Optimal. Leaf size=467 \[ \frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+\frac{\sqrt{c} \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )}{\sqrt{a}}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{30 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{e x \sqrt{a+b x^2+c x^4} \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )}{15 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{e^2 x \sqrt{a+b x^2+c x^4} (15 c d-4 b e)}{15 c^2}+\frac{e^3 x^3 \sqrt{a+b x^2+c x^4}}{5 c} \]
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Rubi [A] time = 0.424683, antiderivative size = 467, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1206, 1679, 1197, 1103, 1195} \[ \frac{e x \sqrt{a+b x^2+c x^4} \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )}{15 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+\frac{\sqrt{c} \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )}{\sqrt{a}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{30 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} e \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{e^2 x \sqrt{a+b x^2+c x^4} (15 c d-4 b e)}{15 c^2}+\frac{e^3 x^3 \sqrt{a+b x^2+c x^4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1197
Rule 1103
Rule 1195
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{a} e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )\right ) \int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+b x^2+c x^4}} \, dx}{15 c^{5/2}}+\frac{\left (15 c^2 d^3-15 a c d e^2+4 a b e^3+\frac{\sqrt{a} e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )}{\sqrt{c}}\right ) \int \frac{1}{\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{e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right ) x \sqrt{a+b x^2+c x^4}}{15 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\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 )}{15 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{\left (15 c^2 d^3-15 a c d e^2+4 a b e^3+\frac{\sqrt{a} e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )}{\sqrt{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}} 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 )}{30 \sqrt [4]{a} c^{9/4} \sqrt{a+b x^2+c x^4}}\\ \end{align*}
Mathematica [C] time = 2.90362, size = 584, normalized size = 1.25 \[ \frac{-i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \left (15 c^2 d e \left (3 d \sqrt{b^2-4 a c}-2 a e-3 b d\right )+c e^2 \left (-30 b d \sqrt{b^2-4 a c}-9 a e \sqrt{b^2-4 a c}+17 a b e+30 b^2 d\right )+8 b^2 e^3 \left (\sqrt{b^2-4 a c}-b\right )+30 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+i e \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+4 c e^2 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (a+b x^2+c x^4\right ) \left (3 c \left (5 d+e x^2\right )-4 b e\right )}{60 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.066, size = 1186, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{3}}{\sqrt{c x^{4} + b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}{\sqrt{c x^{4} + b x^{2} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x^{2}\right )^{3}}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{3}}{\sqrt{c x^{4} + b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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