Optimal. Leaf size=468 \[ \frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a} b \sqrt {c}-6 a c+2 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \sqrt [4]{c} \sqrt {x} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac {2 \sqrt {c} x^{3/2} \left (b^2-3 a c\right ) \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.41, antiderivative size = 468, 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 = {1924, 1951, 1953, 1197, 1103, 1195} \[ \frac {2 \sqrt {c} x^{3/2} \left (b^2-3 a c\right ) \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a} b \sqrt {c}-6 a c+2 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \sqrt [4]{c} \sqrt {x} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1197
Rule 1924
Rule 1951
Rule 1953
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}-\frac {\int \frac {-2 b^2+6 a c-b c x^2}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {\int \frac {\sqrt {x} \left (a b c+2 c \left (b^2-3 a c\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{a^2 \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {a b c+2 c \left (b^2-3 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}-\frac {\left (2 \sqrt {c} \left (b^2-3 a c\right ) \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}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\left (\left (\sqrt {a} b c^{3/2}+2 c \left (b^2-3 a c\right )\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{a^{3/2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\\ &=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}+\frac {2 \sqrt {c} \left (b^2-3 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}-\frac {2 \sqrt [4]{c} \left (b^2-3 a 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}} 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^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{c} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a 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 )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [C] time = 1.35, size = 519, normalized size = 1.11 \[ -\frac {2 \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \left (-4 a^2 c+a \left (b^2-7 b c x^2-6 c^2 x^4\right )+2 b^2 x^2 \left (b+c x^2\right )\right )-i x \left (b^2-3 a c\right ) \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i x \left (b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}+4 a b c-b^3\right ) \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+2 b+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {x} \left (b^2-4 a c\right ) \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}}{c^{2} x^{11} + 2 \, b c x^{9} + {\left (b^{2} + 2 \, a c\right )} x^{7} + 2 \, a b x^{5} + a^{2} x^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1136, normalized size = 2.43 \[ -\frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \left (12 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b \,c^{2} x^{4}-4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{3} c \,x^{4}+12 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, a \,c^{2} x^{4}-4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, b^{2} c \,x^{4}-12 \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^{2} c^{2} 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 )+12 \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^{2} c^{2} 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 )+14 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a \,b^{2} c \,x^{2}+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 \,b^{2} 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 )-3 \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^{2} 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 )-4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{4} x^{2}+14 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, a b c \,x^{2}+\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 b 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 )-4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, b^{3} x^{2}+8 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a^{2} b c -2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a \,b^{3}+8 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, a^{2} c -2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {-4 a c +b^{2}}\, a \,b^{2}\right )}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right ) a^{2} 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 \[ \int \frac {1}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}} \sqrt {x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x}\,{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {x} \left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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