Optimal. Leaf size=112 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right )}{c e (m+1) (b c-a d)} \]
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Rubi [A] time = 0.185098, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right )}{c e (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m/((a + b*x^3)*(c + d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 20.177, size = 83, normalized size = 0.74 \[ \frac{d \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{d x^{3}}{c}} \right )}}{c e \left (m + 1\right ) \left (a d - b c\right )} - \frac{b \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a e \left (m + 1\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m/(b*x**3+a)/(d*x**3+c),x)
[Out]
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Mathematica [A] time = 0.0983938, size = 86, normalized size = 0.77 \[ \frac{x (e x)^m \left (a d \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{d x^3}{c}\right )-b c \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{a c (m+1) (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m/((a + b*x^3)*(c + d*x^3)),x]
[Out]
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Maple [F] time = 0.086, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( b{x}^{3}+a \right ) \left ( d{x}^{3}+c \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m/(b*x^3+a)/(d*x^3+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (e x\right )^{m}}{b d x^{6} +{\left (b c + a d\right )} x^{3} + a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^3 + a)*(d*x^3 + c)),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)**m/(b*x**3+a)/(d*x**3+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{m}}{{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^m/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")
[Out]