Optimal. Leaf size=349 \[ -\frac{2 x \left (-13 a e^2-2 b d e+17 c d^2\right )}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 x \left (91 a e^2+14 b d e+16 c d^2\right )}{405 d^3 e^2 \sqrt{d+e x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (91 a e^2+14 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{405 \sqrt [4]{3} d^3 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]
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Rubi [A] time = 0.641623, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 x \left (-13 a e^2-2 b d e+17 c d^2\right )}{135 d^2 e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{15 d e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 x \left (91 a e^2+14 b d e+16 c d^2\right )}{405 d^3 e^2 \sqrt{d+e x^3}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (91 a e^2+14 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{405 \sqrt [4]{3} d^3 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 50.5503, size = 328, normalized size = 0.94 \[ \frac{2 x \left (a e^{2} - b d e + c d^{2}\right )}{15 d e^{2} \left (d + e x^{3}\right )^{\frac{5}{2}}} + \frac{2 x \left (13 a e^{2} + 2 b d e - 17 c d^{2}\right )}{135 d^{2} e^{2} \left (d + e x^{3}\right )^{\frac{3}{2}}} + \frac{2 x \left (91 a e^{2} + 14 b d e + 16 c d^{2}\right )}{405 d^{3} e^{2} \sqrt{d + e x^{3}}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (91 a e^{2} + 14 b d e + 16 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{1215 d^{3} e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(7/2),x)
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Mathematica [C] time = 0.837752, size = 262, normalized size = 0.75 \[ \frac{2 \left (3 \sqrt [3]{-e} x \left (-3 d \left (d+e x^3\right ) \left (17 c d^2-e (13 a e+2 b d)\right )+\left (d+e x^3\right )^2 \left (7 e (13 a e+2 b d)+16 c d^2\right )+27 d^2 \left (e (a e-b d)+c d^2\right )\right )+i 3^{3/4} \sqrt [3]{d} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (d+e x^3\right )^2 \left (7 e (13 a e+2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{1215 d^3 (-e)^{7/3} \left (d+e x^3\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(7/2),x]
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Maple [B] time = 0.074, size = 1095, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^6+b*x^3+a)/(e*x^3+d)^(7/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(7/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{c x^{6} + b x^{3} + a}{{\left (e^{3} x^{9} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{3} + d^{3}\right )} \sqrt{e x^{3} + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(7/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(7/2),x, algorithm="giac")
[Out]