Optimal. Leaf size=301 \[ -\frac {2 d \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (-\frac {1}{4};\frac {3}{2},\frac {3}{2};\frac {3}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f \sqrt {f x} \sqrt {a+b x^2+c x^4}}+\frac {2 e (f x)^{3/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {3}{4};\frac {3}{2},\frac {3}{2};\frac {7}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 a f^3 \sqrt {a+b x^2+c x^4}} \]
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Rubi [A]
time = 0.25, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1349, 1155,
524} \begin {gather*} \frac {2 e (f x)^{3/2} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} F_1\left (\frac {3}{4};\frac {3}{2},\frac {3}{2};\frac {7}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 a f^3 \sqrt {a+b x^2+c x^4}}-\frac {2 d \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} F_1\left (-\frac {1}{4};\frac {3}{2},\frac {3}{2};\frac {3}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f \sqrt {f x} \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 1155
Rule 1349
Rubi steps
\begin {align*} \int \frac {d+e x^2}{(f x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {d}{(f x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}}+\frac {e \sqrt {f x}}{f^2 \left (a+b x^2+c x^4\right )^{3/2}}\right ) \, dx\\ &=d \int \frac {1}{(f x)^{3/2} \left (a+b x^2+c x^4\right )^{3/2}} \, dx+\frac {e \int \frac {\sqrt {f x}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx}{f^2}\\ &=\frac {\left (d \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {1}{(f x)^{3/2} \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{3/2}} \, dx}{a \sqrt {a+b x^2+c x^4}}+\frac {\left (e \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {\sqrt {f x}}{\left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{3/2}} \, dx}{a f^2 \sqrt {a+b x^2+c x^4}}\\ &=-\frac {2 d \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (-\frac {1}{4};\frac {3}{2},\frac {3}{2};\frac {3}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{a f \sqrt {f x} \sqrt {a+b x^2+c x^4}}+\frac {2 e (f x)^{3/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {3}{4};\frac {3}{2},\frac {3}{2};\frac {7}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{3 a f^3 \sqrt {a+b x^2+c x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.66, size = 460, normalized size = 1.53 \begin {gather*} -\frac {x \left (-21 \left (-3 b^2 d x^2 \left (b+c x^2\right )+a^2 c \left (8 d-2 e x^2\right )+a \left (10 c^2 d x^4+b^2 \left (-2 d+e x^2\right )+b c x^2 \left (11 d+e x^2\right )\right )\right )+7 \left (-3 b^3 d+9 a b c d+a b^2 e+2 a^2 c e\right ) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {3}{4};\frac {1}{2},\frac {1}{2};\frac {7}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )-9 c \left (3 b^2 d-10 a c d-a b e\right ) x^4 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {7}{4};\frac {1}{2},\frac {1}{2};\frac {11}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{21 a^2 \left (b^2-4 a c\right ) (f x)^{3/2} \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {e \,x^{2}+d}{\left (f x \right )^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\, dx\]
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 {d + e x^{2}}{\left (f x\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\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 {e\,x^2+d}{{\left (f\,x\right )}^{3/2}\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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