Optimal. Leaf size=1209 \[ \text{result too large to display} \]
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Rubi [A] time = 6.23576, antiderivative size = 1209, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 18, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818 \[ -\frac{(2 c d-b e) \sqrt [4]{c x^2+b x+a} (b+2 c x)}{\sqrt{c} \sqrt{b^2-4 a c} e^2 \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt [4]{4 a c-b^2} \left (c d^2-b e d+a e^2\right )^{3/4} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{c x^2+b x+a}}-\frac{\sqrt [4]{4 a c-b^2} \left (c d^2-b e d+a e^2\right )^{3/4} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt [4]{4 a c-b^2} \sqrt{e} \sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{c x^2+b x+a}}+\frac{2 \left (c x^2+b x+a\right )^{3/4}}{3 e}+\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right ) E\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 )}{\sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{3/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{c x^2+b x+a}}{\sqrt{b^2-4 a c}}+1\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 )}{2 \sqrt{2} c^{3/4} e^2 (b+2 c x)}-\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \Pi \left (-\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e^3 \sqrt [4]{c x^2+b x+a} (b+2 c x)}+\frac{\sqrt{4 a c-b^2} (2 c d-b e) \sqrt{c d^2-b e d+a e^2} \sqrt{\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \Pi \left (\frac{\sqrt{4 a c-b^2} e}{2 \sqrt{c} \sqrt{c d^2-b e d+a e^2}};\left .\sin ^{-1}\left (\sqrt [4]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}\right )\right |-1\right )}{\sqrt{2} \sqrt{c} e^3 \sqrt [4]{c x^2+b x+a} (b+2 c x)} \]
Warning: Unable to verify antiderivative.
[In] Int[(a + b*x + c*x^2)^(3/4)/(d + e*x),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((c*x**2+b*x+a)**(3/4)/(e*x+d),x)
[Out]
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Mathematica [C] time = 0.533231, size = 180, normalized size = 0.15 \[ \frac{4 \sqrt{2} (a+x (b+c x))^{3/4} F_1\left (-\frac{3}{2};-\frac{3}{4},-\frac{3}{4};-\frac{1}{2};\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{3 e \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x + c*x^2)^(3/4)/(d + e*x),x]
[Out]
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Maple [F] time = 0.154, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(3/4)/(e*x+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/4)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [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^2 + b*x + a)^(3/4)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{4}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(3/4)/(e*x+d),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/4)/(e*x + d),x, algorithm="giac")
[Out]