Optimal. Leaf size=319 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right )}{84 c^3}+\frac{9 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c} \]
[Out]
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Rubi [A] time = 0.894754, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right )}{84 c^3}+\frac{9 e \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x) \left (a+b x+c x^2\right )^{5/4}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a + b*x + c*x^2)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 75.4358, size = 371, normalized size = 1.16 \[ \frac{2 e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{7 c} - \frac{9 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}}{35 c^{2}} + \frac{\left (b + 2 c x\right ) \sqrt [4]{a + b x + c x^{2}} \left (- 8 a c e^{2} + 9 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right )}{84 c^{3}} - \frac{\sqrt{2} \sqrt{- \frac{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}{\left (4 a c - b^{2}\right ) \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right )^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} + 1\right ) \left (- 8 a c e^{2} + 9 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right ) \sqrt{\left (b + 2 c x\right )^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a + b x + c x^{2}}}{\sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | \frac{1}{2}\right )}{336 c^{\frac{13}{4}} \left (b + 2 c x\right ) \sqrt{- 4 a c + b^{2} + c \left (4 a + 4 b x + 4 c x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+b*x+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.947535, size = 274, normalized size = 0.86 \[ \frac{4 c (a+x (b+c x)) \left (4 b c \left (c \left (35 d^2+14 d e x+3 e^2 x^2\right )-37 a e^2\right )+8 c^2 \left (a e (42 d+5 e x)+c x \left (35 d^2+42 d e x+15 e^2 x^2\right )\right )+45 b^3 e^2-2 b^2 c e (70 d+9 e x)\right )-5 \sqrt [4]{2} \left (b^2-4 a c\right ) \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/4} \left (-4 c e (2 a e+7 b d)+9 b^2 e^2+28 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{-b-2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{1680 c^4 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a + b*x + c*x^2)^(1/4),x]
[Out]
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Maple [F] time = 0.125, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2}\sqrt [4]{c{x}^{2}+bx+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+b*x+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{2} \sqrt [4]{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+b*x+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(1/4)*(e*x + d)^2,x, algorithm="giac")
[Out]