+c x^4\right )}{3 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5}-\frac {\sqrt [4]{a} b \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 )}{3 c^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \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 )}{6 c^{3/4} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [C] time = 0.99, size = 452, normalized size = 1.30 \[ \frac {\sqrt {x} \left (-i \left (b \sqrt {b^2-4 a c}+4 a c-b^2\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i b \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+4 c x \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \left (a+b x^2+c x^4\right )\right )}{12 c \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{5} + b x^{3} + a x}}{\sqrt {x}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{5} + b x^{3} + a x}}{\sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 508, normalized size = 1.46 \[ \frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \left (\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b c \,x^{5}+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, c \,x^{5}+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} x^{3}+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, b \,x^{3}+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b x +\sqrt {-\frac {2 \left (-b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}-2 a \right )}{a}}\, \sqrt {\frac {b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}+2 a}{a}}\, a b \EllipticE \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {2}\, \sqrt {\frac {-2 a c +b^{2}+\sqrt {-4 a c +b^{2}}\, b}{a c}}}{2}\right )+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, a x +\sqrt {-\frac {2 \left (-b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}-2 a \right )}{a}}\, \sqrt {\frac {b \,x^{2}+\sqrt {-4 a c +b^{2}}\, x^{2}+2 a}{a}}\, \sqrt {-4 a c +b^{2}}\, a \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {2}\, \sqrt {\frac {-2 a c +b^{2}+\sqrt {-4 a c +b^{2}}\, b}{a c}}}{2}\right )\right )}{3 \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {x}} \]
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 \[ \int \frac {\sqrt {c x^{5} + b x^{3} + a x}}{\sqrt {x}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^5+b\,x^3+a\,x}}{\sqrt {x}} \,d x \]
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 \[ \int \frac {\sqrt {x \left (a + b x^{2} + c x^{4}\right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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