Optimal. Leaf size=297 \[ \frac {2 d (f x)^{5/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {5}{4};-\frac {1}{2},-\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 f \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}+\frac {2 e (f x)^{9/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {9}{4};-\frac {1}{2},-\frac {1}{2};\frac {13}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{9 f^3 \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.39, antiderivative size = 297, 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 = {1335, 1141, 510} \[ \frac {2 d (f x)^{5/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {5}{4};-\frac {1}{2},-\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 f \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}}+\frac {2 e (f x)^{9/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {9}{4};-\frac {1}{2},-\frac {1}{2};\frac {13}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{9 f^3 \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1141
Rule 1335
Rubi steps
\begin {align*} \int (f x)^{3/2} \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4} \, dx &=\int \left (d (f x)^{3/2} \sqrt {a+b x^2+c x^4}+\frac {e (f x)^{7/2} \sqrt {a+b x^2+c x^4}}{f^2}\right ) \, dx\\ &=d \int (f x)^{3/2} \sqrt {a+b x^2+c x^4} \, dx+\frac {e \int (f x)^{7/2} \sqrt {a+b x^2+c x^4} \, dx}{f^2}\\ &=\frac {\left (d \sqrt {a+b x^2+c x^4}\right ) \int (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}}} \, dx}{\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}}}}+\frac {\left (e \sqrt {a+b x^2+c x^4}\right ) \int (f x)^{7/2} \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}}} \, dx}{f^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}}}}\\ &=\frac {2 d (f x)^{5/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {5}{4};-\frac {1}{2},-\frac {1}{2};\frac {9}{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 )}{5 f \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}}}}+\frac {2 e (f x)^{9/2} \sqrt {a+b x^2+c x^4} F_1\left (\frac {9}{4};-\frac {1}{2},-\frac {1}{2};\frac {13}{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 f^3 \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}}}}\\ \end {align*}
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Mathematica [A] time = 0.98, size = 430, normalized size = 1.45 \[ \frac {2 f \sqrt {f x} \left (10 a \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {1}{4};\frac {1}{2},\frac {1}{2};\frac {5}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right ) \left (-18 a c e+7 b^2 e-13 b c d\right )+2 x^2 \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {5}{4};\frac {1}{2},\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right ) \left (-79 a b c e+130 a c^2 d+21 b^3 e-39 b^2 c d\right )+5 \left (a+b x^2+c x^4\right ) \left (c \left (36 a e+65 c d x^2+45 c e x^4\right )-14 b^2 e+2 b c \left (13 d+5 e x^2\right )\right )\right )}{2925 c^2 \sqrt {a+b x^2+c x^4}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e f x^{3} + d f x\right )} \sqrt {c x^{4} + b x^{2} + a} \sqrt {f x}, x\right ) \]
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 \[ \int \sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )} \left (f x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{\frac {3}{2}} \left (e \,x^{2}+d \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, dx \]
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 \[ \int \sqrt {c x^{4} + b x^{2} + a} {\left (e x^{2} + d\right )} \left (f x\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (f\,x\right )}^{3/2}\,\left (e\,x^2+d\right )\,\sqrt {c\,x^4+b\,x^2+a} \,d x \]
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 \[ \int \left (f x\right )^{\frac {3}{2}} \left (d + e x^{2}\right ) \sqrt {a + b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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