Optimal. Leaf size=84 \[ \frac{b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)}-\frac{x \left (c+d x^3\right )^{q+1} (b c-a d (3 q+4)) \, _2F_1\left (1,q+\frac{4}{3};\frac{4}{3};-\frac{d x^3}{c}\right )}{c d (3 q+4)} \]
[Out]
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Rubi [A] time = 0.0984101, antiderivative size = 85, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (a-\frac{b c}{3 d q+4 d}\right ) \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+\frac{b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)*(c + d*x^3)^q,x]
[Out]
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Rubi in Sympy [A] time = 11.049, size = 73, normalized size = 0.87 \[ \frac{b x \left (c + d x^{3}\right )^{q + 1}}{d \left (3 q + 4\right )} - \frac{x \left (1 + \frac{d x^{3}}{c}\right )^{- q} \left (c + d x^{3}\right )^{q} \left (- a d \left (3 q + 4\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{d x^{3}}{c}} \right )}}{d \left (3 q + 4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)*(d*x**3+c)**q,x)
[Out]
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Mathematica [A] time = 0.0312364, size = 75, normalized size = 0.89 \[ \frac{1}{4} x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} \left (4 a \, _2F_1\left (\frac{1}{3},-q;\frac{4}{3};-\frac{d x^3}{c}\right )+b x^3 \, _2F_1\left (\frac{4}{3},-q;\frac{7}{3};-\frac{d x^3}{c}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^3)*(c + d*x^3)^q,x]
[Out]
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Maple [F] time = 0.048, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) \left ( d{x}^{3}+c \right ) ^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)*(d*x^3+c)^q,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)*(d*x^3 + c)^q,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)*(d*x^3 + c)^q,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)*(d*x**3+c)**q,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}{\left (d x^{3} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)*(d*x^3 + c)^q,x, algorithm="giac")
[Out]