Optimal. Leaf size=553 \[ -\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}+\frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {2 c \left (4 a b e^3+15 a c d e^2+15 c^2 d^3\right )}{b-\sqrt {4 a c+b^2}}+e \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c} \]
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Rubi [A] time = 1.28, antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1206, 1679, 1202, 524, 424, 419} \[ \frac {\left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (\frac {2 c \left (4 a b e^3+15 a c d e^2+15 c^2 d^3\right )}{b-\sqrt {4 a c+b^2}}+e \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}-\frac {e \left (b-\sqrt {4 a c+b^2}\right ) \sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}-\frac {e^2 x \sqrt {a+b x^2-c x^4} (4 b e+15 c d)}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1202
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+b x^2-c x^4}} \, dx &=-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}-\frac {\int \frac {-5 c d^3-3 e \left (5 c d^2+a e^2\right ) x^2-e^2 (15 c d+4 b e) x^4}{\sqrt {a+b x^2-c x^4}} \, dx}{5 c}\\ &=-\frac {e^2 (15 c d+4 b e) x \sqrt {a+b x^2-c x^4}}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}+\frac {\int \frac {15 c^2 d^3+15 a c d e^2+4 a b e^3+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) x^2}{\sqrt {a+b x^2-c x^4}} \, dx}{15 c^2}\\ &=-\frac {e^2 (15 c d+4 b e) x \sqrt {a+b x^2-c x^4}}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}+\frac {\left (\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 {15 c^2 d^3+15 a c d e^2+4 a b e^3+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) x^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}{15 c^2 \sqrt {a+b x^2-c x^4}}\\ &=-\frac {e^2 (15 c d+4 b e) x \sqrt {a+b x^2-c x^4}}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}-\frac {\left (\left (b-\sqrt {b^2+4 a c}\right ) e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) \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 {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}{30 c^3 \sqrt {a+b x^2-c x^4}}+\frac {\left (\left (b-\sqrt {b^2+4 a c}\right ) \left (\frac {2 c \left (15 c^2 d^3+15 a c d e^2+4 a b e^3\right )}{b-\sqrt {b^2+4 a c}}+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right )\right ) \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}{\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}{30 c^3 \sqrt {a+b x^2-c x^4}}\\ &=-\frac {e^2 (15 c d+4 b e) x \sqrt {a+b x^2-c x^4}}{15 c^2}-\frac {e^3 x^3 \sqrt {a+b x^2-c x^4}}{5 c}-\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right ) \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}}} E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}+\frac {\left (b-\sqrt {b^2+4 a c}\right ) \sqrt {b+\sqrt {b^2+4 a c}} \left (\frac {2 c \left (15 c^2 d^3+15 a c d e^2+4 a b e^3\right )}{b-\sqrt {b^2+4 a c}}+e \left (45 c^2 d^2+8 b^2 e^2+3 c e (10 b d+3 a e)\right )\right ) \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\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{30 \sqrt {2} c^{7/2} \sqrt {a+b x^2-c x^4}}\\ \end {align*}
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Mathematica [C] time = 2.47, size = 596, normalized size = 1.08 \[ \frac {-i \sqrt {2} e \left (\sqrt {4 a c+b^2}-b\right ) \sqrt {\frac {\sqrt {4 a c+b^2}+b-2 c x^2}{\sqrt {4 a c+b^2}+b}} \sqrt {\frac {\sqrt {4 a c+b^2}-b+2 c x^2}{\sqrt {4 a c+b^2}-b}} \left (3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )+i \sqrt {2} \sqrt {\frac {\sqrt {4 a c+b^2}+b-2 c x^2}{\sqrt {4 a c+b^2}+b}} \sqrt {\frac {\sqrt {4 a c+b^2}-b+2 c x^2}{\sqrt {4 a c+b^2}-b}} \left (15 c^2 d e \left (3 d \sqrt {4 a c+b^2}-2 a e-3 b d\right )+c e^2 \left (30 b d \sqrt {4 a c+b^2}+9 a e \sqrt {4 a c+b^2}-17 a b e-30 b^2 d\right )+8 b^2 e^3 \left (\sqrt {4 a c+b^2}-b\right )-30 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right )|\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )-4 c e^2 x \sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \left (a+b x^2-c x^4\right ) \left (4 b e+3 c \left (5 d+e x^2\right )\right )}{60 c^3 \sqrt {-\frac {c}{\sqrt {4 a c+b^2}+b}} \sqrt {a+b x^2-c x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )} \sqrt {-c x^{4} + b x^{2} + a}}{c x^{4} - b x^{2} - a}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + b x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1195, normalized size = 2.16 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + b x^{2} + a}}\,{d x} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^3}{\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]
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 \[ \int \frac {\left (d + e x^{2}\right )^{3}}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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