Optimal. Leaf size=298 \[ \frac{d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-3 \sqrt{a} e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{3 d e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt{a+c x^4}}-\frac{3 d e^2 x \sqrt{a+c x^4}}{2 a \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]
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Rubi [A] time = 0.342391, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{d \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \left (\sqrt{c} d^2-3 \sqrt{a} e^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt{a+c x^4}}+\frac{3 d e^2 \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt{a+c x^4}}-\frac{a e^3-c x \left (d^3+3 d^2 e x+3 d e^2 x^2\right )}{2 a c \sqrt{a+c x^4}}-\frac{3 d e^2 x \sqrt{a+c x^4}}{2 a \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 37.383, size = 272, normalized size = 0.91 \[ - \frac{a e^{3} - c x \left (d^{3} + 3 d^{2} e x + 3 d e^{2} x^{2}\right )}{2 a c \sqrt{a + c x^{4}}} - \frac{3 d e^{2} x \sqrt{a + c x^{4}}}{2 a \sqrt{c} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{3 d e^{2} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} c^{\frac{3}{4}} \sqrt{a + c x^{4}}} - \frac{d \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (3 \sqrt{a} e^{2} - \sqrt{c} d^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{5}{4}} c^{\frac{3}{4}} \sqrt{a + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**4+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.48482, size = 215, normalized size = 0.72 \[ \frac{\sqrt{c} d \sqrt{\frac{c x^4}{a}+1} \left (3 \sqrt{a} e^2-i \sqrt{c} d^2\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \left (c d x \left (d^2+3 d e x+3 e^2 x^2\right )-a e^3\right )-3 \sqrt{a} \sqrt{c} d e^2 \sqrt{\frac{c x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{2 a c \sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} \sqrt{a+c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a + c*x^4)^(3/2),x]
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Maple [C] time = 0.039, size = 261, normalized size = 0.9 \[{d}^{3} \left ({\frac{x}{2\,a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}+{\frac{1}{2\,a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \right ) -{\frac{{e}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+{\frac{3\,{d}^{2}e{x}^{2}}{2\,a}{\frac{1}{\sqrt{c{x}^{4}+a}}}}+3\,{e}^{2}d \left ( 1/2\,{\frac{{x}^{3}}{a}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{c}} \right ) c}}}}-{\frac{i/2}{\sqrt{a}\sqrt{c{x}^{4}+a}\sqrt{c}}\sqrt{1-{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}}\sqrt{1+{\frac{i\sqrt{c}{x}^{2}}{\sqrt{a}}}} \left ({\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}},i \right ) \right ){\frac{1}{\sqrt{{\frac{i\sqrt{c}}{\sqrt{a}}}}}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(c*x^4+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (a + c x^{4}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^4 + a)^(3/2),x, algorithm="giac")
[Out]