Optimal. Leaf size=79 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (c+d x^3\right )^p \left (\frac{d x^3}{c}+1\right )^{-p} F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right ) \]
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Rubi [A] time = 0.106559, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (c+d x^3\right )^p \left (\frac{d x^3}{c}+1\right )^{-p} F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)^m*(c + d*x^3)^p,x]
[Out]
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Rubi in Sympy [A] time = 21.1481, size = 61, normalized size = 0.77 \[ x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (1 + \frac{d x^{3}}{c}\right )^{- p} \left (a + b x^{3}\right )^{m} \left (c + d x^{3}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{3},- m,- p,\frac{4}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**m*(d*x**3+c)**p,x)
[Out]
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Mathematica [B] time = 0.355429, size = 172, normalized size = 2.18 \[ \frac{4 a c x \left (a+b x^3\right )^m \left (c+d x^3\right )^p F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (b c m F_1\left (\frac{4}{3};1-m,-p;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+a d p F_1\left (\frac{4}{3};-m,1-p;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )+4 a c F_1\left (\frac{1}{3};-m,-p;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3)^m*(c + d*x^3)^p,x]
[Out]
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Maple [F] time = 0.114, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^m*(d*x^3+c)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{m}{\left (d x^{3} + c\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m*(d*x^3 + c)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}^{m}{\left (d x^{3} + c\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m*(d*x^3 + c)^p,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((b*x**3+a)**m*(d*x**3+c)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{m}{\left (d x^{3} + c\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m*(d*x^3 + c)^p,x, algorithm="giac")
[Out]