Optimal. Leaf size=1224 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 3.58935, antiderivative size = 1224, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 (b d-2 a e+(2 c d-b e) x) (d+e x)^2}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt [3]{2} c^{5/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}}}-\frac{3^{3/4} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt{\frac{\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{2^{5/6} c^{5/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}}}+\frac{3 \left (-\left (11 c d^2 e-a e^3\right ) b^2+10 c d \left (c d^2+3 a e^2\right ) b-8 a c e \left (2 c d^2+3 a e^2\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac{3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{5/3} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )} \]
Warning: Unable to verify antiderivative.
[In] Int[(d + e*x)^3/(a + b*x + c*x^2)^(7/3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**2+b*x+a)**(7/3),x)
[Out]
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Mathematica [C] time = 1.3475, size = 400, normalized size = 0.33 \[ -\frac{3 \left (4 \left (b^2-4 a c\right ) \left (2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (3 c d x-a e)+b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )-b^3 e^3 x\right )-4 (a+x (b+c x)) \left (4 c^2 \left (-8 a^2 e^3+9 a c d e^2 x+5 c^2 d^3 x\right )+b^2 c e \left (7 a e^2+3 c d (2 e x-5 d)\right )+2 b c^2 \left (9 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right )+b^4 \left (-e^3\right )+b^3 c e^2 (3 d+2 e x)\right )+2^{2/3} \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [3]{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}} (a+x (b+c x)) (b e-2 c d) \left (c e (5 b d-9 a e)+b^2 e^2-5 c^2 d^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )\right )}{16 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a + b*x + c*x^2)^(7/3),x]
[Out]
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Maple [F] time = 0.136, size = 0, normalized size = 0. \[ \int{ \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{7}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(c*x^2+b*x+a)^(7/3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a)^(7/3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a)^(7/3),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((e*x+d)**3/(c*x**2+b*x+a)**(7/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{7}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x + a)^(7/3),x, algorithm="giac")
[Out]