Optimal. Leaf size=468 \[ \frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}+\frac {2 \sqrt {c} \left (b^2-3 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}-\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{c} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A]
time = 0.27, 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 = {1938, 1965,
1967, 1211, 1117, 1209} \begin {gather*} \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 \text {ArcTan}\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 \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1209
Rule 1211
Rule 1938
Rule 1965
Rule 1967
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] Result contains complex when optimal does not.
time = 10.89, size = 519, normalized size = 1.11 \begin {gather*} -\frac {2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-4 a^2 c+2 b^2 x^2 \left (b+c x^2\right )+a \left (b^2-7 b c x^2-6 c^2 x^4\right )\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+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 \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+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 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x} \sqrt {x \left (a+b x^2+c x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1135\) vs.
\(2(454)=908\).
time = 0.05, size = 1136, normalized size = 2.43
method | result | size |
default | \(-\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (12 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a \,c^{2} x^{4}-4 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} c \,x^{4}+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}+a b c \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) x \sqrt {-4 a c +b^{2}}+12 \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a^{2} c^{2} x -3 a \,b^{2} c \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) x -12 \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a^{2} c^{2} x +4 \sqrt {-\frac {2 \left (x^{2} \sqrt {-4 a c +b^{2}}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {x^{2} \sqrt {-4 a c +b^{2}}+b \,x^{2}+2 a}{a}}\, \EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a \,b^{2} c x +14 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b c \,x^{2}-4 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{3} x^{2}+14 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a \,b^{2} c \,x^{2}-4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{4} x^{2}+8 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a^{2} c -2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a \,b^{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}\right )}{2 x^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right ) a^{2} \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 )}\) | \(1136\) |
risch | \(\text {Expression too large to display}\) | \(1497\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x} \left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {x}\,{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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